Content-Type: multipart/mixed; boundary="-------------0102190745538" This is a multi-part message in MIME format. ---------------0102190745538 Content-Type: text/plain; name="01-69.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-69.keywords" Navier-Stokes stationary wake universality ---------------0102190745538 Content-Type: application/postscript; name="nswake.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nswake.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.47 Copyright 1986-91 Radical Eye Software %%Title: nswake.dvi %%Pages: 47 1 %%BoundingBox: 0 0 596 843 %%DocumentFonts: Times-Roman Times-Bold Times-Italic %%EndComments %%BeginProcSet: tex.pro /TeXDict 200 dict def TeXDict begin /N /def load def /B{bind def}N /S /exch load def /X{S N}B /TR /translate load N /isls false N /vsize 10 N /@rigin{ isls{[0 1 -1 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale Resolution VResolution vsize neg mul TR matrix 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b(17)-9 4633 y(B)70 b(Existence)21 b(of)h(Euler')m(s)e (Solution)2526 b(18)-9 4818 y(C)66 b(Pr)n(oofs)23 b(f)n(or)e(Section)h(3)2856 b(19)118 4918 y Fz(C.1)75 b(Preliminary)22 b(Bounds)84 b(.)42 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h (.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f (.)h(.)76 b(19)118 5017 y(C.2)f(Proof)23 b(of)f(Lemma)f(3.1)77 b(.)42 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)76 b(21)118 5117 y(C.3)f(Proof)23 b(of)f(Lemma)f(3.2)77 b(.)42 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)76 b(22)118 5216 y(C.4)f(Proof)23 b(of)f(Lemma)f(3.5)77 b(.)42 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)76 b(25)118 5316 y(C.5)f(Proof)23 b(of)f(Lemma)f(3.4)77 b(.)42 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)76 b(26)118 5416 y(C.6)f(Proof)23 b(of)f(Proposition)i(3.3)70 b(.)41 b(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h (.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f (.)h(.)76 b(27)118 5515 y(C.7)f(Proof)23 b(of)f(Proposition)i(3.6)70 b(.)41 b(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h (.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f (.)h(.)76 b(29)p -9 5583 1512 4 v 77 5636 a Fx(\003)113 5660 y Fw(Supported)16 b(in)h(part)g(by)g(the)g(F)o(onds)g(National)g(Suisse.)1859 5909 y Fz(1)p eop %%Page: 2 2 bop -9 145 a FA(D)66 b(Pr)n(oof)23 b(of)f(Lemma)e(3.8)2848 b(30)118 244 y Fz(D.1)71 b(Preliminary)22 b(Bounds)84 b(.)42 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h (.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f (.)h(.)76 b(30)118 344 y(D.2)71 b(Bound)23 b(on)29 b Fv(^)-49 b Fy(q)704 356 y Fu(1)763 344 y Fz(and)22 b Fy(D)8 b Fv(^)-48 b Fy(q)1015 356 y Fu(1)1078 344 y Fz(.)42 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h (.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)76 b(34)118 443 y(D.3)71 b(Bound)23 b(on)29 b Fv(^)-49 b Fy(q)704 455 y Fu(2)763 443 y Fz(and)22 b Fy(D)8 b Fv(^)-48 b Fy(q)1015 455 y Fu(2)1078 443 y Fz(.)42 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h (.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)76 b(34)118 543 y(D.4)71 b(Bound)23 b(on)29 b Fv(^)-49 b Fy(q)704 555 y Fu(1)p Ft(;p)817 543 y Fz(and)22 b Fy(D)8 b Fv(^)-48 b Fy(q)1069 555 y Fu(1)p Ft(;p)1204 543 y Fz(.)41 b(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.) h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f (.)h(.)g(.)g(.)f(.)h(.)76 b(35)118 643 y(D.5)71 b(Bound)23 b(on)29 b Fv(^)-49 b Fy(q)704 655 y Fu(2)p Ft(;p)817 643 y Fz(and)22 b Fy(D)8 b Fv(^)-48 b Fy(q)1069 655 y Fu(2)p Ft(;p)1204 643 y Fz(.)41 b(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)76 b(35)-9 827 y FA(E)70 b(Pr)n(oof)23 b(of)f(Lemma)e(3.10) 2806 b(35)118 927 y Fz(E.1)80 b(Preliminary)22 b(Bounds)84 b(.)42 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)76 b(35)118 1027 y(E.2)k(Proof)23 b(of)f(Lemma)f(3.10)35 b(.)42 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)76 b(39)-9 1211 y FA(F)f(Bounds)22 b(on)g(Nonlinear)g(T)-8 b(erms)2547 b(40)118 1311 y Fz(F)-7 b(.1)92 b(Preliminary)22 b(Bounds)84 b(.)42 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f (.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)76 b(41)118 1410 y(F)-7 b(.2)92 b(Proof)23 b(of)f(Proposition)i(4.4)70 b(.)41 b(.)h(.)g(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)76 b(43)313 1510 y(F)-7 b(.2.1)106 b(Bound)23 b(on)29 b Fv(^)-49 b Fy(q)976 1522 y Fu(0)1035 1510 y Fz(and)22 b Fy(D)8 b Fv(^)-48 b Fy(q)1287 1522 y Fu(0)1392 1510 y Fz(.)42 b(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h (.)g(.)g(.)f(.)h(.)76 b(44)313 1610 y(F)-7 b(.2.2)106 b(Bound)23 b(on)940 1588 y Fv(^)939 1610 y Fy(h)987 1622 y Fu(1)1046 1610 y Fz(and)f Fy(D)1262 1588 y Fv(^)1261 1610 y Fy(h)1309 1622 y Fu(1)1392 1610 y Fz(.)42 b(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)76 b(45)313 1709 y(F)-7 b(.2.3)106 b(Bound)23 b(on)940 1687 y Fv(^)939 1709 y Fy(h)987 1721 y Fu(2)1046 1709 y Fz(and)f Fy(D)1262 1687 y Fv(^)1261 1709 y Fy(h)1309 1721 y Fu(2)1392 1709 y Fz(.)42 b(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)76 b(46)-9 1986 y FB(1)122 b(Intr)n(oduction)-9 2173 y Fz(In)24 b(this)f(paper)h(we)g(consider)g(the)g(time)e(independent)j (incompressible)f(Na)n(vier)n(-Stok)o(es)h(equations)f(in)g(dimensionless)g (form)-9 2273 y(in)g(the)h(domain)g Fv(\012)549 2285 y Fu(+)632 2273 y Fv(=)i Fs(f)p Fv(\()p Fy(x;)14 b(y)s Fv(\))28 b Fs(2)g Fr(R)1141 2237 y Fu(2)1203 2273 y Fs(j)d Fy(x)j Fs(\025)f Fv(1)p Fs(g)p Fz(.)40 b(W)-7 b(e)25 b(assume)g(that)f(there)g(is)h(an)g(in\256nite)f (\257ux)i(through)h Fv(\006)3278 2285 y Fu(+)3361 2273 y Fv(=)g Fs(f)p Fv(\()p Fy(x;)14 b(y)s Fv(\))28 b Fs(2)-9 2373 y Fr(R)63 2343 y Fu(2)120 2373 y Fs(j)21 b Fy(x)j Fv(=)e(1)p Fs(g)p Fz(,)e(the)h (boundary)j(of)e Fv(\012)1066 2385 y Fu(+)1121 2373 y Fz(,)f(meaning)g(that)g (at)f(in\256nity)h(the)g(\257uid)g(\257o)n(w)h(is)f(parallel)e(to)i(the)g Fy(x)g Fz(axis,)g(with)g(a)g(constant)-9 2472 y(speed)g Fy(u)253 2484 y Fq(1)346 2472 y Fv(=)i(1)p Fz(.)30 b(This)22 b(leads)e(to)i(the)f (follo)n(wing)h(set)f(of)h(equations)g(for)g(the)f(v)o(ector)h(\256eld)f Fr(u)2736 2442 y Fp(1)2770 2472 y Fz(:)1201 2619 y Fs(\000)p Fv(\()p Fr(u)e Fs(\001)f(r)p Fv(\))p Fr(u)h Fv(+)f Fs(4)p Fr(u)f Fs(\000)i(r)p Fy(p)k Fv(=)f(0)h Fz(,)1404 b(\(1.1\))1824 2744 y Fs(r)19 b(\001)f Fr(u)24 b Fv(=)e(0)h Fz(,)1404 b(\(1.2\))1607 2912 y(lim)1603 2966 y Ft(x)p Fq(\025)p Fu(1)1503 3037 y Ft(x)1541 3012 y Fo(2)1573 3037 y Fu(+)p Ft(y)1660 3012 y Fo(2)1692 3037 y Fq(!1)1838 2912 y Fr(u)p Fv(\()p Fr(x)p Fv(\))25 b(=)d Fr(u)2170 2924 y Fq(1)2264 2912 y Fs(\021)2351 2795 y Fn(\022)2412 2856 y Fv(1)2412 2969 y(0)2454 2795 y Fn(\023)2538 2912 y Fz(,)1048 b(\(1.3\))1832 3152 y Fr(u)p Fs(j)1908 3177 y Fu(\006)1955 3185 y Fo(+)2030 3152 y Fv(=)22 b Fr(u)2170 3164 y Fu(+)2248 3152 y Fz(,)1338 b(\(1.4\))-9 3299 y(with)1568 3399 y(lim)1521 3456 y Fq(j)p Ft(y)r Fq(j)o(!1)1743 3399 y Fr(u)1796 3411 y Fu(+)1851 3399 y Fv(\()p Fy(y)s Fv(\))23 b(=)g Fr(u)2123 3411 y Fq(1)2216 3399 y Fy(:)-9 3573 y Fz(As)33 b(w)o(as)g(noted)g(in)g([Amick])g (\(see)g(also)f([Galdi]\),)37 b(the)32 b(Na)n(vier)n(\261Stok)o(es)j (equations)e(\(in)g(e)o(xterior)g(domains\))h(ha)n(v)o(e)f(better)-9 3672 y(properties)21 b(for)h(symmetric)e(\257o)n(ws)i(around)h(the)d Fy(x)p Fz(\261axis,)i(that)e(is)g(when)i Fy(u)2181 3684 y Fu(1)2239 3672 y Fz(\(the)e(component)j(parallel)c(to)i Fr(u)3205 3684 y Fq(1)3275 3672 y Fz(\))g(is)g(e)n(v)o(en)f(in)h(the)-9 3772 y(`)p Fy(y)s Fz(')g(direction,)h(and)g Fy(u)649 3784 y Fu(2)707 3772 y Fz(is)e(odd.)33 b(W)-7 b(e)21 b(will)g(restrict)f(ourselv)o(es)i(to)f (this)g(particular)g(case.)115 3871 y(The)j(choice)f Fy(x)k Fv(=)e(1)e Fz(for)i(the)e(position)h(of)h Fv(\006)1447 3883 y Fu(+)1525 3871 y Fz(is)e(arbitrary)-6 b(.)39 b(In)24 b(a)g(second)g(part)g (of)g(this)f(introduction,)j(we)d(will)g(indicate)-9 3971 y(what)c(our)i (results)f(imply)g(for)h(the)e(Na)n(vier)n(\261Stok)o(es)j(equations)e(in)g (an)g(e)o(xterior)f(domain)i(with)f(physical)f(parameters)h(dif)n(ferent)-9 4071 y(from)i(the)f(ones)h(we)f(choose,)i(that)d(is)h(when)i(the)e(dynamic)h (viscosity)-6 b(,)22 b(the)f(density)g(and)h Fy(u)2671 4083 y Fq(1)2762 4071 y Fz(are)f(dif)n(ferent)h(from)h Fv(1)p Fz(.)115 4170 y(W)-7 b(e)34 b(are)g(going)i(to)e(sho)n(w)i(e)o(xistence)d(and)i (asymptotic)f(properties)h(of)g(the)f(solutions)h(to)f(\(1.1\)\261\(1.4\))k (under)e(certain)-9 4270 y(conditions)c(on)g(the)f(boundary)k(v)n(alue)c Fr(u)1246 4282 y Fu(+)1301 4270 y Fz(.)61 b(These)31 b(restrictions)g(are)g (best)g(e)o(xpressed)h(in)g(terms)f(of)h(the)f(in)m(v)o(erse)g(F)o(ourier)-9 4370 y(transform)22 b(of)g Fr(u)487 4382 y Fu(+)564 4370 y Fz(with)f(respect)g(to)g Fy(y)s Fz(.)31 b(T)-7 b(o)22 b(\256x)f(notation,)h (we)g(de\256ne)g(the)f(F)o(ourier)h(transform)g Fy(f)30 b Fz(of)22 b(a)f(function)3362 4348 y Fv(^)3344 4370 y Fy(f)29 b Fz(by)1331 4557 y Fy(f)9 b Fv(\()p Fy(x;)14 b(y)s Fv(\))23 b(=)1719 4501 y(1)p 1694 4538 92 4 v 1694 4614 a(2)p Fy(\031)1809 4444 y Fn(Z)1892 4557 y Fy(dk)h(e)2041 4523 y Ft(ik)q(y)2159 4535 y Fv(^)2141 4557 y Fy(f)9 b Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fz(,)-9 4756 y(where)220 4689 y Fn(R)275 4756 y Fy(dk)i Fz(is)d(a)f (shorthand)j(notation)f(for)1303 4689 y Fn(R)1358 4709 y Fq(1)1342 4785 y(\0001)1478 4756 y Fy(dk)s Fz(.)33 b(T)-7 b(o)23 b(formulate)f(our)h (theorem,)g(we)f(de\256ne)h(the)e(tw)o(o)i(follo)n(wing)g(Banach)-9 4855 y(spaces.)-9 5002 y FA(De\256nition)f(1.1)42 b Fm(Let)25 b Fy(\013)30 b(>)e Fv(3)p Fm(,)e Fy(\015)960 5014 y Fu(1)997 5002 y Fy(;)14 b(\015)1077 5014 y Ft(E)1162 5002 y Fs(2)29 b Fv([0)p Fy(;)14 b Fv(1\))p Fm(,)26 b Fy(\015)1512 5014 y Fu(2)1578 5002 y Fs(2)j Fv([0)p Fy(;)1774 4969 y Fu(1)p 1774 4983 34 4 v 1774 5031 a(2)1817 5002 y Fv(])c Fm(and)i Fy(\016)32 b(>)c Fs(\000)p Fv(1)p Fm(.)43 b(W)-8 b(e)25 b(denote)h(by)g Fs(W)2917 5014 y Ft(\013;\015)3015 5022 y Fo(1)3047 5014 y Ft(;\015)3102 5022 y Fo(2)3164 5002 y Fm(the)f(Banac)o(h)i(space)-9 5102 y(obtained)21 b(by)f(completing)g Fy(C)870 5071 y Fq(1)864 5122 y Fu(0)940 5102 y Fv(\()p Fr(R)p Fy(;)14 b Fr(R)1153 5071 y Fu(2)1190 5102 y Fv(\))p Fm(,)21 b(the)e(space)h(of)g(smooth)h(compactly)f (supported)i(functions)e Fr(R)j Fs(!)g Fr(R)3284 5071 y Fu(2)3321 5102 y Fm(,)d(with)f(r)m(espect)-9 5201 y(to)i(the)g(norm)487 5378 y Fs(jj)p Fr(u)p Fs(jj)632 5390 y Ft(\013;\015)730 5398 y Fo(1)762 5390 y Ft(;\015)817 5398 y Fo(2)876 5378 y Fv(=)30 b Fz(max)964 5431 y Ft(i)p Fu(=1)p Ft(;)p Fu(2)1138 5261 y Fn(\032)1210 5378 y Fz(sup)1201 5450 y Ft(k)q Fq(2)p Fl(R)1338 5310 y Fn(\000)1376 5378 y Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1611 5310 y Fn(\001)1649 5328 y Ft(\013)1717 5378 y Fs(j)5 b Fv(^)-47 b Fy(u)1788 5390 y Ft(i)1816 5378 y Fv(\()p Fy(k)s Fv(\))p Fs(j)23 b Fm(,)45 b Fz(sup)2028 5450 y Ft(k)q Fq(2)p Fl(R)2166 5310 y Fn(\000)2204 5378 y Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2439 5310 y Fn(\001)2477 5328 y Ft(\013)p Fq(\000)p Ft(\015)2607 5336 y Fk(i)2658 5378 y Fs(j)p Fy(k)s Fs(j)2750 5343 y Ft(\015)2785 5351 y Fk(i)2837 5378 y Fs(j)p Fy(@)2904 5390 y Ft(k)2950 5378 y Fv(^)-47 b Fy(u)2993 5390 y Ft(i)3020 5378 y Fv(\()p Fy(k)s Fv(\))p Fs(j)3153 5261 y Fn(\033)3253 5378 y Fm(,)p -9 5502 1512 4 v 83 5558 a Fj(1)113 5581 y Fw(W)-5 b(e)18 b(will)g(adopt)h(the)f (follo)n(wing)g(notation,)h(e)o(xcept)f(otherwise)h(stated.)29 b(V)-8 b(ectors)19 b(will)f(be)h(denoted)f(by)h(boldf)o(ace)f(letters,)h(lik) o(e)g Fi(u)p Fw(,)g(and)f(components)h(by)-9 5660 y(the)e(same)g(letter)g (with)g(a)g(subscript,)g(lik)o(e)h Fh(u)1028 5669 y Fg(1)1062 5660 y Fw(.)1859 5909 y Fz(2)p eop %%Page: 3 3 bop -9 145 a Fm(and)22 b(by)g Fs(A)307 157 y Ft(\013;\015)405 165 y Fk(E)453 157 y Ft(;\016)530 145 y Fm(the)f(Banac)o(h)i(space)f (obtained)g(by)g(completing)f Fy(C)2033 114 y Fq(1)2027 165 y Fu(0)2104 145 y Fv(\()p Fr(R)p Fy(;)14 b Fr(R)p Fv(\))21 b Fm(with)g(r)m(espect)f(to)h(the)g(norm)502 333 y Fs(j)p Fy(f)9 b Fs(j)598 345 y Ft(\013;\015)696 353 y Fk(E)743 345 y Ft(;\016)823 333 y Fv(=)22 b Fz(max)q Fs(f)10 b Fz(sup)1099 406 y Ft(k)q Fq(2)p Fl(R)1249 333 y Fy(e)1288 299 y Fq(j)p Ft(k)q Fq(j)p Fu(\(1+)p Ft(\016)r Fu(\))1536 333 y Fs(j)p Fy(f)f Fv(\()p Fy(k)s Fv(\))p Fs(j)24 b Fm(,)d Fv(\(1)d(+)g Fy(\016)s Fv(\))2055 299 y Ft(\015)2090 307 y Fk(E)2139 299 y Fq(\000)p Fu(1)2272 333 y Fz(sup)2263 406 y Ft(k)q Fq(2)p Fl(R)2414 333 y Fs(j)p Fy(k)s Fs(j)2506 299 y Ft(\015)2541 307 y Fk(E)2593 333 y Fy(e)2632 299 y Fq(j)p Ft(k)q Fq(j)p Fu(\(1+)p Ft(\016)r Fu(\))2880 333 y Fs(j)p 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Fo(1)2535 1474 y Ft(;\015)2590 1482 y Fo(2)2653 1462 y Fs(\024)27 b Fy(")p Fz(,)f(meaning)f(that)f(Condition)h(1.2)-9 1562 y(is)h(satis\256ed)g(with)g Fy(f)39 b Fv(=)29 b(0)p Fz(.)46 b(Roughly)29 b(speaking,)g(this)d(implies)f(that)h Fr(u)2142 1574 y Fu(+)2219 1562 y Fs(\000)21 b Fr(u)2358 1574 y Fq(1)2454 1562 y Fz(has)27 b(deri)n(v)n(ati)n(v)o(es)f(of)h(order)h Fy(\013)21 b Fs(\000)g Fv(1)26 b Fz(and)h(the)-9 1662 y Fy(i)20 1632 y Fu(th)113 1662 y Fz(component)h(of)g Fr(u)21 b Fs(\000)g Fr(u)828 1674 y Fq(1)925 1662 y Fz(decays)27 b(at)f(in\256nity)h(at)g(least)e(lik)o(e) h Fs(j)p Fy(y)s Fs(j)2057 1632 y Ft(\015)2092 1640 y Fk(i)2118 1632 y Fq(\000)p Fu(2)2207 1662 y Fz(,)j(see)d(Appendix)j(A.)48 b(Thus,)29 b(the)e(\256rst)f(component)-9 1761 y(should)20 b(decay)f(f)o(aster)g(than)g Fs(j)p Fy(y)s Fs(j)915 1731 y Fq(\000)p Fu(1)1004 1761 y Fz(,)h(whereas)f(the)g(second)h(should)g(decay)f (at)g(least)e(lik)o(e)i Fs(j)p Fy(y)s Fs(j)2673 1731 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2829 1761 y Fz(.)30 b(In)20 b(vie)n(w)f(of)g(applications) g(for)-9 1861 y(the)j(e)o(xterior)h(problem,)i Fs(j)p Fy(y)s Fs(j)821 1831 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1000 1861 y Fz(is)d(too)h (restricti)n(v)o(e)f(\(see)g(belo)n(w\).)37 b(If)23 b(the)g(second)h (component)g(of)g(the)e(boundary)k(condition)-9 1961 y(decays)g(slo)n(wer)g (than)h Fs(j)p Fy(y)s Fs(j)761 1930 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)917 1961 y Fz(,)g(e)n(v)o(en)f(lik)o(e)g Fs(j)p Fy(y)s Fs(j)1391 1930 y Fq(\000)p Fu(1)1480 1961 y Fz(,)h(our)h(technique)e(can)g (still)f(be)h(applied,)h(b)n(ut)g(we)g(ha)n(v)o(e)f(to)g(capture)g(all)f (this)-9 2060 y(slo)n(w)e(decay)g(in)g(a)g(non\261zero)i Fr(u)935 2072 y Ft(f)978 2060 y Fz(.)37 b(From)24 b(a)f(more)g(technical)f(point)i(of) f(vie)n(w)-6 b(,)25 b(we)e(note)g(that)28 b Fv(^)-47 b Fy(u)2815 2072 y Fu(+)p Ft(;)p Fu(2)2945 2060 y Fz(is)23 b(an)g(odd)i(function.)37 b(This)-9 2160 y(implies)20 b(that)h Fy(f)9 b Fv(\(0\))21 b Fz(has)g(to)g(be)h(equal)f(to)27 b Fv(^)-47 b Fy(u)1252 2172 y 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b(we)g(de\256ne)517 5386 y Fn(e)515 5387 y Fr(x)p Fy(;)606 5386 y Fn(e)602 5387 y Fr(u)g Fz(and)37 b Fn(e)-51 b Fy(p)31 b Fz(by)i(the)e(equations)1526 5386 y Fn(e)1524 5387 y Fr(x)37 b Fv(=)g Fy(R)32 b Fr(x)p Fy(;)14 b Fr(u)p Fv(\()p Fr(x)p Fv(\))39 b(=)d Fy(u)2250 5399 y Fq(1)2324 5386 y Fn(e)2320 5387 y Fr(u)p Fv(\()2407 5386 y Fn(e)2405 5387 y Fr(x)q Fv(\))c Fz(and)g Fy(p)p Fv(\()p Fr(x)p Fv(\))38 b(=)2970 5320 y Fn(\000)3008 5387 y Fy(\045u)3099 5357 y Fu(2)3099 5408 y Fq(1)3168 5320 y Fn(\001)3225 5387 y(e)-51 b Fy(p)p Fv(\()3296 5386 y Fn(e)3294 5387 y Fr(x)q Fv(\))p Fz(.)62 b(Then)32 b(the)-9 5487 y(equations)21 b(\(1.10\)\261\(1.12\))27 b(become)21 b(identical)f(to)h(\(1.1\)\261\(1.3\),)26 b(namely)1317 5660 y Fs(\000)p Fv(\()1417 5659 y Fn(e)1414 5660 y Fr(u)19 b Fs(\001)f(r)p Fv(\))1631 5659 y Fn(e)1628 5660 y Fr(u)h Fv(+)f Fs(4)1861 5659 y Fn(e)1858 5660 y Fr(u)g Fs(\000)g(r)5 b Fn(e)-51 b Fy(p)23 b Fv(=)f(0)h Fz(,)1246 b(\(1.14\))1859 5909 y(4)p eop %%Page: 5 5 bop 1940 145 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2191 y(Condition)g(1.2.)-9 2474 y FB(2)122 b(Ref)m(ormulation)32 b(as)e(a)h(Dynamical)g(System)-9 2662 y Fz(In)37 b(this)g(section,)k(we)c(transform)i(\(1.1\))g(and)e(\(1.2\)) i(to)e(a)g Fv(4)p Fz(\261dimensional)g(system)h(of)f(\256rst)h(order)g(in)f Fy(@)3252 2674 y Ft(x)3294 2662 y Fz(,)k(of)d(the)f(form)-9 2761 y Fy(@)35 2773 y Ft(x)76 2761 y Fr(w)25 b Fv(=)d Fy(L)p Fr(w)e Fv(+)e Fy(F)12 b Fv(\()p Fr(w)q Fv(\))p Fz(,)22 b(where)g Fy(L)e Fz(is)h(a)g(dif)n(ferential)g(operator)h(in)g Fy(@)2004 2773 y Ft(y)2065 2761 y Fz(and)g Fy(F)33 b Fz(a)21 b(nonlinear)h(map)g (depending)h(on)f Fr(w)h Fz(and)f Fy(@)3625 2773 y Ft(y)3664 2761 y Fr(w)q Fz(.)115 2861 y(Suppose)27 b(that)d(Condition)h(1.2)h(is)e (satis\256ed.)41 b(Lemma)24 b(1.3)i(pro)o(vides)g(a)e(solution)2595 2860 y Fv(~)2589 2861 y Fr(u)2642 2873 y Ft(E)2723 2861 y Fz(of)h (\(1.1\)\261\(1.3\))k(which,)d(ho)n(we)n(v)o(er)m(,)-9 2961 y(does)c(not)f(satisfy)g(the)h(boundary)i(condition)e(\(1.4\).)33 b(T)-7 b(o)21 b(remedy)i(this,)e(we)g(mak)o(e)h(an)f(ansatz)g(of)h(the)f (form)1645 3143 y Fr(u)i Fv(=)1812 3142 y Fn(e)1809 3143 y Fr(u)1862 3155 y Ft(E)1936 3143 y Fv(+)18 b Fr(v)25 b Fz(,)1656 3268 y Fy(p)e Fv(=)28 b Fn(e)-51 b Fy(p)1851 3280 y Ft(E)1925 3268 y Fv(+)25 b(~)-49 b Fy(p)23 b(:)-9 3451 y Fz(Then)e(we)h(\256nd)g(for)g Fr(v)h Fz(the)e(equations)1374 3633 y Fy(@)1418 3645 y Ft(x)1459 3633 y Fr(v)k Fv(=)e Fs(4)p Fr(v)c Fs(\000)f Fr(Q)g Fs(\000)g(r)7 b Fv(~)-49 b Fy(p)23 b Fz(,)1430 b(\(2.1\))1330 3758 y Fs(r)19 b(\001)f Fr(v)25 b Fv(=)e(0)f Fz(,)1900 b(\(2.2\))1128 3882 y(lim)1116 3934 y Ft(x)p Fq(!)p Fu(1)1267 3882 y Fr(v)q Fv(\()p Fy(x;)14 b(y)s Fv(\))24 b(=)f Fr(u)1675 3894 y Fu(+)1730 3882 y Fv(\()p Fy(y)s Fv(\))c Fs(\000)f Fr(u)1993 3894 y Fq(1)2082 3882 y Fs(\000)g Fr(u)2218 3894 y Ft(E)2274 3882 y Fv(\(1)p Fy(;)c(y)s Fv(\))22 b Fs(\021)h Fr(v)2621 3894 y Fu(+)2677 3882 y Fv(\()p Fy(y)s Fv(\))g Fz(,)778 b(\(2.3\))1036 4037 y(lim)931 4094 y Ft(x)969 4078 y Fo(2)1002 4094 y Fu(+)p 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Fy(v)s(v)1884 5545 y Ft(E)1940 5533 y Fv(\))23 b Fy(:)1589 b Fz(\(2.8\))1859 5909 y(6)p eop %%Page: 7 7 bop -9 145 a Fz(Exploiting)22 b(again)e(the)i(incompressibility)f(equation)h Fy(@)1670 157 y Ft(x)1712 145 y Fy(u)c Fv(+)g Fy(@)1905 157 y Ft(y)1945 145 y Fy(v)26 b Fv(=)c(0)p Fz(,)f(the)h(tw)o(o)f(components)i(of) f(\(2.1\))h(read)1160 327 y Fy(@)1204 339 y Ft(y)1257 327 y Fv(\()q Fs(\000)p Fy(@)1399 339 y Ft(x)1440 327 y Fy(v)f Fv(+)c Fy(v)k Fv(+)c Fy(@)1774 339 y Ft(y)1814 327 y Fy(u)g Fv(+)g Fy(q)2000 339 y Fu(0)2037 327 y Fv(\))23 b(=)87 b Fy(h)2292 339 y Fu(1)2348 327 y Fv(+)18 b Fy(@)2475 339 y Ft(x)2524 327 y Fv(~)-49 b Fy(p)23 b Fz(,)1004 b(\(2.9\))1158 451 y Fy(@)1202 463 y Ft(x)1257 451 y Fv(\()q Fs(\000)p Fy(@)1399 463 y Ft(x)1440 451 y Fy(v)22 b Fv(+)c Fy(v)k Fv(+)c Fy(@)1774 463 y Ft(y)1814 451 y Fy(u)g Fv(+)g Fy(q)2000 463 y Fu(0)2037 451 y Fv(\))23 b(=)g Fs(\000)p Fy(h)2293 463 y Fu(2)2348 451 y Fs(\000)18 b Fy(@)2475 463 y Ft(y)2522 451 y Fv(~)-49 b Fy(p)23 b(:)962 b Fz(\(2.10\))-9 633 y(This)23 b(suggest)h(that)f(instead)g(of)i(using)f(an)g (auxiliary)f(v)n(ariable)g(for)i Fy(@)2051 645 y Ft(x)2093 633 y Fy(v)h Fz(to)e(reduce)g(the)f(problem)i(to)e(a)g(\256rst)h(order)h (system,)f(it)-9 733 y(is)c(more)i(adv)n(antageous)h(to)e(introduce)h(a)f(ne) n(w)h(v)n(ariable)27 b Fv(~)-48 b Fy(\021)24 b Fz(de\256ned)f(by)1444 915 y Fv(~)-47 b Fy(\021)26 b Fv(=)c Fy(@)1637 927 y Ft(x)1679 915 y Fy(v)g Fs(\000)c Fy(v)k Fs(\000)c Fy(@)2013 927 y Ft(y)2053 915 y Fy(u)g Fs(\000)g Fy(q)2239 927 y Fu(0)2299 915 y Fy(:)1243 b Fz(\(2.11\))-9 1097 y(One)21 b(sho)n(ws)i(easily)d(no)n(w)j(that)d(\(2.1\)) j(and)f(\(2.2\),)i(together)d(with)h(\(2.11\),)h(are)e(equi)n(v)n(alent)g(to) 1437 1279 y Fy(@)1481 1291 y Ft(x)1523 1279 y Fy(u)i Fv(=)f Fs(\000)p Fy(@)1790 1291 y Ft(y)1830 1279 y Fy(v)k Fz(,)1648 b(\(2.12\))1442 1403 y Fy(@)1486 1415 y Ft(x)1527 1403 y Fy(v)27 b Fv(=)22 b Fy(v)g Fv(+)c Fy(@)1870 1415 y Ft(y)1910 1403 y Fy(u)g Fv(+)g Fy(q)2096 1415 y Fu(0)2152 1403 y Fv(+)24 b(~)-48 b Fy(\021)26 b Fz(,)1242 b(\(2.13\))1443 1528 y Fy(@)1487 1540 y Ft(x)1536 1528 y Fv(~)-49 b Fy(p)23 b Fv(=)f Fs(\000)p Fy(@)1790 1540 y Ft(y)1836 1528 y Fv(~)-48 b Fy(\021)22 b Fs(\000)c Fy(h)2024 1540 y Fu(1)2084 1528 y Fz(,)1460 b(\(2.14\))1441 1652 y Fy(@)1485 1664 y Ft(x)1532 1652 y Fv(~)-48 b Fy(\021)27 b Fv(=)22 b Fy(@)1725 1664 y Ft(y)1772 1652 y Fv(~)-49 b Fy(p)19 b Fv(+)f Fy(h)1957 1664 y Fu(2)2017 1652 y Fy(:)1525 b Fz(\(2.15\))-9 1834 y(These)18 b(equations)i(will)e(be)h(interpreted)f(as)h(a)g(dynamical)f(system,)i(where) f(the)g(coordinate)g(parallel)f(to)h(the)f(\257o)n(w)-6 b(,)21 b Fy(x)p Fz(,)f(plays)f(the)-9 1934 y(role)i(of)h(time.)30 b(Indeed,)22 b(as)f(is)g(easily)g(seen)g(from)h(\(2.6\),)i(\(2.7\))f(and)e (\(2.8\),)j(the)d(right)g(hand)i(side)e(of)h(these)f(equations)h(does)f(not) -9 2033 y(contain)g Fy(x)h Fz(deri)n(v)n(ati)n(v)o(es)e(an)o(ymore.)115 2133 y(There)g(is)e(another)j(ar)n(gument)f(for)h(the)e(introduction)h(of)h (a)e(ne)n(w)g(v)n(ariable)g(for)h Fy(@)2456 2145 y Ft(x)2498 2133 y Fy(v)e Fs(\000)c Fy(v)j Fs(\000)d Fy(@)2815 2145 y Ft(y)2855 2133 y Fy(u)g Fs(\000)f Fy(q)3032 2145 y Fu(0)3070 2133 y Fz(.)30 b(A)20 b(priori,)g(we)g(e)o(xpect)-9 2233 y Fs(\000)p Fy(@)100 2245 y Ft(x)141 2233 y Fy(v)k Fv(+)c Fy(v)k Fv(+)c Fy(@)483 2245 y Ft(y)523 2233 y Fy(u)25 b Fz(to)g(be)g(of)h(order)h Fy(")p Fz(,)f(with)f Fy(")g Fz(small,)h(whereas)f Fy(q)1991 2245 y Fu(0)2029 2233 y Fz(,)h Fy(h)2124 2245 y Fu(1)2187 2233 y Fz(and)f Fy(h)2382 2245 y Fu(2)2445 2233 y Fz(should)i(be)e(of)h(order)g Fy(")3141 2203 y Fu(2)3178 2233 y Fz(.)43 b(Equations)26 b(\(2.9\))-9 2332 y(and)c(\(2.10\))i(indicate)d(that)g(there)g(is)h(a)f(compensation)i (between)e Fy(@)1973 2344 y Ft(x)2015 2332 y Fy(v)s Fz(,)i Fy(v)i Fz(and)d Fy(@)2355 2344 y Ft(y)2395 2332 y Fy(u)p Fz(.)32 b(The)22 b(introduction)h(of)28 b Fv(~)-48 b Fy(\021)25 b Fz(as)c(ne)n(w)i(v) n(ariable)-9 2432 y(allo)n(ws)e(us)g(to)h(e)o(xplicitly)e(control)i(this)f 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b(of)g(these)e(tw)o(o)i(linear)f(inhomogeneous)j(equations)e(will)e(tak)o(e)h (place)f(in)i(the)f(Sections)g(3.2)h(and)g(3.3)g(where)g(we)-9 3401 y(will)h(start)g(by)i(making)g(v)o(ery)g(fe)n(w)f(assumptions)h(on)g (the)f(`inhomogeneous)j(terms'.)54 b(Finally)-6 b(,)32 b(we)d(will)f(mak)o(e) h(additional)-9 3500 y(hypotheses)22 b(on)g(them)f(in)h(Section)f(3.4)h(and)g (see)f(their)g(implications)f(for)j(the)e(solutions)h(of)g(the)f(Sections)g (3.2)i(and)e(3.3.)-9 3741 y Fd(3.1)102 b(Pr)n(eamble)-9 3898 y Fz(If)29 b(we)f(try)h(to)g(interpret)g(\(2.12\)\261\(2.15\))k(as)28 b(a)h(dynamical)f(system)h(where)g Fy(x)g Fz(plays)g(the)g(role)f(of)i(time,) f(we)g(are)f(f)o(aced)h(with)-9 3997 y(the)c(follo)n(wing)i(problem.)44 b(F)o(or)27 b(the)e(moment,)i(we)f(did)g(not)g(specify)g(an)o(y)g(boundary)i (condition)f(for)33 b Fv(~)-49 b Fy(p)25 b Fz(\(let)g(alone)h(for)32 b Fv(~)-48 b Fy(\021)s Fz(\))26 b(on)-9 4097 y Fv(\006)51 4109 y Fu(+)106 4097 y Fz(.)35 b(A)23 b(priori,)i(there)d(is)h(no)h(ob)o(vious)g (reason)g(why)f(we)g(should)i(be)e(able)f(to)h(impose)g(e)o(xactly)f(tw)o(o)i (out)f(of)h(the)e(four)j(possible)-9 4197 y(boundary)e(conditions,)f(neither) e(why)h(we)g(should)h(be)e(able)g(to)h(satisfy)f(those)h(for)g Fy(u;)14 b(v)s Fz(,)20 b(and)h(not,)h(say)-6 b(,)21 b(those)g(of)g Fy(u)f Fz(and)h Fy(p)f Fz(only)-6 b(.)-9 4296 y(W)f(e)22 b(will)f(no)n(w)j (moti)n(v)n(ate)d(by)i(a)f(linear)g(analysis)g(that)g(we)g(can)g(impose)h Fy(u)f Fz(and)h Fy(v)i Fz(on)e Fv(\006)2565 4308 y Fu(+)2620 4296 y Fz(.)34 b(T)-7 b(o)23 b(do)g(this,)g(we)f(set)g Fy(q)3334 4308 y Fu(0)3371 4296 y Fy(;)14 b(h)3456 4308 y Fu(1)3493 4296 y Fy(;)g(h)3578 4308 y Fu(2)3640 4296 y Fv(=)24 b(0)-9 4396 y Fz(in)d(\(2.12\)\261\(2.15\),)26 b(which)c(gi)n(v)o(es)1525 4578 y Fy(@)1569 4590 y Ft(x)1611 4578 y Fy(u)h Fv(=)f Fs(\000)p Fy(@)1878 4590 y Ft(y)1918 4578 y Fy(v)k Fz(,)1602 b(\(3.1\))1529 4702 y Fy(@)1573 4714 y Ft(x)1615 4702 y Fy(v)27 b Fv(=)22 b Fy(v)g Fv(+)c Fy(@)1958 4714 y Ft(y)1998 4702 y Fy(u)g Fv(+)24 b(~)-48 b Fy(\021)26 b Fz(,)1372 b(\(3.2\))1531 4827 y Fy(@)1575 4839 y Ft(x)1624 4827 y Fv(~)-49 b Fy(p)23 b Fv(=)f Fs(\000)p Fy(@)1878 4839 y Ft(y)1924 4827 y Fv(~)-48 b Fy(\021)26 b Fz(,)1601 b(\(3.3\))1529 4951 y Fy(@)1573 4963 y Ft(x)1620 4951 y Fv(~)-48 b Fy(\021)27 b Fv(=)22 b Fy(@)1813 4963 y Ft(y)1860 4951 y Fv(~)-49 b Fy(p)23 b(:)1666 b Fz(\(3.4\))-9 5133 y(Suppose)24 b(no)n(w)g(that)29 b Fv(~)-49 b Fy(p)o Fv(\()p Fy(x;)14 b(y)s Fv(\))26 b(=)e Fy(a)e(e)1071 5103 y Ft(\025)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\)+)p Ft(ik)q(y)1436 5133 y Fz(,)h(then)g(by)g(\(3.4\),)i(we)e (get)28 b Fv(~)-48 b Fy(\021)s Fv(\()p Fy(x;)14 b(y)s Fv(\))26 b(=)e Fy(ik)s(\025)2685 5103 y Fq(\000)p Fu(1)2781 5133 y Fv(~)-49 b Fy(p)p Fv(\()p Fy(x;)14 b(y)s Fv(\))p Fz(.)35 b(Substituting)23 b(in)g(\(3.3\),)-9 5233 y(we)f(get)g(the)g(characteristic)f(equation)i Fy(\025)1203 5203 y Fu(2)1259 5233 y Fs(\000)c Fy(k)1389 5203 y Fu(2)1450 5233 y Fv(=)25 b(0)p 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Fz(,)h(then)f(by)h(\(3.1\),)i(we)d(get)789 324 y Fy(v)s Fv(\()p Fy(x;)14 b(y)s Fv(\))24 b(=)f Fs(\000)p Fy(i)34 b Fz(s)-5 b(i)g(g)g(n)q Fv(\()p Fy(k)s Fv(\))22 b Fy(b)f(e)1619 289 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\)+)p Ft(ik)q(y)2091 324 y Fv(+)d Fy(i)i(c)i Fv(\003)f Fy(k)2406 289 y Fq(\000)p Fu(1)2516 324 y Fy(e)2555 289 y Fu(\003\()p Ft(x)p Fq(\000)p Fu(1\)+)p Ft(ik)q(y)2948 324 y Fy(:)-9 503 y Fz(Substituting)h(in)f(\(3.2\),)i(we)f(get)733 682 y Fy(i)f(k)j Fv(\()p Fy(b)18 b Fs(\000)g Fy(a)p Fv(\))k Fy(e)1156 648 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\)+)p Ft(ik)q(y)1632 682 y Fv(=)h Fs(\000)p Fy(i)d(c)h(k)1937 648 y Fq(\000)p Fu(1)2047 682 y Fv(\(\003)2137 648 y Fu(2)2193 682 y Fs(\000)d Fv(\003)g Fs(\000)g Fy(k)2481 648 y Fu(2)2518 682 y Fv(\))k Fy(e)2611 648 y Fu(\003\()p Ft(x)p Fq(\000)p Fu(1\)+)p Ft(ik)q(y)3005 682 y Fy(:)-9 861 y Fz(T)-7 b(o)21 b(solv)o(e)h(this)f(equation,)h(we)f(set)g Fy(a)i Fv(=)f Fy(b)p 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Fu(+)2031 5428 y Fv(\()p Fy(y)s Fv(\))2143 5254 y Fn(\023)2227 5371 y Fz(,)1317 b(\(3.14\))p -9 5502 1512 4 v 83 5558 a Fj(3)113 5581 y Fw(At)24 b(the)f(linear)h(le)n(v)o(el,)31 b Ff(^)-40 b Fh(\021)697 5590 y Fg(+)748 5581 y Ff(\()p Fh(k)r Ff(\))30 b(=)f Fh(i)35 b Fw(s)-5 b(i)g(g)g(n)r Ff(\()p Fh(k)r Ff(\))24 b Fh(a)p Ff(\()p Fh(k)r Ff(\))p Fw(,)i(where)e Fh(a)p Ff(\()p Fh(k)r Ff(\))g Fw(is)h(determined)e(by)h(\(3.9\),)h(and)e(in)h(the)g (nonlinear)f(case,)k(we)d(will)f(gi)n(v)o(e)h(the)g(good)-9 5660 y(e)o(xpression)17 b(for)22 b Ff(^)-41 b Fh(\021)428 5669 y Fg(+)480 5660 y Ff(\()p Fh(k)r Ff(\))17 b Fw(later)g(on.)1859 5909 y Fz(8)p eop %%Page: 9 9 bop -9 145 a Fz(where)21 b Fy(\021)260 157 y Fu(0)319 145 y Fz(is)g(de\256ned)h(by)h(\(3.10\))g(for)f(a)f(gi)n(v)o(en)28 b Fv(^)-48 b Fy(\021)1428 157 y Fu(+)1483 145 y Fz(,)22 b(and)g Fy(q)k Fv(:)d(\012)1839 157 y Fu(+)1917 145 y Fs(!)g Fr(R)e Fz(is)g(some)h(gi)n(v)o(en)f(function,)i(and)f(the)f(second)h(one)g(is)1679 327 y Fy(@)1723 339 y Ft(x)1765 327 y Fy(p)h Fv(=)g Fs(\000)p Fy(@)2027 339 y Ft(y)2066 327 y Fy(\021)f Fs(\000)c Fy(h)2260 339 y Fu(1)2320 327 y Fz(,)1224 b(\(3.15\))1677 452 y Fy(@)1721 464 y Ft(x)1763 452 y Fy(\021)26 b Fv(=)d Fy(@)1962 464 y Ft(y)2002 452 y Fy(p)18 b Fv(+)g Fy(h)2193 464 y Fu(2)2253 452 y Fz(,)1291 b(\(3.16\))1431 576 y(lim)1419 628 y Ft(x)p Fq(!)p Fu(1)1570 576 y Fy(\021)s Fv(\()p Fy(x;)14 b(y)s Fv(\))24 b(=)f(0)f Fz(,)1562 b(\(3.17\))-9 785 y(where)31 b Fy(h)277 797 y Fu(1)314 785 y Fy(;)14 b(h)399 797 y Fu(2)471 785 y Fv(:)36 b(\012)590 797 y Fu(+)681 785 y Fs(!)g Fr(R)30 b Fz(are)g(some)h(gi)n(v)o(en)g(functions.)60 b(The)30 b(strate)o(gy)h(of)g(the)f(proof)j(of)e(e)o(xistence,)h(uniqueness)g (and)-9 884 y(dif)n(ferent)27 b(properties)g(of)h(the)f(solutions)g(of)h(the) f(equations)g(\(2.12\)\261\(2.15\))32 b(will)26 b(be)h(as)g(follo)n(ws.)48 b(W)-7 b(e)27 b(will)f(\256rst)i(sho)n(w)g(that)-9 984 y(under)d(relati)n(v)o (ely)d(weak)i(assumptions)g(on)h Fy(q)s Fz(,)f Fy(h)1437 996 y Fu(1)1498 984 y Fz(and)g Fy(h)1692 996 y Fu(2)1729 984 y Fz(,)h(the)e(tw)o(o)h(linear)f(inhomogeneous)k(problems)e (\(3.12\)\261\(3.14\))j(and)-9 1084 y(\(3.15\)\261\(3.17\))i(admit)25 b(strong)i(solutions.)45 b(Then)26 b(we)g(will)f(construct)h(a)f(sequence)h (of)g(functions)h Fr(v)3006 1096 y Ft(n)3052 1084 y Fy(;)14 b(p)3131 1096 y Ft(n)3201 1084 y Fz(and)26 b Fy(\021)3390 1096 y Ft(n)3461 1084 y Fz(such)h(that)-9 1183 y Fr(v)41 1195 y Ft(n)86 1183 y Fy(;)21 b Fv(~)-49 b Fy(p)165 1195 y Ft(n)233 1183 y Fv(=)23 b Fy(p)363 1195 y Fu(0)417 1183 y Fv(+)17 b Fy(p)541 1195 y Ft(n)607 1183 y Fz(and)27 b Fv(~)-48 b Fy(\021)791 1195 y Ft(n)860 1183 y Fv(=)22 b Fy(\021)988 1195 y Fu(0)1043 1183 y Fv(+)17 b Fy(\021)1166 1195 y Ft(n)1232 1183 y Fz(con)m(v)o(er)n(ges) 22 b(to)f(the)g(solution)g(of)h(the)e(nonlinear)i(equations)f (\(2.12\)\261\(2.15\).)36 b(W)-7 b(e)21 b(will)-9 1283 y(essentially)f(ask)j (that)e Fr(v)706 1295 y Ft(n)p Fu(+1)836 1283 y Fy(;)14 b(p)915 1295 y Ft(n)p Fu(+1)1065 1283 y Fz(and)23 b Fy(\021)1251 1295 y Ft(n)p Fu(+1)1403 1283 y Fz(satisfy)e(\(3.12\)\261\(3.14\))27 b(and)c(\(3.15\))h(\261\(3.17\))h(where)e(the)f Fy(q)s Fz(,)g Fy(h)3250 1295 y Fu(1)3287 1283 y Fz(,)g Fy(h)3378 1295 y Fu(2)3416 1283 y Fz(,)g(are)g(gi)n(v)o(en)-9 1383 y(in)g(terms)h(of)h Fr(v)435 1395 y Ft(n)480 1383 y Fy(;)14 b(p)559 1395 y Ft(n)627 1383 y Fz(and)23 b Fy(\021)813 1395 y Ft(n)859 1383 y Fz(.)35 b(W)-7 b(e)23 b(will)f(see)g(ho)n(w)i(the)f(sequence)g Fr(v)2013 1395 y Ft(n)2081 1383 y Fz(can)g(be)g(interpreted)g(as)f(the)h(iteration)f (of)h(a)g(contracti)n(v)o(e)-9 1482 y(map)i(in)f(a)h(Banach)f(space)h (leading)g(to)f(a)h(\256x)o(ed)g(point)g Fr(v)1678 1494 y Ft(?)1717 1482 y Fz(,)h(and)f(corresponding)34 b Fv(~)-48 b Fy(\021)2463 1494 y Ft(?)2526 1482 y Fz(and)33 b Fv(~)-50 b Fy(p)2715 1494 y Ft(?)2778 1482 y Fz(which)25 b(will)f(be)h(the)g(solution)g(of)-9 1582 y(the)c(nonlinear)h(problem.)115 1681 y(Our)e(technique)g(is)f(mainly)g (an)g(e)o(xtension)h(of)f(the)g(one)h(used)g(in)f([Bricmont)j(et)e(al.])f(in) h(the)e(case)h(of)h(P)o(arabolic)e(PDE')-5 b(s,)21 b(and)-9 1781 y(relies)14 b(on)j(estimates)d(on)i(the)f(F)o(ourier)i(transform)g(of)f (the)g(semi\261groups)h(generated)f(by)h(the)e(linear)g(operator)i(on)g(the)e (right\261hand)-9 1881 y(side)h(of)i(\(3.12\))h(,)e(\(3.13\),)k(and)c (\(3.15\))i(,)f(\(3.16\),)i(together)d(with)g(an)g(e)o(xtension)h(of)f(the)g (space)g Fs(W)2779 1893 y Ft(\013;\015)2877 1901 y Fo(1)2909 1893 y Ft(;\015)2964 1901 y Fo(2)3017 1881 y Fz(to)g(functions)h Fv(\012)3493 1893 y Fu(+)3571 1881 y Fs(!)23 b Fr(R)p Fz(.)115 1980 y(W)-7 b(e)24 b(will)g(\256rst)g(concentrate)g(on)h(the)f(equations)h (for)h Fy(u)d Fz(and)i Fy(v)s Fz(,)h(the)e(ones)h(for)g Fy(\021)i Fz(and)e Fy(p)f Fz(being)h(simpler)f(are)g(postponed)j(to)-9 2080 y(the)21 b(ne)o(xt)g(section.)-9 2320 y Fd(3.2)102 b(Linear)26 b(Inhomogeneous)e(Equations)g(f)m(or)j Fb(u)e Fd(and)f Fb(v)-9 2477 y Fz(In)h(order)g(to)g(pro)o(v)o(e)g(e)o(xistence)e(of)i(solutions)g(to) g(\(3.12\)\261\(3.14\))k(for)c Fy(u)f Fz(and)h Fy(v)s Fz(,)h(we)e(will)f(gi)n (v)o(e)i(an)f(inte)o(gral)g(representation)g(of)-9 2577 y(the)17 b(solution)i(in)g(F)o(ourier)g(space)e(using)j(Duhamel')-5 b(s)19 b(formula.)31 b(Under)19 b(quite)f(general)g(hypotheses)i(on)f Fy(q)26 b Fv(:)d(\012)3225 2589 y Fu(+)3303 2577 y Fs(!)g Fr(R)p Fz(,)c(we)f(will)-9 2677 y(sho)n(w)26 b(that)e(the)g(F)o(ourier)i(transform)g (of)f(this)g(inte)o(gral)f(solution)h(e)o(xists)f(and)i(is)e(a)h(strong)h (solution)f(of)h(\(3.12\)\261\(3.14\).)46 b(Since)-9 2776 y(the)22 b(inte)o(gral)f(formulation)j(in)m(v)n(olv)o(es)f(rather)f(lengthy)h (formulas,)h(we)e(will)g(separate)g(their)g(e)o(xpression)h(and)g(their)f (properties)-9 2876 y(in)d(the)f(tw)o(o)i(ne)o(xt)f(lemmas.)30 b(The)19 b(\256rst)g(will)f(deal)h(with)g(the)g(part)g(of)h(Duhamel')-5 b(s)20 b(formula)g(in)m(v)n(olving)g(the)f(boundary)j(condition)-9 2976 y(\(the)29 b(solution)h(of)h(\(3.12\)\261\(3.14\))j(with)c Fy(q)37 b Fv(=)d(0)p Fz(\),)d(and)f(the)g(second)g(with)g(the)f(one)h(in)m(v) n(olving)h(the)e(function)i Fy(q)s Fz(.)56 b(Then,)32 b(we)-9 3075 y(will)22 b(sho)n(w)i(ho)n(w)g(to)f(use)g(the)g(results)g(of)h(these)e (lemmas)g(to)h(b)n(uild)h(the)f(solution.)37 b(The)23 b(follo)n(wing)h(lemma) f(is)f(a)h(more)g(precise)-9 3175 y(statement)d(than)i(the)g(discussion)h(of) g(the)e(preceding)i(section)f(for)h(the)f(case)f Fy(q)27 b Fv(=)c(0)p Fz(.)33 b(It)22 b(consists)g(essentially)e(in)i(in)m(v)o(erting)g (the)-9 3274 y(relation)e(\(3.9\).)-9 3440 y FA(Lemma)g(3.1)43 b Fm(Assume)23 b(that)f Fr(v)924 3452 y Fu(+)1004 3440 y Fv(=)i(\()p Fy(u)1173 3452 y Fu(+)1228 3440 y Fy(;)14 b(v)1305 3452 y Fu(+)1360 3440 y Fv(\))25 b Fs(2)f(W)1578 3452 y Ft(\013;\015)1676 3460 y Fo(1)1708 3452 y Ft(;\015)1763 3460 y Fo(2)1822 3440 y Fm(for)e(some)h Fy(\013)h(>)g Fv(3)p Fm(,)e Fy(\015)2434 3452 y Fu(1)2496 3440 y Fs(2)j Fv([0)p Fy(;)14 b Fv(1\))21 b Fm(and)j Fy(\015)2966 3452 y Fu(2)3027 3440 y Fs(2)h Fv([0)p Fy(;)3218 3408 y Fu(1)p 3218 3422 34 4 v 3218 3469 a(2)3261 3440 y Fv(])e Fm(with)e Fy(\015)3518 3452 y Fu(2)3580 3440 y Fs(\024)j Fy(\015)3712 3452 y Fu(1)3749 3440 y Fm(.)-9 3540 y(Assume)d(that)h Fy(u)473 3552 y Fu(+)549 3540 y Fm(and)g Fy(v)737 3552 y Fu(+)814 3540 y Fm(ar)m(e)f(e)o(ven)g(and)i(odd)f(in)g Fy(y)i Fm(r)m(espectively)-5 b(.)29 b(De\256ne)22 b Fy(\021)2305 3552 y Fu(0)p Ft(;)p Fu(0)2395 3540 y Fm(,)f Fy(u)2485 3552 y Fu(0)2543 3540 y Fm(and)i Fy(v)2732 3552 y Fu(0)2791 3540 y Fm(as)e(the)h(F)-9 b(ourier)21 b(tr)o(ansforms)h(of) 1166 3733 y Fv(^)-48 b Fy(\021)1201 3745 y Fu(0)p Ft(;)p Fu(0)1291 3733 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)k(^)-48 b Fy(\021)1637 3745 y Fu(0)p Ft(;)p Fu(+)1746 3733 y Fv(\()p Fy(k)s Fv(\))p Fy(e)1895 3699 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2224 3733 y Fm(,)1320 b Fz(\(3.18\))1084 3793 y Fn(\022)1150 3854 y Fv(^)-47 b Fy(u)1193 3866 y Fu(0)1152 3967 y Fv(^)i Fy(v)1189 3979 y Fu(0)1230 3793 y Fn(\023)1291 3910 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1596 3793 y Fn(\022)1658 3860 y Fy(P)1711 3872 y Fu(1)p Ft(;)p Fu(1)1884 3860 y Fy(P)1937 3872 y Fu(1)p Ft(;)p Fu(2)1658 3959 y Fy(P)1711 3971 y Fu(2)p Ft(;)p Fu(1)1884 3959 y Fy(P)1937 3971 y Fu(2)p Ft(;)p Fu(2)2027 3793 y Fn(\023)2102 3910 y Fv(\()p Fy(x;)14 b(k)s Fv(\))2296 3793 y Fn(\022)2363 3854 y Fv(^)-47 b Fy(u)2406 3866 y Fu(+)2365 3967 y Fv(^)i Fy(v)2402 3979 y Fu(+)2461 3793 y Fn(\023)2522 3910 y Fv(\()p Fy(k)s Fv(\))23 b Fm(,)889 b Fz(\(3.19\))-9 4138 y Fm(wher)m(e)788 4349 y Fv(^)-48 b Fy(\021)823 4361 y Fu(0)p Ft(;)p Fu(+)931 4349 y Fv(\()p Fy(k)s Fv(\))24 b(=)e Fs(\000)p Fy(ik)1302 4293 y Fv(\003)1360 4305 y Fu(+)1414 4293 y Fv(\()p Fy(k)s Fv(\))d(+)f Fs(j)p Fy(k)s Fs(j)p 1302 4330 417 4 v 1398 4406 a Fv(\003)1456 4418 y Fu(+)1511 4406 y Fv(\()p Fy(k)s Fv(\))1733 4349 y(^)-47 b Fy(u)1776 4361 y Fu(+)1831 4349 y Fv(\()p Fy(k)s Fv(\))19 b Fs(\000)f Fv(\(\003)2133 4361 y Fu(+)2188 4349 y Fv(\()p Fy(k)s Fv(\))g(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))s(^)-45 b Fy(v)2563 4361 y Fu(+)2619 4349 y Fv(\()p Fy(k)s Fv(\))24 b Fm(,)791 b Fz(\(3.20\))704 4578 y Fy(P)757 4590 y Fu(1)p Ft(;)p Fu(1)847 4578 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e Fy(e)1191 4544 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1516 4578 y Fv(+)1772 4522 y Fs(j)p Fy(k)s Fs(j)p 1609 4559 418 4 v 1609 4635 a Fv(\003)1667 4647 y Fq(\000)1723 4635 y Fv(\()p Fy(k)s Fv(\))d(+)f Fs(j)p Fy(k)s Fs(j)2051 4486 y Fn(\020)2100 4578 y Fy(e)2139 4544 y Fu(\003)2184 4552 y Fc(\000)2233 4544 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2519 4578 y Fs(\000)g Fy(e)2641 4544 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2947 4486 y Fn(\021)3034 4578 y Fm(,)704 4803 y Fy(P)757 4815 y Fu(1)p Ft(;)p Fu(2)847 4803 y Fv(\()p Fy(x;)c(k)s Fv(\))24 b(=)1302 4747 y Fs(\000)p Fy(ik)p 1162 4784 V 1162 4860 a Fv(\003)1220 4872 y Fq(\000)1276 4860 y Fv(\()p Fy(k)s Fv(\))19 b(+)f Fs(j)p Fy(k)s Fs(j)1604 4711 y Fn(\020)1653 4803 y Fy(e)1692 4769 y Fu(\003)1737 4777 y Fc(\000)1787 4769 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2072 4803 y Fs(\000)g Fy(e)2194 4769 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2501 4711 y Fn(\021)2587 4803 y Fm(,)704 5033 y Fy(P)757 5045 y Fu(2)p Ft(;)p Fu(1)847 5033 y Fv(\()p Fy(x;)c(k)s Fv(\))24 b(=)1162 4976 y Fy(i)35 b Fz(s)-5 b(i)g(g)g(n)r Fv(\()p Fy(k)s Fv(\)\003)1522 4988 y Fq(\000)1578 4976 y Fv(\()p Fy(k)s Fv(\))p 1162 5013 526 4 v 1216 5090 a(\003)1274 5102 y Fq(\000)1330 5090 y Fv(\()p Fy(k)s Fv(\))19 b(+)f Fs(j)p Fy(k)s Fs(j)1712 4940 y Fn(\020)1761 5033 y Fy(e)1800 4998 y Fu(\003)1845 5006 y Fc(\000)1895 4998 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2180 5033 y Fs(\000)g Fy(e)2302 4998 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2609 4940 y Fn(\021)2695 5033 y Fm(,)704 5262 y Fy(P)757 5274 y Fu(2)p Ft(;)p Fu(2)847 5262 y Fv(\()p Fy(x;)c(k)s Fv(\))24 b(=)e Fy(e)1191 5228 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1516 5262 y Fv(+)1706 5206 y(\003)1764 5218 y Fq(\000)1820 5206 y Fv(\()p Fy(k)s Fv(\))p 1609 5243 418 4 v 1609 5319 a(\003)1667 5331 y Fq(\000)1723 5319 y Fv(\()p Fy(k)s Fv(\))d(+)f Fs(j)p Fy(k)s Fs(j)2051 5170 y Fn(\020)2100 5262 y Fy(e)2139 5228 y Fu(\003)2184 5236 y Fc(\000)2233 5228 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2519 5262 y Fs(\000)g Fy(e)2641 5228 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2947 5170 y Fn(\021)3034 5262 y Fy(:)-9 5489 y Fm(Then)h(the)e(functions)i Fy(\021)671 5501 y Fu(0)p Ft(;)p Fu(0)761 5489 y Fm(,)g Fy(u)849 5501 y Fu(0)886 5489 y Fm(,)g Fy(v)966 5501 y Fu(0)1003 5489 y Fm(,)g Fy(@)1087 5501 y Ft(y)1127 5489 y Fy(u)1175 5501 y Fu(0)1212 5489 y Fm(,)g Fy(@)1296 5501 y Ft(y)1336 5489 y Fy(v)1376 5501 y Fu(0)1432 5489 y Fm(ar)m(e)f(bounded)i(and)g(continuous)g(in)e Fy(x;)c(y)s Fm(,)19 b(and)g Fy(u)2852 5501 y Fu(0)2907 5489 y Fm(is)f(e)o(ven)f(in)i Fy(y)s Fm(,)f(while)f Fy(v)3564 5501 y Fu(0)3602 5489 y Fy(;)d(\021)3680 5501 y Fu(0)p Ft(;)p Fu(0)-9 5588 y Fm(ar)m(e)21 b(odd)i(in)e Fy(y)s Fm(.)1859 5909 y Fz(9)p eop %%Page: 10 10 bop -9 145 a Fm(Pr)l(oof)o(.)31 b Fz(See)21 b(Appendix)i(C.)p 861 145 43 43 v -9 244 a(W)-7 b(e)23 b(no)n(w)i(proceed)f(to)g(the)f(part)h (of)g(Duhamel')-5 b(s)24 b(formula)h(containing)f Fy(q)s Fz(.)38 b(This)23 b(part)h(in)m(v)n(olv)o(es)g(auxiliary)f(functions)i Fy(N)3584 256 y Fu(1)3621 244 y Fz(,)f Fy(N)3733 256 y Fu(2)-9 344 y Fz(and)e Fy(\021)176 356 y Fu(0)p Ft(;q)266 344 y Fz(,)h(whose)g (de\256nition)g(and)f(properties)h(are)f(gi)n(v)o(en)h(in)f(the)g(ne)o(xt)g (lemma.)33 b(This)22 b(is)g(an)g(e)o(xtension)g(of)h(the)f(discussion)h(of)-9 443 y(the)e(preceding)h(section)f(to)g(the)g(nonlinear)h(case.)-9 608 y FA(Lemma)e(3.2)43 b Fm(Let)34 b Fv(^)-48 b Fy(q)36 b Fv(:)d(\012)764 620 y Fu(+)852 608 y Fs(!)g Fr(R)28 b Fm(be)g(bounded.)56 b(Assume)29 b(that)35 b Fv(^)-49 b Fy(q)32 b Fm(is)c(odd)i(in)e Fy(k)s Fm(,)j(that)k Fv(^)-49 b Fy(q)36 b Fs(2)e Fy(L)2872 578 y Fu(1)2908 608 y Fv(\(\012)3000 620 y Fu(+)3056 608 y Fy(;)14 b Fs(j)p Fy(k)s Fs(j)28 b Fy(dk)k(dx)p Fv(\))d Fm(and)h(that)-9 708 y Fz(sup)109 730 y Ft(x)p Fq(\025)p Fu(1)249 641 y Fn(R)304 708 y Fy(dk)24 b Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))p Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b Fy(<)f Fs(1)p Fm(.)31 b(De\256ne)22 b Fy(N)1551 720 y Fu(1)1609 708 y Fm(and)g Fy(N)1824 720 y Fu(2)1883 708 y Fm(and)g Fy(\021)2072 720 y Fu(0)p Ft(;q)2183 708 y Fm(as)g(the)f(F)-9 b(ourier)22 b(tr)o(ansforms)g(of)361 927 y Fv(^)337 948 y Fy(N)404 960 y Fu(1)441 948 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)809 892 y Fy(ik)p 756 929 180 4 v 756 1005 a Fv(\001\()p Fy(k)s Fv(\))960 835 y Fn(Z)1043 856 y Ft(x)1006 1024 y Fu(1)1063 948 y Fy(dx)1153 960 y Fu(0)1212 948 y Fy(e)1251 914 y Fu(\003)1296 922 y Fc(\000)1345 914 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)1587 922 y Fo(0)1619 914 y Fu(\))1656 948 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)1769 960 y Fu(0)1807 948 y Fy(;)14 b(k)s Fv(\))k(+)2086 892 y Fy(ik)p 2033 929 V 2033 1005 a Fv(\001\()p Fy(k)s Fv(\))2236 835 y Fn(Z)2319 856 y Fq(1)2283 1024 y Ft(x)2368 948 y Fy(dx)2458 960 y Fu(0)2517 948 y Fy(e)2556 914 y Fu(\003)2601 922 y Fo(+)2648 914 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)2890 922 y Fo(0)2922 914 y Fu(\))2959 948 y Fv(^)-49 b Fy(q)t Fv(\()p Fy(x)3072 960 y Fu(0)3109 948 y Fy(;)14 b(k)s Fv(\))719 1180 y Fs(\000)k Fv(\(\003)892 1192 y Fu(+)947 1180 y Fv(\()p Fy(k)s Fv(\))h(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1297 1062 y Fn(\024)1644 1123 y Fy(ik)p 1351 1160 662 4 v 1351 1236 a Fv(\001\()p Fy(k)s Fv(\)\(\003)1620 1248 y Fq(\000)1676 1236 y Fv(\()p Fy(k)s Fv(\))h(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))2022 1180 y Fy(e)2061 1145 y Fu(\003)2106 1153 y Fc(\000)2155 1145 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2441 1180 y Fs(\000)g Fy(i)35 b Fz(s)-5 b(i)g(g)g(n)q Fv(\()p Fy(k)s Fv(\))p Fy(e)2864 1145 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))3171 1062 y Fn(\025)3250 1180 y Fy(\034)9 b Fv(\()p Fy(k)s Fv(\))24 b Fm(,)115 b Fz(\(3.21\))361 1389 y Fv(^)337 1410 y Fy(N)404 1422 y Fu(2)441 1410 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e Fs(\000)821 1354 y Fv(\003)879 1366 y Fq(\000)935 1354 y Fv(\()p Fy(k)s Fv(\))p 821 1391 225 4 v 843 1467 a(\001\()p Fy(k)s Fv(\))1069 1297 y Fn(Z)1152 1318 y Ft(x)1115 1486 y Fu(1)1172 1410 y Fy(dx)1262 1422 y Fu(0)1321 1410 y Fy(e)1360 1376 y Fu(\003)1405 1384 y Fc(\000)1454 1376 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)1696 1384 y Fo(0)1729 1376 y Fu(\))1765 1410 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)1878 1422 y Fu(0)1916 1410 y Fy(;)14 b(k)s Fv(\))k Fs(\000)2142 1354 y Fv(\003)2200 1366 y Fu(+)2255 1354 y Fv(\()p Fy(k)s Fv(\))p 2142 1391 224 4 v 2164 1467 a(\001\()p Fy(k)s Fv(\))2389 1297 y Fn(Z)2472 1318 y Fq(1)2435 1486 y Ft(x)2521 1410 y Fy(dx)2611 1422 y Fu(0)2670 1410 y Fy(e)2709 1376 y Fu(\003)2754 1384 y Fo(+)2801 1376 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)3043 1384 y Fo(0)3075 1376 y Fu(\))3111 1410 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)3224 1422 y Fu(0)3262 1410 y Fy(;)14 b(k)s Fv(\))719 1641 y(+)k(\(\003)892 1653 y Fu(+)947 1641 y Fv(\()p Fy(k)s Fv(\))h(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1297 1524 y Fn(\024)1569 1585 y Fv(\003)1627 1597 y Fq(\000)1683 1585 y Fv(\()p Fy(k)s Fv(\))p 1351 1622 662 4 v 1351 1698 a(\001\()p Fy(k)s Fv(\)\(\003)1620 1710 y Fq(\000)1676 1698 y Fv(\()p Fy(k)s Fv(\))h(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))2022 1641 y Fy(e)2061 1607 y Fu(\003)2106 1615 y Fc(\000)2155 1607 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2441 1641 y Fv(+)g Fy(e)2563 1607 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2870 1524 y Fn(\025)2948 1641 y Fy(\034)9 b Fv(\()p Fy(k)s Fv(\))25 b Fm(,)416 b Fz(\(3.22\))316 1831 y Fv(^)-48 b Fy(\021)351 1843 y Fu(0)p Ft(;q)441 1831 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)k(^)-48 b Fy(\021)787 1843 y Ft(q)r(;)p Fu(+)895 1831 y Fv(\()p Fy(k)s Fv(\))p Fy(e)1044 1797 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1374 1831 y Fm(,)2170 b Fz(\(3.23\))-9 2012 y Fm(wher)m(e)1284 2221 y Fy(\034)9 b Fv(\()p Fy(k)s Fv(\))25 b(=)1551 2108 y Fn(Z)1634 2128 y Fq(1)1597 2296 y Fu(1)1683 2221 y Fy(dx)1773 2233 y Fu(0)1832 2221 y Fy(e)1871 2187 y Fu(\003)1916 2195 y Fo(+)1963 2187 y Fu(\()p Ft(k)q Fu(\)\(1)p Fq(\000)p Ft(x)2200 2195 y Fo(0)2233 2187 y Fu(\))2269 2221 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2382 2233 y Fu(0)2420 2221 y Fy(;)14 b(k)s Fv(\))23 b Fm(,)986 b Fz(\(3.24\))1188 2392 y Fv(^)-48 b Fy(\021)1223 2404 y Ft(q)r(;)p Fu(+)1330 2392 y Fv(\()p Fy(k)s Fv(\))24 b(=)e Fs(\000)p Fv(\(\003)1706 2404 y Fu(+)1761 2392 y Fv(\()p Fy(k)s Fv(\))d(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))p Fy(\034)9 b Fv(\()p Fy(k)s Fv(\))24 b Fy(:)1266 b Fz(\(3.25\))-9 2573 y Fm(Then)27 b(the)g(functions)g Fy(N)723 2585 y Fu(1)760 2573 y Fm(,)h Fy(N)876 2585 y Fu(2)913 2573 y Fm(,)g Fy(@)1006 2585 y Ft(y)1046 2573 y Fy(N)1113 2585 y Fu(1)1150 2573 y Fy(;)14 b(@)1231 2585 y Ft(y)1271 2573 y Fy(N)1338 2585 y Fu(2)1402 2573 y Fm(and)28 b Fy(\021)1597 2585 y Fu(0)p Ft(;q)1713 2573 y Fm(ar)m(e)f(bounded)i(and)f(continuous)h(in)e Fy(x;)14 b(y)s Fm(,)28 b(and)g(the)f(function)g Fy(N)3650 2585 y Fu(1)3713 2573 y Fm(is)-9 2673 y(e)o(ven)20 b(in)i Fy(y)s Fm(,)f(while)f(the)h (functions)h Fy(N)1070 2685 y Fu(2)1107 2673 y Fy(;)14 b(\021)1185 2685 y Fu(0)p Ft(;q)1296 2673 y Fm(ar)m(e)21 b(odd)i(in)e Fy(y)s Fm(.)-9 2837 y(Pr)l(oof)o(.)31 b Fz(See)21 b(Appendix)i(C.)p 861 2837 43 43 v -9 2937 a(W)-7 b(e)31 b(ha)n(v)o(e)h(no)n(w)h(all)d(we)h (need)h(to)g(write)f(do)n(wn)i(Duhamel')-5 b(s)33 b(formula)f(and)g(sho)n(w)h (that)e(it)f(pro)o(vides)j(a)e(strong)i(solution)f(of)-9 3037 y(\(3.12\)\261\(3.14\).)-9 3201 y FA(Pr)n(oposition)22 b(3.3)43 b Fm(Let)20 b(the)g(conditions)h(of)f(Lemmas)h(3.1)h(and)f(3.2)h(be)e (satis\256ed.)31 b(Assume)21 b(that)f Fy(q)j Fm(is)d(continuous)i(in)e Fy(x;)14 b(y)s Fm(,)21 b(and)-9 3301 y(de\256ne)h(the)f(functions)h Fy(u)e Fm(and)j Fy(v)h Fm(as)e(the)f(F)-9 b(ourier)22 b(tr)o(ansforms)g(of) 1347 3482 y Fv(^)-47 b Fy(u)o Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)k(^)-47 b Fy(u)1743 3494 y Fu(0)1780 3482 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(N)2143 3494 y Fu(1)2180 3482 y Fv(\()p Fy(x;)c(k)s Fv(\))24 b Fm(,)1146 b Fz(\(3.26\))1349 3607 y Fv(^)-45 b Fy(v)s Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)i(^)-45 b Fy(v)1735 3619 y Fu(0)1772 3607 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)g Fy(N)2136 3619 y Fu(2)2172 3607 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)1152 b Fz(\(3.27\))-9 3788 y Fm(Then)25 b(the)f(functions)h Fy(u)p Fm(,)f Fy(v)s Fm(,)i Fy(@)876 3800 y Ft(y)916 3788 y Fy(u)e Fm(and)h Fy(@)1183 3800 y Ft(y)1223 3788 y Fy(v)j Fm(ar)m(e)c(bounded)j(and)f(continuous)g(in)e Fy(x;)14 b(y)s Fm(.)40 b(The)25 b(function)g Fy(u)f Fm(is)g(e)o(ven)f(in)i Fy(y)i Fm(and)e(the)-9 3887 y(functions)c Fy(v)s(;)14 b(\021)444 3899 y Fu(0)500 3887 y Fv(+)k Fy(\021)624 3899 y Ft(q)683 3887 y Fm(ar)m(e)j(odd.)32 b(The)m(y)22 b(ar)m(e)f(str)l(ong)i(solutions)f(of)g (equations)g(\(3.12\))i(\261)d(\(3.14\))j(with)d Fy(\021)3036 3899 y Fu(0)3096 3887 y Fv(=)i Fy(\021)3225 3899 y Fu(0)p Ft(;)p Fu(0)3334 3887 y Fv(+)18 b Fy(\021)3458 3899 y Fu(0)p Ft(;q)3547 3887 y Fm(.)-9 4052 y(Pr)l(oof)o(.)31 b Fz(See)21 b(Appendix)i(C.)p 861 4052 V -9 4152 a(The)f(question)h(of)g(con)m(v)o(er)n(gence)g(of)g(the)f (solution)h(to)f Fv(0)g Fz(as)g Fy(y)27 b Fs(!)d(1)e Fz(or)h Fy(x)i Fs(!)f(1)e Fz(will)g(be)g(a)g(simple)f(consequence)j(of)f(bounds)-9 4251 y(that)17 b(we)h(will)f(pro)o(v)o(e)i(in)f(Section)g(3.4.)31 b(W)m(ithout)19 b(making)f(additional)g(assumptions)h(on)g(the)f(function)i Fy(q)s Fz(,)e(we)h(can)f(e)o(xtract)e(more)-9 4351 y(information)k(on)f Fv(\()p Fy(u)594 4363 y Fu(0)631 4351 y Fy(;)14 b(v)708 4363 y Fu(0)746 4351 y Fv(\))19 b Fz(than)g(what)g(we)g(wrote)g(in)g(Lemma)g(3.1)h (by)g(e)o(xploiting)f(the)g(f)o(act)f(that)g Fr(v)2895 4363 y Fu(+)2974 4351 y Fs(2)23 b(W)3134 4363 y Ft(\013;\015)3232 4371 y Fo(1)3264 4363 y Ft(;\015)3319 4371 y Fo(2)3356 4351 y Fz(.)30 b(As)19 b(we)g(will)-9 4461 y(see,)g(the)g(contrib)n(ution)h(of)g (\()838 4440 y Fv(^)813 4461 y Fy(N)880 4473 y Fu(1)917 4461 y Fy(;)978 4440 y Fv(^)954 4461 y Fy(N)1021 4473 y Fu(2)1058 4461 y Fv(\))g Fz(to)f Fv(\()p Fy(u)1275 4473 y Fu(0)1312 4461 y Fy(;)14 b(v)1389 4473 y Fu(0)1426 4461 y Fv(\))20 b Fz(will)e(be)h(smaller) f(than)i(the)f(one)g(of)25 b Fv(^)-47 b Fy(u)2565 4473 y Fu(0)2621 4461 y Fz(and)23 b Fv(^)-45 b Fy(v)2803 4473 y Fu(0)2840 4461 y Fz(.)30 b(This)19 b(will)g(guide)h(us)f(later)f(on)-9 4561 y(for)23 b(de\256ning)h(the)e(Banach)h(space)f(in)h(which)g(we)f(will)g(pro)o (v)o(e)h(that)f(the)h(nonlinear)g(problem)h(has)e(a)h(solution.)35 b(The)23 b(follo)n(wing)-9 4661 y(lemma)d(gi)n(v)o(es)h(these)g (informations.)-9 4825 y FA(Lemma)f(3.4)43 b Fm(Let)24 b Fr(v)621 4837 y Fu(+)703 4825 y Fv(=)i(\()p Fy(u)874 4837 y Fu(+)929 4825 y Fy(;)14 b(v)1006 4837 y Fu(+)1062 4825 y Fv(\))27 b Fs(2)g(W)1285 4837 y Ft(\013;\015)1383 4845 y Fo(1)1415 4837 y Ft(;\015)1470 4845 y Fo(2)1530 4825 y Fm(for)e(some)f Fy(\013)k(>)e Fv(3)p Fm(,)f Fy(\015)2155 4837 y Fu(1)2219 4825 y Fs(2)i Fv([0)p Fy(;)14 b Fv(1\))23 b Fm(and)j Fy(\015)2695 4837 y Fu(2)2759 4825 y Fs(2)h Fv([0)p Fy(;)2953 4793 y Fu(1)p 2953 4807 34 4 v 2953 4854 a(2)2996 4825 y Fv(])d Fm(with)f Fy(\015)3256 4837 y Fu(2)3321 4825 y Fs(\024)j Fy(\015)3455 4837 y Fu(1)3492 4825 y Fm(.)40 b(Let)29 b Fv(^)-48 b Fy(u)3733 4837 y Fu(0)-9 4925 y Fm(and)25 b Fv(^)-45 b Fy(v)179 4937 y Fu(0)238 4925 y Fm(be)21 b(de\256ned)i(by)e(\(3.19\).)34 b(Then)22 b(ther)m(e)f(e)n(xists)f (a)i(constant)g Fy(C)27 b Fm(suc)o(h)22 b(that)f(for)h(all)f Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)2933 4937 y Fu(+)2988 4925 y Fm(,)1153 5145 y Fs(j)5 b Fv(^)-47 b Fy(u)1224 5157 y Fu(0)1261 5145 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fs(k)p Fr(v)1768 5157 y Fu(+)1823 5145 y Fs(k)1865 5157 y Ft(\013;\015)1963 5165 y Fo(1)1994 5157 y Ft(;\015)2049 5165 y Fo(2)2328 5089 y Fv(1)p 2117 5126 464 4 v 2117 5202 a(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2384 5143 y(p)p 2453 5143 48 4 v 59 x Fy(x)p Fv(\))2532 5179 y Ft(\013)2613 5145 y Fm(,)931 b Fz(\(3.28\))1160 5366 y Fs(j)s Fv(^)-45 b Fy(v)1223 5378 y Fu(0)1261 5366 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fs(k)p Fr(v)1768 5378 y Fu(+)1823 5366 y Fs(k)1865 5378 y Ft(\013;\015)1963 5386 y Fo(1)1994 5378 y Ft(;\015)2049 5386 y Fo(2)2154 5310 y Fv(1)p 2117 5347 117 4 v 2117 5364 a Fs(p)p 2186 5364 48 4 v 59 x Fy(x)2464 5310 y Fv(1)p 2253 5347 464 4 v 2253 5423 a(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2520 5364 y(p)p 2590 5364 48 4 v 2590 5423 a Fy(x)p Fv(\))2669 5399 y Ft(\013)2750 5366 y Fm(,)794 b Fz(\(3.29\))1068 5623 y Fs(j)p Fy(@)1135 5635 y Ft(k)1181 5623 y Fv(^)-47 b Fy(u)1224 5635 y Fu(0)1261 5623 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fs(k)p Fr(v)1768 5635 y Fu(+)1823 5623 y Fs(k)1865 5635 y Ft(\013;\015)1963 5643 y Fo(1)1994 5635 y Ft(;\015)2049 5643 y Fo(2)2244 5566 y Fy(x)2301 5505 y Fo(1)p Fc(\000)p Fk(\015)2405 5517 y Fo(1)p 2302 5523 137 3 v 2356 5556 a(2)p 2117 5603 464 4 v 2117 5680 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2384 5620 y(p)p 2453 5620 48 4 v 60 x Fy(x)p Fv(\))2532 5656 y Ft(\013)2604 5506 y Fn(\022)2675 5566 y Fv(1)g(+)g Fs(j)p Fy(k)s Fs(j)p 2675 5603 235 4 v 2746 5680 a(j)p Fy(k)s Fs(j)2920 5506 y Fn(\023)2981 5523 y Ft(\015)3016 5531 y Fo(1)3089 5623 y Fm(,)455 b Fz(\(3.30\))1838 5909 y(10)p eop %%Page: 11 11 bop 1075 184 a Fs(j)p Fy(@)1142 196 y Ft(k)1186 184 y Fv(^)-45 b Fy(v)1223 196 y Fu(0)1261 184 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fs(k)p Fr(v)1768 196 y Fu(+)1823 184 y Fs(k)1865 196 y Ft(\013;\015)1963 204 y Fo(1)1994 196 y Ft(;\015)2049 204 y Fo(2)2370 128 y Fv(1)p 2117 165 549 4 v 2117 241 a(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2384 182 y(p)p 2453 182 48 4 v 59 x Fy(x)p Fv(\))2532 217 y Ft(\013)p Fq(\000)p Fu(1)2853 68 y Fs(p)p 2922 68 V 60 x Fy(x)p 2685 165 452 4 v 2685 241 a Fs(j)p Fy(k)s(x)p Fs(j)2824 217 y Ft(\015)2859 225 y Fo(2)2914 241 y Fv(+)g Fs(j)p Fy(k)s(x)p Fs(j)3170 184 y Fm(,)374 b Fz(\(3.31\))570 345 y Fn(\014)570 395 y(\014)570 445 y(\014)603 441 y Fv(^)-48 b Fy(u)645 453 y Fu(0)682 441 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b Fs(\000)f Fy(e)1017 406 y Fu(\003)1062 414 y Fc(\000)1112 406 y Fu(\()p Ft(k)q Fu(\))p Ft(x)1242 441 y Fy(u)1290 453 y Fu(+)1345 441 y Fv(\(0\))1451 345 y Fn(\014)1451 395 y(\014)1451 445 y(\014)1501 441 y Fs(\024)23 b Fy(C)28 b Fs(k)p Fr(v)1768 453 y Fu(+)1823 441 y Fs(k)1865 453 y Ft(\013;\015)1963 461 y Fo(1)1994 453 y Ft(;\015)2049 461 y Fo(2)2244 384 y Fy(x)2301 323 y Fk(\015)2332 335 y Fo(1)2365 323 y Fc(\000)p Fo(1)p 2302 341 137 3 v 2356 374 a(2)p 2117 421 464 4 v 2117 498 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2384 438 y(p)p 2453 438 48 4 v 60 x Fy(x)p Fv(\))2532 474 y Ft(\013)2613 441 y Fy(:)929 b Fz(\(3.32\))-9 667 y Fm(Pr)l(oof)o(.)31 b Fz(See)21 b(Appendix)i(C.)p 861 667 43 43 v -9 908 a Fd(3.3)102 b(Linear)26 b(Inhomogeneous)e(Equations)g(f)m(or)j Fb(p)e Fd(and)g Fb(\021)-9 1065 y Fz(This)k(section)h(is)f(v)o(ery)h(similar)f(to)h(the)f (pre)n(vious)j(one.)56 b(W)-7 b(e)30 b(will)f(gi)n(v)o(e)h(an)g(inte)o(gral)f (representation)h(in)f(F)o(ourier)i(space)e(of)-9 1164 y(the)j(solution)h(to) g(the)f(equations)i(\(3.15\)\261\(3.17\))j(for)d Fy(\021)i Fz(and)d Fy(p)p Fz(.)65 b(W)-7 b(e)32 b(will)g(sho)n(w)i(that)e(under)i (quite)e(general)h(hypotheses)-9 1264 y(on)e Fy(h)154 1276 y Fu(1)221 1264 y Fz(and)g Fy(h)422 1276 y Fu(2)459 1264 y Fz(,)h(the)e(F)o(ourier)h(transform)h(of)f(the)f(inte)o(gral)f(solution)i(is) f(a)g(strong)i(solution)f(of)g(\(3.15\)\261\(3.14\).)62 b(Since)30 b(the)-9 1364 y(boundary)24 b(condition)e(\(3.17\))i(is)d Fy(\021)s Fv(\(1)p Fy(;)14 b(y)s Fv(\))24 b(=)f(0)p Fz(,)e(the)g(equi)n(v)n(alent)g(of) i(Lemma)e(3.1)h(is)g Fy(\021)2524 1376 y Fu(0)2584 1364 y Fv(=)h(0)p Fy(;)14 b(p)2793 1376 y Fu(0)2853 1364 y Fv(=)23 b(0)p Fz(.)31 b(The)22 b(follo)n(wing)h(lemma)-9 1463 y(corresponds)g(to)f(Lemma)e(3.2)j (of)f(the)f(pre)n(vious)i(section.)-9 1629 y FA(Lemma)d(3.5)43 b Fm(Let)580 1608 y Fv(^)579 1629 y Fy(h)627 1641 y Fu(1)664 1629 y Fy(;)702 1608 y Fv(^)701 1629 y Fy(h)749 1641 y Fu(2)825 1629 y Fv(:)38 b(\012)946 1641 y Fu(+)1040 1629 y Fs(!)g Fr(R)32 b Fm(be)h(bounded,)38 b(and)c(assume)g(that)2336 1608 y Fv(^)2335 1629 y Fy(h)2383 1641 y Ft(i)2449 1629 y Fs(2)k Fy(L)2599 1599 y Fu(1)2636 1629 y Fv(\(\012)2728 1641 y Fu(+)2783 1629 y Fy(;)14 b Fv(\(1)25 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))3132 1599 y Ft(\026)3177 1629 y Fy(dk)35 b(dx)p Fv(\))p Fm(,)i(and)d(that)-9 1738 y Fz(sup)109 1761 y Ft(x)p Fq(\025)p Fu(1)249 1671 y Fn(R)304 1738 y Fy(dk)24 b Fv(\(1)16 b(+)h Fs(j)p Fy(k)s Fs(j)p Fv(\))710 1708 y Ft(\026)755 1738 y Fs(j)779 1716 y Fv(^)778 1738 y Fy(h)826 1750 y Ft(i)853 1738 y Fv(\()p Fy(x;)d(k)s Fv(\))p Fs(j)24 b Fy(<)f Fs(1)d Fm(for)h(some)g Fy(\026)i(>)g Fv(0)p Fm(.)30 b(Assume)21 b(furthermor)m(e)g(that)2716 1716 y Fv(^)2715 1738 y Fy(h)2763 1750 y Fu(1)2820 1738 y Fm(is)f(e)o(ven)g(in)h Fy(k)i Fm(and)3374 1716 y Fv(^)3373 1738 y Fy(h)3421 1750 y Fu(2)3479 1738 y Fm(is)d(odd)i(in)-9 1838 y Fy(k)s Fm(.)30 b(De\256ne)22 b Fy(q)368 1850 y Fu(1)406 1838 y Fm(,)f Fy(q)485 1850 y Fu(2)522 1838 y Fm(,)h Fy(q)602 1850 y Fu(1)p Ft(;p)714 1838 y Fm(and)h Fy(q)900 1850 y Fu(2)p Ft(;p)1013 1838 y Fm(as)e(the)g(F)-9 b(ourier)22 b(tr)o(ansforms)g(of)526 2073 y Fv(^)-48 b Fy(q)557 2085 y Fu(1)594 2073 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)900 1960 y Fn(Z)983 1981 y Fq(1)946 2149 y Fu(1)1031 2073 y Fy(dx)1121 2085 y Fu(0)1194 1956 y Fn(\024)1238 2073 y Fs(\000)1313 2017 y Fv(1)p 1313 2054 42 4 v 1313 2130 a(2)1364 2073 y Fy(i)35 b Fz(s)-5 b(i)g(g)g(n)q Fv(\()p Fy(k)s Fv(\))1679 1981 y Fn(\020)1729 2073 y Fy(e)1768 2039 y Fq(\000j)p Ft(x)p Fq(\000)p Ft(x)1968 2047 y Fo(0)1999 2039 y Fq(jj)p Ft(k)q Fq(j)2117 2073 y Fs(\000)19 b Fy(e)2240 2039 y Fq(\000)p Fu(\()p Ft(x)p Fu(+)p Ft(x)2445 2047 y Fo(0)2476 2039 y Fq(\000)p Fu(2\))p Fq(j)p Ft(k)q Fq(j)2667 1981 y Fn(\021)2716 1956 y(\025)2796 2051 y Fv(^)2795 2073 y Fy(h)2843 2085 y Fu(1)2880 2073 y Fv(\()p Fy(x)2959 2085 y Fu(0)2997 2073 y Fy(;)14 b(k)s Fv(\))24 b Fm(,)526 2306 y Fv(^)-48 b Fy(q)557 2318 y Fu(2)594 2306 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)900 2193 y Fn(Z)983 2213 y Fq(1)946 2381 y Fu(1)1031 2306 y Fy(dx)1121 2318 y Fu(0)1194 2189 y Fn(\024)1248 2249 y Fv(1)p 1248 2286 V 1248 2363 a(2)1313 2213 y Fn(\020)1363 2306 y Fy(e)1402 2271 y Fq(\000j)p Ft(x)p Fq(\000)p Ft(x)1602 2279 y Fo(0)1633 2271 y Fq(jj)p Ft(k)q Fq(j)1747 2306 y Fz(s)-5 b(i)g(g)g(n)q Fv(\()p Fy(x)19 b Fs(\000)f Fy(x)2102 2318 y Fu(0)2140 2306 y Fv(\))h(+)f Fy(e)2313 2271 y Fq(\000)p Fu(\()p Ft(x)p Fu(+)p Ft(x)2518 2279 y Fo(0)2549 2271 y Fq(\000)p Fu(2\))p Fq(j)p Ft(k)q Fq(j)2740 2213 y Fn(\021)2790 2189 y(\025)2870 2284 y Fv(^)2869 2306 y Fy(h)2917 2318 y Fu(2)2954 2306 y Fv(\()p Fy(x)3033 2318 y Fu(0)3071 2306 y Fy(;)c(k)s Fv(\))23 b Fm(,)472 2538 y Fv(^)-48 b Fy(q)503 2550 y Fu(1)p Ft(;p)594 2538 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)900 2425 y Fn(Z)983 2446 y Fq(1)946 2614 y Fu(1)1031 2538 y Fy(dx)1121 2550 y Fu(0)1194 2421 y Fn(\024)1238 2538 y Fs(\000)1313 2482 y Fv(1)p 1313 2519 V 1313 2595 a(2)1378 2446 y Fn(\020)1428 2538 y Fy(e)1467 2504 y Fq(\000j)p Ft(x)p Fq(\000)p Ft(x)1667 2512 y Fo(0)1697 2504 y Fq(jj)p Ft(k)q Fq(j)1811 2538 y Fz(s)-5 b(i)g(g)g(n)r Fv(\()p Fy(x)19 b Fs(\000)f Fy(x)2167 2550 y Fu(0)2205 2538 y Fv(\))g Fs(\000)g Fy(e)2377 2504 y Fq(\000)p Fu(\()p Ft(x)p Fu(+)p Ft(x)2582 2512 y Fo(0)2614 2504 y Fq(\000)p Fu(2\))p Fq(j)p Ft(k)q Fq(j)2805 2446 y Fn(\021)2854 2421 y(\025)2934 2516 y Fv(^)2933 2538 y Fy(h)2981 2550 y Fu(1)3018 2538 y Fv(\()p Fy(x)3097 2550 y Fu(0)3135 2538 y Fy(;)c(k)s Fv(\))23 b Fm(,)472 2771 y Fv(^)-48 b Fy(q)503 2783 y Fu(2)p Ft(;p)594 2771 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)900 2658 y Fn(Z)983 2678 y Fq(1)946 2846 y Fu(1)1031 2771 y Fy(dx)1121 2783 y Fu(0)1194 2653 y Fn(\024)1238 2771 y Fs(\000)1313 2714 y Fv(1)p 1313 2751 V 1313 2827 a(2)1364 2771 y Fy(i)35 b Fz(s)-5 b(i)g(g)g(n)q Fv(\()p Fy(k)s Fv(\))1679 2678 y Fn(\020)1729 2771 y Fy(e)1768 2736 y Fq(\000j)p Ft(x)p Fq(\000)p Ft(x)1968 2744 y Fo(0)1999 2736 y Fq(jj)p Ft(k)q Fq(j)2117 2771 y Fv(+)19 b Fy(e)2240 2736 y Fq(\000)p Fu(\()p Ft(x)p Fu(+)p Ft(x)2445 2744 y Fo(0)2476 2736 y Fq(\000)p Fu(2\))p Fq(j)p Ft(k)q Fq(j)2667 2678 y Fn(\021)2716 2653 y(\025)2796 2749 y Fv(^)2795 2771 y Fy(h)2843 2783 y Fu(2)2880 2771 y Fv(\()p Fy(x)2959 2783 y Fu(0)2997 2771 y Fy(;)14 b(k)s Fv(\))24 b Fy(:)-9 2998 y Fm(Then)e(the)g(functions)g Fy(q)678 3010 y Ft(i)705 2998 y Fm(,)g Fy(q)785 3010 y Ft(i;p)867 2998 y Fm(,)g Fy(@)954 3010 y Ft(y)994 2998 y Fy(q)1031 3010 y Ft(i)1080 2998 y Fm(and)h Fy(@)1273 3010 y Ft(y)1313 2998 y Fy(q)1350 3010 y Ft(i;p)1454 2998 y Fm(ar)m(e)e(bounded)k(and)e(continuous)g(in)f Fy(x;)14 b(y)s Fm(,)22 b(and)h(the)f(functions)g Fy(q)3347 3010 y Ft(i)3396 2998 y Fm(ar)m(e)g(e)o(ven)f(in)-9 3098 y Fy(y)i Fm(while)e(the)g(functions)h Fy(q)756 3110 y Ft(i;p)859 3098 y Fm(ar)m(e)f(odd)i(in)e Fy(y)s Fm(.)-9 3264 y(Pr)l(oof)o(.)31 b Fz(See)21 b(Appendix)i(C.)p 861 3264 43 43 v -9 3364 a(W)-7 b(e)21 b(can)g(write)g(do)n(wn)i(Duhamel')-5 b(s)22 b(formula)g(and)g(sho)n (w)h(that)e(it)f(pro)o(vides)j(a)e(strong)h(solution)g(to)g (\(3.15\)\261\(3.17\).)-9 3530 y FA(Pr)n(oposition)g(3.6)43 b Fm(Assume)25 b(that)f(the)g(hypotheses)h(of)f(Lemma)h(3.5)g(ar)m(e)f (satis\256ed,)i(and)f(that)f Fy(h)2851 3542 y Fu(1)2912 3530 y Fm(and)i Fy(h)3112 3542 y Fu(2)3173 3530 y Fm(ar)m(e)e(continuous)i(in)-9 3629 y Fy(x;)14 b(y)s Fm(,)21 b(and)i(de\256ne)f(the)f(functions)h Fy(\021)i Fm(and)f Fy(p)e Fm(as)h(the)f(F)-9 b(ourier)21 b(tr)o(ansforms)i (of)1315 3812 y Fv(^)-48 b Fy(\021)s Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)29 b(^)-49 b Fy(q)1695 3824 y Fu(1)1733 3812 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)24 b(^)-48 b Fy(q)2066 3824 y Fu(2)2103 3812 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)1223 b Fz(\(3.33\))1318 3937 y Fv(^)-49 b Fy(p)p Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)29 b(^)-49 b Fy(q)1695 3949 y Fu(1)p Ft(;p)1787 3937 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)24 b(^)-48 b Fy(q)2120 3949 y Fu(2)p Ft(;p)2211 3937 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)1113 b Fz(\(3.34\))-9 4119 y Fm(Then)24 b(the)e(functions)i Fy(\021)s(;)14 b(p;)g(@)849 4131 y Ft(y)889 4119 y Fy(\021)26 b Fm(and)f Fy(@)1151 4131 y Ft(y)1191 4119 y Fy(p)e Fm(ar)m(e)g(bounded,)j (continuous)f(in)e Fy(x;)14 b(y)s Fm(,)24 b(e)o(ven)f(and)h(odd)h(in)e Fy(y)j Fm(r)m(espectively)-5 b(.)34 b(The)m(y)24 b(ar)m(e)-9 4219 y(str)l(ong)e(solutions)g(of)g(\(3.15\)\261\(3.17\).)-9 4385 y(Pr)l(oof)o(.)31 b Fz(See)21 b(Appendix)i(C.)p 861 4385 V -9 4625 a Fd(3.4)102 b(Linear)26 b(Inhomogeneous)e(Equations,)h(Additional) g(Pr)n(operties)-9 4782 y Fz(In)19 b(this)g(section,)g(we)g(pro)o(v)o(e)h (that)e(the)h(inte)o(gral)f(solutions)i(of)g(\(3.12\)\261\(3.14\))k(and)19 b(\(3.15\)\261\(3.17\))24 b(of)c(the)e(Sections)h(3.2)h(and)g(3.3)-9 4882 y(obe)o(y)h(certain)g(estimates)e(when)j Fy(q)1007 4894 y Fu(0)1044 4882 y Fz(,)f Fy(h)1134 4894 y Fu(1)1171 4882 y Fz(,)h Fy(h)1262 4894 y Fu(2)1320 4882 y Fz(and)f Fy(q)j Fz(satisfy)d (additional)g(properties.)32 b(The)21 b(bounds)i(on)f(the)f(F)o(ourier)h (transform)-9 4982 y(of)i Fy(q)122 4994 y Fu(0)159 4982 y Fz(,)g Fy(h)252 4994 y Fu(1)312 4982 y Fz(and)g Fy(h)506 4994 y Fu(2)566 4982 y Fz(are)f(suggested)i(by)f(the)f(bounds)j(obtained)e(when)g (substituting)g Fy(u;)14 b(v)26 b Fz(by)e(the)f(bounds)j(on)e Fy(u)3310 4994 y Fu(0)3347 4982 y Fy(;)14 b(v)3424 4994 y Fu(0)3484 4982 y Fz(in)24 b Fy(q)3611 4994 y Fu(0)3648 4982 y Fy(;)14 b(h)3733 4994 y Fu(1)-9 5081 y Fz(and)23 b Fy(h)184 5093 y Fu(2)221 5081 y Fz(,)g(see)f(Appendix)j(F.)35 b(W)-7 b(e)23 b(will)f(pro)o(v)o(e)i(that)e(when)h Fy(h)1764 5093 y Fu(1)1824 5081 y Fz(and)g Fy(h)2017 5093 y Fu(2)2077 5081 y Fz(satisfy)g(these)f (bounds,)k(then)d Fy(q)3015 5093 y Fu(1)3052 5081 y Fy(;)14 b(q)3126 5093 y Fu(2)3186 5081 y Fz(satisfy)22 b(additional)-9 5181 y(properties.)41 b(This)24 b(will)g(suggest)h(appropriate)g(bounds)i(on) f Fy(q)k Fv(=)d Fy(q)1990 5193 y Fu(0)2047 5181 y Fv(+)20 b Fy(\021)31 b Fv(=)c Fy(q)2333 5193 y Fu(0)2390 5181 y Fv(+)20 b Fy(q)2512 5193 y Fu(1)2569 5181 y Fv(+)g Fy(q)2691 5193 y Fu(2)2728 5181 y Fz(.)41 b(All)24 b(these)g(bounds)j(will)c(ensure)-9 5280 y(that)e Fy(u;)14 b(v)s(;)g(p)22 b Fz(and)h Fy(\021)i Fz(tend)d(to)h Fv(0)e Fz(as)h Fy(x)j Fs(!)g(1)d Fz(or)h Fy(y)k Fs(!)d(1)p Fz(.)34 b(Finally)-6 b(,)23 b(if)f Fy(q)2074 5292 y Fu(0)2111 5280 y Fy(;)14 b(h)2196 5292 y Fu(1)2256 5280 y Fz(and)22 b Fy(h)2448 5292 y Fu(2)2508 5280 y Fz(satisfy)g(the)g(conditions)h (belo)n(w)-6 b(,)24 b(then)e(the)-9 5380 y(assumptions)g(of)g(Lemmas)f(3.2)h (and)g(3.5)h(are)e(satis\256ed.)115 5480 y(W)-7 b(e)21 b(be)o(gin)h(with)g (the)f(conditions)h(on)g Fy(h)1283 5492 y Fu(1)1341 5480 y Fz(and)g Fy(h)1533 5492 y Fu(2)1570 5480 y Fz(.)1838 5909 y(11)p eop %%Page: 12 12 bop -9 145 a FA(Condition)22 b(3.7)43 b Fm(Let)32 b Fy(\013)39 b(>)f Fv(3)p Fm(,)d Fy(\015)44 b Fs(2)38 b Fv([0)p Fy(;)14 b Fv(1\))32 b Fm(and)i Fy(h)1553 157 y Fu(1)1590 145 y Fy(;)14 b(h)1675 157 y Fu(2)1751 145 y Fv(:)38 b(\012)1872 157 y Fu(+)1966 145 y Fs(!)h Fr(R)p Fm(.)64 b(Ther)m(e)33 b(e)n(xists)f(a)h(constant)g Fy(H)40 b Fm(suc)o(h)33 b(that)g(for)f(all)-9 244 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(2)h Fv(\012)347 256 y Fu(+)402 244 y Fm(,)d(the)g(functions)h Fy(h)950 256 y Fu(1)1009 244 y Fm(and)g Fy(h)1205 256 y Fu(2)1264 244 y Fm(satisfy)1349 482 y Fs(j)1373 460 y Fv(^)1372 482 y Fy(h)1420 494 y Fu(1)1457 482 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(H)2065 426 y(x)2112 396 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)p 1892 463 549 4 v 1892 539 a Fv(\(1)d(+)f Fs(j)p Fy(k)s Fs(j)2160 479 y(p)p 2229 479 48 4 v 60 x Fy(x)p Fv(\))2308 515 y Ft(\013)p Fq(\000)p Fu(1)2474 482 y Fm(,)1264 734 y Fs(j)p Fy(@)1331 746 y Ft(k)1373 712 y Fv(^)1372 734 y Fy(h)1420 746 y Fu(1)1457 734 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)24 b(\024)e Fy(H)1978 678 y Fs(j)p Fy(k)s Fs(j)f Fy(x)2148 621 y Fk(\015)s Fc(\000)p Fo(1)p 2149 635 108 3 v 2188 668 a(2)p 1892 715 464 4 v 1892 791 a Fv(\(1)e(+)f Fs(j)p Fy(k)s Fs(j)2160 731 y(p)p 2229 731 48 4 v 60 x Fy(x)p Fv(\))2308 767 y Ft(\013)2389 734 y Fm(,)1349 975 y Fs(j)1373 953 y Fv(^)1372 975 y Fy(h)1420 987 y Fu(2)1457 975 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)24 b(\024)e Fy(H)1966 919 y Fs(j)p Fy(k)s Fs(j)f Fy(x)2126 889 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)p 1892 956 464 4 v 1892 1032 a Fv(\(1)e(+)f Fs(j)p Fy(k)s Fs(j)2160 973 y(p)p 2229 973 48 4 v 59 x Fy(x)p Fv(\))2308 1008 y Ft(\013)2389 975 y Fm(,)1264 1216 y Fs(j)p Fy(@)1331 1228 y Ft(k)1373 1194 y Fv(^)1372 1216 y Fy(h)1420 1228 y Fu(2)1457 1216 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)24 b(\024)e Fy(H)2065 1160 y(x)2112 1130 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)p 1892 1197 549 4 v 1892 1274 a Fv(\(1)d(+)f Fs(j)p Fy(k)s Fs(j)2160 1214 y(p)p 2229 1214 48 4 v 60 x Fy(x)p Fv(\))2308 1250 y Ft(\013)p Fq(\000)p Fu(1)2474 1216 y Fy(:)-9 1438 y Fz(Under)k(these)f(conditions,)h (we)g(can)f(pro)o(v)o(e)h(the)f(follo)n(wing)i(lemma.)-9 1601 y FA(Lemma)d(3.8)43 b Fm(Let)31 b Fy(\021)j Fm(and)f Fy(p)e Fm(be)h(de\256ned)g(by)g(\(3.33\))i(and)e(\(3.34\).)64 b(Assume)32 b(that)f Fy(h)2570 1613 y Fu(1)2638 1601 y Fm(and)i Fy(h)2845 1613 y Fu(2)2914 1601 y Fm(satisfy)d(Condition)i(3.7)h(for)-9 1701 y(some)25 b(constants)g Fy(H)7 b Fm(,)25 b Fy(\013)j(>)f Fv(3)d Fm(and)i Fy(\015)32 b Fs(2)c Fv([0)p Fy(;)14 b Fv(1\))p Fm(.)39 b(Then)26 b(for)e(all)g Fy(\027)33 b Fs(2)28 b Fv(\(1)p Fy(=)p Fv(2)p Fy(;)14 b Fv(1\))p Fm(,)24 b(ther)m(e)g(e)n(xists)f(a)i (constant)h Fy(C)k Fm(suc)o(h)25 b(that)g(for)f(all)-9 1800 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(2)h Fv(\012)347 1812 y Fu(+)402 1800 y Fm(,)262 2038 y Fs(j)6 b Fv(^)-48 b Fy(\021)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)k(H)966 1982 y Fs(j)p Fy(k)s Fs(j)22 b Fy(x)1127 1952 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)p 851 2019 548 4 v 851 2095 a Fv(\(1)c(+)g Fs(j)p Fy(k)s Fs(j)1118 2036 y(p)p 1187 2036 48 4 v 59 x Fy(x)q Fv(\))1267 2071 y Ft(\013)p Fu(+1)1448 2038 y Fv(+)39 b Fy(C)28 b(H)1888 1982 y Fs(j)p Fy(k)s Fs(j)21 b Fy(x)2048 1952 y Fu(1)p Ft(=)p Fu(2)p 1746 2019 549 4 v 1746 2095 a Fv(\(1)d(+)g Fs(j)p Fy(k)s Fs(j)2013 2036 y(p)p 2082 2036 48 4 v 59 x Fy(x)q Fv(\))2162 2071 y Ft(\013)p Fq(\000)p Fu(1)2449 1982 y Fv(1)p 2314 2019 311 4 v 2314 2095 a(1)g(+)g Fy(k)2503 2071 y Fu(2)2540 2095 y Fy(x)2587 2071 y Fu(2)2658 2038 y Fm(,)177 2290 y Fs(j)p Fy(@)244 2302 y Ft(k)291 2290 y Fv(^)-48 b Fy(\021)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)k(H)993 2234 y(x)1050 2177 y Fk(\015)s Fc(\000)p Fo(1)p 1051 2191 108 3 v 1090 2224 a(2)p 851 2271 464 4 v 851 2347 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1118 2288 y(p)p 1187 2288 48 4 v 59 x Fy(x)q Fv(\))1267 2323 y Ft(\013)1364 2290 y Fv(+)39 b Fy(C)28 b(H)1860 2234 y(x)1907 2204 y Fu(1)p Ft(=)p Fu(2)p 1662 2271 549 4 v 1662 2347 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1929 2288 y(p)p 1998 2288 48 4 v 59 x Fy(x)q Fv(\))2078 2323 y Ft(\013)p Fq(\000)p Fu(1)2365 2234 y Fv(1)p 2230 2271 311 4 v 2230 2347 a(1)g(+)g Fy(k)2419 2323 y Fu(2)2456 2347 y Fy(x)2503 2323 y Fu(2)2574 2290 y Fm(,)282 2531 y Fs(j)7 b Fv(^)-49 b Fy(p)p Fv(\()p Fy(k)s(;)14 b(t)p Fv(\))p Fs(j)23 b(\024)g Fy(C)k(H)981 2475 y(x)1028 2445 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)p 851 2512 464 4 v 851 2589 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1118 2529 y(p)p 1187 2529 48 4 v 60 x Fy(x)q Fv(\))1267 2565 y Ft(\013)1348 2531 y Fm(,)179 2783 y Fs(j)p Fy(@)246 2795 y Ft(k)294 2783 y Fv(^)-49 b Fy(p)p Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)k(H)993 2727 y(x)1050 2670 y Fk(\015)s Fc(\000)p Fo(1)p 1051 2684 108 3 v 1090 2717 a(2)p 851 2764 464 4 v 851 2841 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1118 2781 y(p)p 1187 2781 48 4 v 60 x Fy(x)q Fv(\))1267 2817 y Ft(\013)1364 2783 y Fv(+)39 b Fy(C)28 b(H)1860 2727 y(x)1907 2697 y Fu(1)p Ft(=)p Fu(2)p 1662 2764 549 4 v 1662 2841 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1929 2781 y(p)p 1998 2781 48 4 v 60 x Fy(x)q Fv(\))2078 2817 y Ft(\013)p Fq(\000)p Fu(1)2365 2727 y Fv(1)p 2230 2764 311 4 v 2230 2840 a(1)g(+)g Fy(k)2419 2816 y Fu(2)2456 2840 y Fy(x)2503 2816 y Fu(2)2569 2783 y Fv(+)g Fy(C)28 b(H)3066 2727 y(x)3113 2697 y Fu(1)p Ft(=)p Fu(2)p Fq(\000)p Ft(\027)p 2846 2764 682 4 v 2846 2841 a Fs(j)p Fy(k)s Fs(j)2938 2817 y Ft(\027)2979 2841 y Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)3246 2781 y(p)p 3316 2781 48 4 v 3316 2841 a Fy(x)p Fv(\))3395 2817 y Ft(\013)p Fq(\000)p Fu(1)3561 2783 y Fy(:)-9 3006 y Fm(Pr)l(oof)o(.)31 b Fz(See)21 b(Appendix)i(D.)p 866 3006 43 43 v -9 3105 a(No)n(w)d(we)f(formulate)h(the)f(conditions)h(on)g Fy(q)s Fz(.)31 b(Recall)17 b(that)i Fy(q)26 b Fv(=)c Fy(q)1903 3117 y Fu(0)1955 3105 y Fv(+)13 b Fy(\021)s Fz(,)21 b(the)e(follo)n(wing)h (requirements)g(are)f(certainly)g(satis\256ed)-9 3205 y(by)j Fy(\021)i Fz(due)e(to)f(Lemma)g(3.8,)i(so)f(this)f(condition)h(is)f(really)g (a)g(condition)h(on)g Fy(q)2243 3217 y Fu(0)2280 3205 y Fz(.)-9 3368 y FA(Condition)g(3.9)43 b Fm(Let)22 b Fy(\013)i(>)g Fv(3)p Fm(,)d Fy(\015)29 b Fs(2)24 b Fv([0)p Fy(;)14 b Fv(1\))21 b Fm(and)j Fy(q)j Fv(:)d(\012)1573 3380 y Fu(+)1652 3368 y Fs(!)g Fr(R)p Fm(.)33 b(Ther)m(e)22 b(e)n(xists)f(a)h(constant)h Fy(Q)e Fm(suc)o(h)i(that)f(for)g(all)g Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)h Fv(\012)3694 3380 y Fu(+)3749 3368 y Fm(,)-9 3467 y(the)c(function)g Fy(q)k Fm(satis\256es)385 3717 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(Q)35 b Fz(min)1023 3575 y Fn(\()1228 3661 y Fs(j)p Fy(k)s Fs(j)21 b Fy(x)1398 3604 y Fk(\015)s Fc(\000)p Fo(1)p 1398 3618 108 3 v 1438 3651 a(2)p 1100 3698 548 4 v 1100 3774 a Fv(\(1)d(+)g Fs(j)p Fy(k)s Fs(j)1367 3715 y(p)p 1436 3715 48 4 v 59 x Fy(x)q Fv(\))1516 3750 y Ft(\013)p Fu(+1)1680 3717 y Fm(,)1896 3661 y Fy(x)1943 3631 y Fq(\000)p Fu(1)p 1733 3698 464 4 v 1733 3774 a Fv(\(1)g(+)g Fs(j)p Fy(k)s Fs(j)2000 3715 y(p)p 2069 3715 48 4 v 59 x Fy(x)q Fv(\))2149 3750 y Ft(\013)2206 3575 y Fn(\))2327 3717 y Fv(+)39 b Fy(Q)2669 3661 y Fs(j)p Fy(k)s Fs(j)22 b Fy(x)2830 3631 y Fu(1)p Ft(=)p Fu(2)p 2528 3698 549 4 v 2528 3774 a Fv(\(1)c(+)g Fs(j)p Fy(k)s Fs(j)2795 3715 y(p)p 2864 3715 48 4 v 59 x Fy(x)q Fv(\))2944 3750 y Ft(\013)p Fq(\000)p Fu(1)3231 3661 y Fv(1)p 3096 3698 311 4 v 3096 3774 a(1)g(+)g Fy(k)3285 3750 y Fu(2)3322 3774 y Fy(x)3369 3750 y Fu(2)3440 3717 y Fm(,)104 b Fz(\(3.35\))300 3995 y Fs(j)p Fy(@)367 4007 y Ft(k)414 3995 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(Q)1015 3939 y(x)1072 3882 y Fk(\015)s Fc(\000)p Fo(1)p 1073 3896 108 3 v 1112 3929 a(2)p 873 3976 464 4 v 873 4052 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1140 3992 y(p)p 1209 3992 48 4 v 60 x Fy(x)q Fv(\))1289 4028 y Ft(\013)1386 3995 y Fv(+)39 b Fy(Q)1786 3939 y(x)1833 3909 y Fu(1)p Ft(=)p Fu(2)p 1587 3976 549 4 v 1587 4052 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1854 3992 y(p)p 1924 3992 48 4 v 1924 4052 a Fy(x)p Fv(\))2003 4028 y Ft(\013)p Fq(\000)p Fu(1)2290 3939 y Fv(1)p 2156 3976 311 4 v 2156 4052 a(1)g(+)g Fy(k)2345 4028 y Fu(2)2382 4052 y Fy(x)2429 4028 y Fu(2)2499 3995 y Fy(:)1043 b Fz(\(3.36\))-9 4217 y(Under)22 b(these)f(hypotheses,)i(we)e(can)g(pro)o(v)o(e)h(the)f(follo) n(wing)i(lemma.)-9 4380 y FA(Lemma)d(3.10)43 b Fm(Let)26 b Fy(N)682 4392 y Fu(1)745 4380 y Fm(and)h Fy(N)965 4392 y Fu(2)1028 4380 y Fm(be)f(de\256ned)i(by)e(\(3.21\))i(and)f(\(3.22\).)48 b(Assume)26 b(that)g Fy(q)j Fm(satis\256es)d(Condition)h(3.9)g(for)f(some)-9 4479 y(constants)19 b Fy(Q)p Fm(,)g Fy(\013)24 b(>)f Fv(3)18 b Fm(and)i Fy(\015)28 b Fs(2)23 b Fv([0)p Fy(;)14 b Fv(1\))p Fm(,)19 b(then)g(for)g(all)f Fy(\015)1606 4491 y Fu(2)1667 4479 y Fs(2)23 b Fv([0)p Fy(;)14 b Fv(1\))k Fm(ther)m(e)g(e)n(xists)g(a)h (constant)h Fy(C)25 b Fm(suc)o(h)19 b(that)g(for)g(all)g Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(2)h Fv(\012)3694 4491 y Fu(+)3749 4479 y Fm(,)1057 4713 y Fs(j)1105 4692 y Fv(^)1080 4713 y Fy(N)1147 4725 y Fu(1)1184 4713 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)k(Q)1866 4656 y(t)1896 4626 y Fq(\000)s Fu(~)-36 b Ft(\015)p 1696 4694 464 4 v 1696 4770 a Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)1964 4710 y(p)p 2033 4710 48 4 v 60 x Fy(x)p Fv(\))2112 4746 y Ft(\013)2193 4713 y Fm(,)1075 4934 y Fs(j)1122 4913 y Fv(^)1098 4934 y Fy(N)1165 4946 y Fu(2)1202 4934 y Fv(\()p Fy(k)s(;)c(t)p Fv(\))p Fs(j)23 b(\024)g Fy(C)k(Q)1734 4877 y Fv(1)p 1696 4914 117 4 v 1696 4931 a Fs(p)p 1765 4931 48 4 v 60 x Fy(x)2044 4877 y Fv(1)p 1833 4914 464 4 v 1833 4991 a(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2100 4931 y(p)p 2169 4931 48 4 v 60 x Fy(x)q Fv(\))2249 4967 y Ft(\013)2329 4934 y Fm(,)990 5186 y Fs(j)p Fy(@)1057 5198 y Ft(k)1122 5165 y Fv(^)1098 5186 y Fy(N)1165 5198 y Fu(1)1202 5186 y Fv(\()p Fy(k)s(;)c(t)p Fv(\))p Fs(j)23 b(\024)g Fy(C)k(Q)1839 5129 y(x)1896 5073 y Fo(1)p Fc(\000)p Fk(\015)p 1896 5087 108 3 v 1936 5119 a Fo(2)p 1696 5167 464 4 v 1696 5243 a Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)1964 5183 y(p)p 2033 5183 48 4 v 60 x Fy(x)p Fv(\))2112 5219 y Ft(\013)2184 5069 y Fn(\022)2255 5129 y Fv(1)g(+)g Fs(j)p Fy(k)s Fs(j)p 2255 5167 235 4 v 2326 5243 a(j)p Fy(k)s Fs(j)2500 5069 y Fn(\023)2561 5086 y Ft(\015)2640 5186 y Fm(,)990 5420 y Fs(j)p Fy(@)1057 5432 y Ft(k)1122 5399 y Fv(^)1098 5420 y Fy(N)1165 5432 y Fu(2)1202 5420 y Fv(\()p Fy(k)s(;)c(t)p Fv(\))p Fs(j)23 b(\024)g Fy(C)k(Q)1950 5364 y Fv(1)p 1696 5401 549 4 v 1696 5478 a(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)1964 5418 y(p)p 2033 5418 48 4 v 60 x Fy(x)p Fv(\))2112 5454 y Ft(\013)p Fq(\000)p Fu(1)2432 5304 y Fs(p)p 2501 5304 V 60 x Fy(x)p 2265 5401 452 4 v 2265 5477 a Fs(j)p Fy(k)s(x)p Fs(j)2404 5453 y Ft(\015)2439 5461 y Fo(2)2494 5477 y Fv(+)g Fs(j)p Fy(k)s(x)p Fs(j)2749 5420 y Fm(,)-9 5660 y(wher)m(e)23 b Fv(~)-45 b Fy(\015)28 b Fv(=)23 b Fz(min)521 5593 y Fn(\010)579 5627 y Fu(1)p 579 5641 34 4 v 579 5689 a(4)645 5660 y Fm(,)698 5623 y Fu(1)p Fq(\000)p Ft(\015)p 698 5641 124 4 v 743 5689 a Fu(2)831 5593 y Fn(\011)879 5660 y Fm(.)1838 5909 y Fz(12)p eop %%Page: 13 13 bop -9 145 a Fm(Pr)l(oof)o(.)31 b Fz(See)21 b(Appendix)i(E.)p 856 145 43 43 v -9 244 a(This)f(lemma)f(implies)g(that)h(the)g(contrib)n (ution)h(of)g Fy(N)1560 256 y Fu(1)1619 244 y Fz(and)g Fy(N)1831 256 y Fu(2)1890 244 y Fz(to)f Fy(u;)14 b(v)25 b Fz(in)d(\(3.26\))i(and)f (\(3.27\))h(are)e(dominated)h(by)g(the)f(one)g(of)-9 344 y Fy(u)39 356 y Fu(0)97 344 y Fz(and)f Fy(v)280 356 y Fu(0)339 344 y Fz(if)g Fy(q)k Fz(satis\256es)20 b(Condition)i(3.9:)-9 510 y FA(Cor)n(ollary)g(3.11)43 b Fm(Assume)27 b(that)32 b Fv(^)-48 b Fy(q)29 b Fm(satis\256es)d(Condition)h(3.9)g(with)e Fy(\013)30 b(>)f Fv(3)d Fm(and)h Fy(\015)34 b Fv(=)29 b Fy(\015)2641 522 y Fu(1)2678 510 y Fm(.)45 b(Then)27 b(for)f(all)g Fy(\015)3224 522 y Fu(2)3290 510 y Fs(2)k Fv([0)p Fy(;)14 b Fv(1\))p Fm(,)27 b(ther)m(e)-9 609 y(e)n(xists)20 b(a)i(constant)g Fy(C)27 b Fm(suc)o(h)22 b(that)f(for)h(all)f Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)1575 621 y Fu(+)1630 609 y Fm(,)1387 831 y Fs(j)5 b Fv(^)-47 b Fy(u)o Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)1805 810 y Fv(~)1786 831 y Fy(C)2093 775 y Fv(1)p 1882 812 464 4 v 1882 889 a(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)2150 829 y(p)p 2219 829 48 4 v 60 x Fy(x)p Fv(\))2298 865 y Ft(\013)2379 831 y Fm(,)1165 b Fz(\(3.37\))1391 1052 y Fs(j)s Fv(^)-45 b Fy(v)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)1805 1031 y Fv(~)1786 1052 y Fy(C)1920 996 y Fv(1)p 1882 1033 117 4 v 1882 1050 a Fs(p)p 1952 1050 48 4 v 1952 1109 a Fy(x)2230 996 y Fv(1)p 2019 1033 464 4 v 2019 1109 a(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2286 1050 y(p)p 2355 1050 48 4 v 59 x Fy(x)q Fv(\))2435 1085 y Ft(\013)2515 1052 y Fm(,)1029 b Fz(\(3.38\))1302 1309 y Fs(j)p Fy(@)1369 1321 y Ft(k)1415 1309 y Fv(^)-47 b Fy(u)o Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)1805 1288 y Fv(~)1786 1309 y Fy(C)2010 1252 y(x)2067 1191 y Fo(1)p Fc(\000)p Fk(\015)2171 1203 y Fo(1)p 2068 1209 137 3 v 2122 1242 a(2)p 1882 1290 464 4 v 1882 1366 a Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)2150 1306 y(p)p 2219 1306 48 4 v 60 x Fy(x)p Fv(\))2298 1342 y Ft(\013)2370 1192 y Fn(\022)2441 1252 y Fv(1)g(+)g Fs(j)p Fy(k)s Fs(j)p 2441 1290 235 4 v 2512 1366 a(j)p Fy(k)s Fs(j)2686 1192 y Fn(\023)2747 1209 y Ft(\015)2782 1217 y Fo(1)2855 1309 y Fm(,)689 b Fz(\(3.39\))1306 1543 y Fs(j)p Fy(@)1373 1555 y Ft(k)1417 1543 y Fv(^)-45 b Fy(v)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)1805 1522 y Fv(~)1786 1543 y Fy(C)2136 1487 y Fv(1)p 1882 1524 549 4 v 1882 1601 a(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)2150 1541 y(p)p 2219 1541 48 4 v 60 x Fy(x)p Fv(\))2298 1577 y Ft(\013)p Fq(\000)p Fu(1)2618 1427 y Fs(p)p 2688 1427 V 2688 1487 a Fy(x)p 2451 1524 452 4 v 2451 1600 a Fs(j)p Fy(k)s(x)p Fs(j)2590 1576 y Ft(\015)2625 1584 y Fo(2)2680 1600 y Fv(+)g Fs(j)p Fy(k)s(x)p Fs(j)2936 1543 y Fm(,)608 b Fz(\(3.40\))804 1684 y Fn(\014)804 1734 y(\014)804 1784 y(\014)837 1780 y Fv(^)-47 b Fy(u)o Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b Fs(\000)f Fy(e)1214 1746 y Fu(\003)1259 1754 y Fc(\000)1308 1746 y Fu(\()p Ft(k)q Fu(\))p Ft(x)1439 1780 y Fy(u)1487 1792 y Fu(+)1542 1780 y Fv(\(0\))1648 1684 y Fn(\014)1648 1734 y(\014)1648 1784 y(\014)1698 1780 y Fs(\024)1805 1759 y Fv(~)1786 1780 y Fy(C)2043 1724 y(x)2090 1694 y Fq(\000)s Fu(~)-36 b Ft(\015)p 1882 1761 464 4 v 1882 1837 a Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)2150 1777 y(p)p 2219 1777 48 4 v 60 x Fy(x)p Fv(\))2298 1813 y Ft(\013)2379 1780 y Fm(,)1165 b Fz(\(3.41\))-9 2026 y Fm(wher)m(e)23 b Fv(~)-45 b Fy(\015)28 b Fv(=)23 b Fz(min)521 1959 y Fn(\010)579 1993 y Fu(1)p 579 2007 34 4 v 579 2054 a(4)645 2026 y Fm(,)698 1989 y Fu(1)p Fq(\000)p Ft(\015)818 1997 y Fo(1)p 698 2007 153 4 v 757 2054 a Fu(2)860 1959 y Fn(\011)929 2026 y Fm(and)1097 2005 y Fv(~)1078 2026 y Fy(C)30 b Fv(=)22 b Fy(C)6 b Fv(\()p Fy(Q)19 b Fv(+)f Fs(k)p Fr(v)1611 2038 y Fu(+)1666 2026 y Fs(k)1708 2038 y Ft(\013;\015)1806 2046 y Fo(1)1837 2038 y Ft(;\015)1892 2046 y Fo(2)1929 2026 y Fv(\))23 b Fy(<)g Fs(1)p Fm(.)-9 2208 y FA(Remark)e(3.12)43 b Fm(Inequalities)22 b(\(3.37\)\261\(3.40\))k(ar)m(e)21 b(the)g(k)o(e)m(y)g (ingr)m(edient)g(for)g(applying)h(the)f(contr)o(action)h(mapping)h(principle) o(.)-9 2308 y(W)-8 b(e)26 b(can)i(de\256ne)g(a)g(Banac)o(h)g(space)g(by)f (asking)h(that)g(eac)o(h)f(element)g(satis\256es)g(the)g(bounds)i (\(3.37\)\261\(3.40\),)34 b(see)27 b(in)g(the)g(ne)n(xt)-9 2408 y(section.)53 b(W)-8 b(e)28 b(will)g(show)h(that)957 2386 y Fv(^)956 2408 y Fy(h)1004 2420 y Fu(1)1041 2408 y Fm(,)1094 2386 y Fv(^)1093 2408 y Fy(h)1141 2420 y Fu(2)1207 2408 y Fm(and)37 b Fv(^)-48 b Fy(q)1401 2420 y Fu(0)1467 2408 y Fm(satisfy)28 b(Conditions)i(3.7)g(and)g(3.9)g(when)35 b Fv(^)-48 b Fy(u)29 b Fm(and)k Fv(^)-45 b Fy(v)32 b Fm(satisfy)c(these)g(bounds.)-9 2507 y(Inequality)22 b(\(3.41\))i(will)d(be)h(suf)n(\256cient)g(to)g(pr)l(o)o (ve)h(Theor)m(em)g(1.7,)h(see)d(below)-6 b(.)33 b(W)-8 b(e)22 b(see)f(also)i(that)f Fs(j)p Fy(u)p Fs(j)p Fy(;)14 b Fs(j)p Fy(v)s Fs(j)p Fy(;)g Fs(j)p Fy(\021)s Fs(j)p Fy(;)g Fs(j)p Fy(p)p Fs(j)24 b(\024)g Fy(C)k(w)3616 2519 y Ft(\013)p Fq(\000)p Fu(1)3749 2507 y Fm(.)-9 2607 y(By)d(inte)m(gr)o(ating)h(this)g(bound)j(with) d(r)m(espect)f(to)h Fy(k)s Fm(,)i(we)d(g)o(et)h(that)h Fy(u;)14 b(v)s(;)g(p)25 b Fm(and)j Fy(\021)h Fm(tend)e(to)f Fv(0)f Fm(at)i(least)e (lik)o(e)g Fy(x)3191 2577 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)3374 2607 y Fm(as)i Fy(x)j Fs(!)g(1)p Fm(.)-9 2707 y(Furthermor)m(e)o(,)25 b(since)d(the)i(derivatives)e(of)28 b Fv(^)-47 b Fy(u;)17 b Fv(^)-45 b Fy(v)s(;)19 b Fv(^)-47 b Fy(\021)26 b Fm(and)32 b Fv(^)-49 b Fy(p)23 b Fm(with)f(r)m(espect)h(to)g Fy(k)j Fm(ar)m(e)e(in)f Fy(L)2602 2676 y Fu(1)2639 2707 y Fv(\()p Fr(R)p Fy(;)14 b(dk)s Fv(\))p Fm(,)24 b(we)f(g)o(et)g(that)h Fy(u;)14 b(v)s(;)g(\021)26 b Fm(and)e Fy(p)-9 2806 y Fm(decay)d(at)g(least)g(lik)o(e)f Fs(j)p Fy(y)s Fs(j)712 2776 y Fq(\000)p Fu(1)822 2806 y Fm(as)i Fy(y)k Fs(!)d(1)p Fm(.)31 b(The)22 b(boundary)h(conditions)g(at)e(in\256nity) g(ar)m(e)h(thus)g(all)f(satis\256ed.)-9 3089 y FB(4)122 b(Existence)31 b(and)g(Uniqueness)f(f)m(or)h(the)g(Nonlinear)h(Pr)n(oblem)-9 3277 y Fz(In)20 b(this)g(section,)g(we)g(will)f(use)h(the)g(results)g(of)g (the)g(preceding)h(section)e(to)h(pro)o(v)o(e)h(that)e(the)h(nonlinear)h (problem)g(\(2.12\)\261\(2.15\))-9 3377 y(has)h(a)g(unique)i(solution)f (which)g(satis\256es)f(the)g(boundary)j(conditions.)36 b(W)-7 b(e)22 b(will)g(\256rst)h(de\256ne)g(a)f(Banach)g(space)g Fs(B)j Fz(of)e(\(v)o(ector)-9 3476 y(v)n(alued\))f(functions)h(which)f(is)g(an)g(e)o (xtension)g(of)g Fs(W)1533 3488 y Ft(\013;\015)1631 3496 y Fo(1)1664 3488 y Ft(;\015)1719 3496 y Fo(2)1776 3476 y Fz(to)g(functions)h Fv(\012)2262 3488 y Fu(+)2341 3476 y Fs(!)h Fr(R)2520 3446 y Fu(2)2557 3476 y Fz(.)32 b(Then)22 b(we)g(will)f(sho)n(w)i(that)e Fy(q)3466 3488 y Fu(0)3504 3476 y Fy(;)14 b(h)3589 3488 y Fu(1)3647 3476 y Fz(and)-9 3576 y Fy(h)39 3588 y Fu(2)95 3576 y Fz(satisfy)19 b(the)h(conditions)g(3.7)h(and)f(3.9)h(when)f Fv(\()p Fy(u;)14 b(v)s Fv(\))24 b Fs(2)f(B)s Fz(.)29 b(After)20 b(this,)g(we)g(will)e (construct)i(a)f(sequence)h(of)g(approximating)-9 3675 y(solutions)j(to)g (\(1.1\)\261\(1.4\),)k(and)d(sho)n(w)g(that)e(this)g(sequence)i(con)m(v)o(er) n(ges)g(to)e(the)h(solution)h(of)f(these)f(equations.)36 b(In)24 b(the)e(whole)-9 3775 y(section,)33 b(it)d(is)g(understood)k(that)c (Condition)i(1.2)g(on)f(the)g(boundary)j(condition)e(on)f Fv(\006)2655 3787 y Fu(+)2741 3775 y Fz(is)g(satis\256ed.)59 b(In)32 b(particular)m(,)g (the)-9 3875 y(parameters)21 b Fy(\013)p Fz(,)g Fy(\015)523 3887 y Fu(1)561 3875 y Fy(;)14 b(\015)641 3887 y Fu(2)678 3875 y Fy(;)g(\015)758 3887 y Ft(E)835 3875 y Fz(and)22 b(the)f(function)h Fy(f)32 b Fs(2)23 b(A)1625 3887 y Ft(\013;\015)1723 3895 y Fk(E)1772 3887 y Ft(;\016)1849 3875 y Fz(are)e(considered)h(as)f(gi)n(v)o(en) h(and)g(\256x)o(ed)g(once)f(and)h(for)h(all.)-9 4115 y Fd(4.1)102 b(Choice)26 b(of)f(Function)g(Spaces)-9 4272 y Fz(In)19 b(this)f(section,)h (we)f(construct)h(the)f(Banach)g(space)g Fs(B)s Fz(.)29 b(As)19 b(we)f(noted)i(in)e(Remark)g(3.12,)j(Corollary)e(3.11)h(pro)o(vides)g(a)e (natural)-9 4372 y(w)o(ay)j(to)h(de\256ne)g(a)f(Banach)g(space.)30 b(W)-7 b(e)22 b(be)o(gin)g(with)f(a)g(de\256nition.)-9 4538 y FA(De\256nition)h(4.1)42 b Fm(F)-9 b(or)22 b Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)1014 4550 y Fu(+)1069 4538 y Fm(,)f(we)f(de\256ne)h(the)f(following)g(functions.)991 4721 y Fy(p)1033 4733 y Ft(u;)p Fu(1)1129 4721 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e(\(1)c(+)g Fs(j)p Fy(k)s Fs(j)1701 4657 y(p)p 1771 4657 48 4 v 1771 4721 a Fy(x)p Fv(\))1850 4686 y Ft(\013)1921 4721 y Fm(,)995 4855 y Fy(p)1037 4867 y Ft(v)r(;)p Fu(1)1129 4855 y Fv(\()p Fy(x;)c(k)s Fv(\))24 b(=)e Fy(x)1481 4821 y Fu(1)p Ft(=)p Fu(2)1607 4855 y Fv(\(1)c(+)g Fs(j)p Fy(k)s Fs(j)1874 4792 y(p)p 1944 4792 V 1944 4855 a Fy(x)p Fv(\))2023 4821 y Ft(\013)2094 4855 y Fm(,)991 5037 y Fy(p)1033 5049 y Ft(u;)p Fu(2)1129 5037 y Fv(\()p Fy(x;)c(k)s Fv(\))24 b(=)e Fy(x)1491 4972 y Fk(\015)1522 4984 y Fo(1)1555 4972 y Fc(\000)p Fo(1)p 1492 4990 137 3 v 1546 5023 a(2)1664 5037 y Fv(\(1)c(+)g Fs(j)p Fy(k)s Fs(j)1931 4973 y(p)p 2000 4973 48 4 v 64 x Fy(x)p Fv(\))2079 5003 y Ft(\013)2141 4920 y Fn(\022)2283 4981 y Fs(j)p Fy(k)s Fs(j)p 2212 5018 235 4 v 2212 5094 a Fv(1)g(+)g Fs(j)p Fy(k)s Fs(j)2457 4920 y Fn(\023)2518 4937 y Ft(\015)2553 4945 y Fo(1)2626 5037 y Fm(,)995 5227 y Fy(p)1037 5239 y Ft(v)r(;)p Fu(2)1129 5227 y Fv(\()p Fy(x;)c(k)s Fv(\))24 b(=)e Fy(x)1481 5193 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1659 5227 y Fv(\(1)c(+)g Fs(j)p Fy(k)s Fs(j)1926 5163 y(p)p 1995 5163 48 4 v 64 x Fy(x)q Fv(\))2075 5193 y Ft(\013)p Fq(\000)p Fu(1)2208 5227 y Fv(\()p Fs(j)p Fy(k)s(x)p Fs(j)2379 5193 y Ft(\015)2414 5201 y Fo(2)2469 5227 y Fv(+)g Fs(j)p Fy(k)s(x)p Fs(j)p Fv(\))24 b Fy(:)-9 5410 y Fz(W)m(ith)c(these)h(functions,)i(we)e(can)g(de\256ne)h(the)f(follo)n(wing) i(Banach)e(space.)1838 5909 y(13)p eop %%Page: 14 14 bop -9 145 a FA(De\256nition)22 b(4.2)42 b Fm(W)-8 b(e)18 b(denote)i(by)e Fs(B)j Fm(the)d(Banac)o(h)i(space)f(obtained)h(by)f(completing)g Fy(C)2544 114 y Fq(1)2538 165 y Fu(0)2614 145 y Fv(\(\012)2706 157 y Fu(+)2762 145 y Fy(;)14 b Fr(R)2871 114 y Fu(2)2907 145 y Fv(\))19 b Fm(with)f(r)m(espect)g(to)g(the)h(norm)750 319 y Fs(k)p Fr(u)p Fs(k)j Fv(=)h Fz(max)1157 252 y Fn(\010)1206 319 y Fs(j)5 b Fv(^)-47 b Fy(u)20 b(p)1339 331 y Ft(u;)p Fu(1)1435 319 y Fs(j)1459 347 y Fq(1)1566 319 y Fm(,)35 b Fs(j)s Fv(^)-45 b Fy(v)25 b(p)1752 331 y Ft(v)r(;)p Fu(1)1844 319 y Fs(j)1867 347 y Fq(1)1974 319 y Fm(,)35 b Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(u)21 b(p)2235 331 y Ft(u;)p Fu(2)2331 319 y Fs(j)2354 347 y Fq(1)2461 319 y Fm(,)36 b Fs(j)p Fy(D)5 b Fv(^)-45 b Fy(v)24 b(p)2718 331 y Ft(v)r(;)p Fu(2)2810 319 y Fs(j)2833 347 y Fq(1)2904 252 y Fn(\011)2989 319 y Fm(,)-9 494 y(wher)m(e)o(,)d(for)g(a)h(function)g Fy(g)j Fv(:)e(\012)892 506 y Fu(+)970 494 y Fs(!)h Fr(R)p Fm(,)d Fs(j)p Fy(g)s Fs(j)1280 506 y Fq(1)1371 494 y Fm(and)i Fy(D)r(g)g Fm(denote)1507 668 y Fs(j)p Fy(g)s Fs(j)1596 680 y Fq(1)1689 668 y Fv(=)106 b Fz(sup)1777 744 y Fu(\()p Ft(x;k)q Fu(\))p Fq(2)p Fu(\012)2015 752 y Fo(+)2075 668 y Fs(j)p Fy(g)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b Fm(,)1358 863 y Fy(D)r(g)s Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b(=)g Fy(@)1821 875 y Ft(k)1862 863 y Fy(g)s Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fy(:)-9 1038 y Fz(Note)j(here)h(that)f(the)h(restriction)f(of)h(a)f(function)i(in)f Fs(B)h Fz(on)g(an)o(y)f(line)f Fy(x)31 b Fv(=)k(~)-47 b Fy(x)30 b(>)g Fv(1)c Fz(is)g(in)h Fs(W)2743 1050 y Ft(\013;\015)2841 1058 y Fo(1)2873 1050 y Ft(;\015)2928 1058 y Fo(2)2964 1038 y Fz(.)47 b(In)28 b(this)e(sense,)i(the)f(line)-9 1137 y Fy(x)c Fv(=)g(1)d Fz(is)h(by)i(no)f(means)f(particular)-5 b(.)-9 1376 y Fd(4.2)102 b(Bounds)24 b(on)h(Nonlinear)h(T)-9 b(erms)-9 1533 y Fz(In)20 b(this)f(section,)h(we)f(pro)o(v)o(e)h(that)f(the)h(in)m(v)o (erse)f(F)o(ourier)h(transforms)h(of)f(the)f(functions)i Fy(q)2601 1545 y Fu(0)2638 1533 y Fz(,)f Fy(h)2727 1545 y Fu(1)2784 1533 y Fz(and)g Fy(h)2974 1545 y Fu(2)3030 1533 y Fz(satisfy)f(Conditions)i(3.7)-9 1633 y(and)g(3.9)i(when)f Fv(\()p Fy(u;)14 b(v)s Fv(\))23 b Fs(2)h(B)s Fz(.)30 b(W)-7 b(e)21 b(gi)n(v)o(e)g(\256rst)h(the)f(e)o (xpression)h(for)29 b Fv(^)-48 b Fy(q)1986 1645 y Fu(0)2023 1633 y Fz(,)2066 1611 y Fv(^)2065 1633 y Fy(h)2113 1645 y Fu(1)2172 1633 y Fz(and)2316 1611 y Fv(^)2316 1633 y Fy(h)2364 1645 y Fu(2)2401 1633 y Fz(.)-9 1793 y FA(De\256nition)22 b(4.3)42 b Fm(Let)18 b Fr(v)25 b Fv(=)e(\()p Fy(u;)14 b(v)s Fv(\))23 b Fs(2)g(B)s Fm(,)18 b(and)i(let)1453 1792 y Fv(^)1447 1793 y Fr(u)1500 1805 y Ft(E)1579 1793 y Fv(=)j(\()5 b(^)-47 b Fy(u)1747 1805 y Ft(E)1803 1793 y Fy(;)17 b Fv(^)-45 b Fy(v)1880 1805 y Ft(E)1936 1793 y Fv(\))19 b Fm(denotes)f(the)h(F)-9 b(ourier)18 b(tr)o(ansform)h(of)g Fr(u)3150 1805 y Ft(E)3224 1793 y Fm(de\256ned)h(by)e (\(1.6\).)-9 1892 y(W)-8 b(e)20 b(de\256ne)j(the)e(following)g(maps)1113 2103 y Fv(^)-48 b Fy(q)1144 2115 y Fu(0)1181 2103 y Fv(\()p Fr(v)q Fv(\))25 b(=)1444 2046 y(1)p 1418 2084 92 4 v 1418 2160 a(2)p Fy(\031)1534 2103 y Fv(\()6 b(^)-48 b Fy(u)18 b Fs(\003)j Fv(^)-45 b Fy(v)22 b Fv(+)h(^)-47 b Fy(u)1885 2115 y Ft(E)1959 2103 y Fs(\003)21 b Fv(^)-45 b Fy(v)22 b Fv(+)h(^)-47 b Fy(u)18 b Fs(\003)j Fv(^)-45 b Fy(v)2330 2115 y Ft(E)2386 2103 y Fv(\))37 b Fm(,)1131 b Fz(\(4.1\))1097 2282 y Fv(^)1096 2304 y Fy(h)1144 2316 y Fu(1)1181 2304 y Fv(\()p Fr(v)q Fv(\))25 b(=)1429 2248 y Fy(i)p 1418 2285 51 4 v 1418 2361 a(\031)1493 2304 y Fv(\()s(^)-45 b Fy(v)22 b Fs(\003)17 b Fy(K)11 b Fv(^)-47 b Fy(u)18 b Fv(+)j(^)-45 b Fy(v)22 b Fs(\003)c Fy(K)11 b Fv(^)-47 b Fy(u)2119 2316 y Ft(E)2192 2304 y Fv(+)21 b(^)-45 b Fy(v)2315 2316 y Ft(E)2390 2304 y Fs(\003)18 b Fy(K)11 b Fv(^)-47 b Fy(u)o Fv(\))37 b Fm(,)943 b Fz(\(4.2\))1097 2483 y Fv(^)1096 2505 y Fy(h)1144 2517 y Fu(2)1181 2505 y Fv(\()p Fr(v)q Fv(\))25 b(=)1450 2449 y Fy(i)p 1418 2486 92 4 v 1418 2562 a Fv(2)p Fy(\031)1520 2505 y(K)20 b Fv(\()s(^)-45 b Fy(v)21 b Fs(\003)h Fv(^)-46 b Fy(v)22 b Fv(+)c(2)s(^)-45 b Fy(v)21 b Fs(\003)g Fv(^)-45 b Fy(v)2112 2517 y Ft(E)2169 2505 y Fv(\))37 b Fm(,)1348 b Fz(\(4.3\))-9 2703 y Fm(wher)m(e)20 b Fs(\003)h Fm(denotes)h(the)f(con)m(volution)h(and)h (for)e Fy(h)i Fv(:)g(\012)1550 2715 y Fu(+)1628 2703 y Fs(!)g Fr(R)e Fm(the)g(function)h Fy(K)6 b(h)21 b Fm(is)f(de\256ned)j(by)1498 2877 y Fy(K)6 b(h)p Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b(=)g Fy(k)s(h)p Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fy(:)-9 3052 y Fz(The)e(follo)n(wing)h (proposition)i(pro)o(v)o(es)e(that)f(Conditions)h(3.7)g(and)g(3.9)h(are)e (satis\256ed.)-9 3212 y FA(Pr)n(oposition)h(4.4)43 b Fm(Let)28 b(Condition)h(1.2)h(be)f(satis\256ed)f(for)h(some)g Fy(f)37 b Fm(and)30 b Fy(")p Fm(,)g(and)g(let)d Fr(u)2610 3224 y Ft(E)2694 3212 y Fm(be)i(de\256ned)h(by)e(\(1.6\).)54 b(Let)34 b Fv(^)-48 b Fy(q)3595 3224 y Fu(0)3633 3212 y Fv(\()p Fr(v)q Fv(\))p Fm(,)-9 3298 y Fv(^)-9 3320 y Fy(h)39 3332 y Fu(1)76 3320 y Fv(\()p Fr(v)q Fv(\))21 b Fm(and)361 3298 y Fv(^)360 3320 y Fy(h)408 3332 y Fu(2)445 3320 y Fv(\()p Fr(v)q Fv(\))h Fm(be)f(de\256ned)g (by)g(\(4.1\)\261\(4.3\).)35 b(Then)21 b(ther)m(e)f(e)n(xists)f(a)i(constant) h Fy(C)k Fm(suc)o(h)21 b(that)g(for)g(all)f Fr(v)k Fs(2)g(B)s Fm(,)c(the)g(following)-9 3420 y(bounds)j(hold)g(for)e(all)g Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)1017 3432 y Fu(+)882 3619 y Fs(j)6 b Fv(^)-48 b Fy(q)942 3631 y Fu(0)979 3619 y Fv(\()p Fr(v)q Fv(\))p Fs(j)25 b(\024)44 b Fy(C)27 b Fv(\()p Fy(")19 b Fv(+)f Fs(k)p Fr(v)q Fs(k)p Fv(\))j Fs(k)p Fr(v)q Fs(k)35 b Fz(min)2014 3527 y Fn(n)2069 3524 y(\014)2069 3574 y(\014)2069 3623 y(\014)2097 3619 y Fy(K)6 b(W)2252 3650 y Ft(\013)p Fu(+1)p Ft(;)2409 3618 y Fo(1)p Fc(\000)p Fk(\015)2513 3630 y Fo(1)p 2409 3636 137 3 v 2463 3670 a(2)2559 3524 y Fn(\014)2559 3574 y(\014)2559 3623 y(\014)2624 3619 y Fm(,)23 b Fy(W)2746 3631 y Ft(\013;)p Fu(1)2847 3527 y Fn(o)2939 3619 y Fm(,)647 b Fz(\(4.4\))811 3769 y Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)942 3781 y Fu(0)979 3769 y Fv(\()p Fr(v)q Fv(\))p Fs(j)25 b(\024)44 b Fy(C)27 b Fv(\()p Fy(")19 b Fv(+)f Fs(k)p Fr(v)q Fs(k)p Fv(\))j Fs(k)p Fr(v)q Fs(k)g Fy(W)1932 3799 y Ft(\013;)2005 3768 y Fo(1)p Fc(\000)p Fk(\015)2109 3780 y Fo(1)p 2005 3786 V 2059 3819 a(2)2179 3769 y Fm(,)1407 b Fz(\(4.5\))871 3932 y Fs(j)895 3910 y Fv(^)894 3932 y Fy(h)942 3944 y Fu(1)979 3932 y Fv(\()p Fr(v)q Fv(\))p Fs(j)25 b(\024)44 b Fy(C)27 b Fv(\()p Fy(")19 b Fv(+)f Fs(k)p Fr(v)q Fs(k)p Fv(\))j Fs(k)p Fr(v)q Fs(k)g Fy(W)1932 3952 y Ft(\013)p Fq(\000)p Fu(1)p Ft(;)2090 3930 y Fo(3)p 2090 3939 29 3 v 2090 3972 a(2)2156 3932 y Fm(,)1430 b Fz(\(4.6\))800 4101 y Fs(j)p Fy(D)895 4079 y Fv(^)894 4101 y Fy(h)942 4113 y Fu(1)979 4101 y Fv(\()p Fr(v)q Fv(\))p Fs(j)25 b(\024)44 b Fy(C)27 b Fv(\()p Fy(")19 b Fv(+)f Fs(k)p Fr(v)q Fs(k)p Fv(\))j Fs(k)p Fr(v)q Fs(k)1868 4005 y Fn(\014)1868 4055 y(\014)1868 4105 y(\014)1896 4101 y Fy(K)6 b(W)2051 4131 y Ft(\013;)2123 4100 y Fo(1)p Fc(\000)p Fk(\015)2227 4112 y Fo(1)p 2123 4118 137 3 v 2177 4151 a(2)2274 4005 y Fn(\014)2274 4055 y(\014)2274 4105 y(\014)2339 4101 y Fm(,)1247 b Fz(\(4.7\))871 4284 y Fs(j)895 4262 y Fv(^)894 4284 y Fy(h)942 4296 y Fu(2)979 4284 y Fv(\()p Fr(v)q Fv(\))p Fs(j)25 b(\024)44 b Fy(C)27 b Fv(\()p Fy(")19 b Fv(+)f Fs(k)p Fr(v)q Fs(k)p Fv(\))j Fs(k)p Fr(v)q Fs(k)1868 4188 y Fn(\014)1868 4238 y(\014)1868 4288 y(\014)1896 4284 y Fy(K)6 b(W)2051 4304 y Ft(\013;)2123 4282 y Fo(3)p 2123 4291 29 3 v 2123 4324 a(2)2166 4188 y Fn(\014)2166 4238 y(\014)2166 4288 y(\014)2230 4284 y Fm(,)1356 b Fz(\(4.8\))800 4450 y Fs(j)p Fy(D)895 4428 y Fv(^)894 4450 y Fy(h)942 4462 y Fu(2)979 4450 y Fv(\()p Fr(v)q Fv(\))p Fs(j)25 b(\024)44 b Fy(C)27 b Fv(\()p Fy(")19 b Fv(+)f Fs(k)p Fr(v)q Fs(k)p Fv(\))j Fs(k)p Fr(v)q Fs(k)g Fy(W)1932 4471 y Ft(\013)p Fq(\000)p Fu(1)p Ft(;)2090 4449 y Fo(3)p 2090 4458 V 2090 4491 a(2)2156 4450 y Fm(,)1430 b Fz(\(4.9\))-9 4625 y Fm(wher)m(e)1370 4761 y Fy(W)1448 4773 y Ft(\013;\014)1556 4761 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)2031 4705 y Fy(x)2078 4675 y Fq(\000)p Ft(\014)p 1871 4742 464 4 v 1871 4819 a Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)2139 4759 y(p)p 2208 4759 48 4 v 60 x Fy(x)p Fv(\))2287 4795 y Ft(\013)2368 4761 y Fy(:)-9 4965 y Fm(Pr)l(oof)o(.)31 b Fz(See)21 b(Appendix)i(F.)p 852 4965 43 43 v -9 5204 a Fd(4.3)102 b(Pr)n(oof)26 b(of)g(Lemma)f(1.5)h(and)f(Theor)n(em)h(1.7)-9 5361 y Fz(In)e(this)g(section,)g(we)g(construct)h(a)f(sequence)g(of)h (functions)g Fr(v)1859 5373 y Ft(n)1904 5361 y Fz(,)g Fy(\021)1991 5373 y Ft(n)2060 5361 y Fz(and)g Fy(p)2249 5373 y Ft(n)2318 5361 y Fz(and)g(sho)n(w)g(that)e(the)h(sequence)g(con)m(v)o(er)n(ges)h(to)f (a)-9 5461 y(solution)e(of)g(the)f(nonlinear)h(problem.)115 5560 y(W)-7 b(e)21 b(be)o(gin)h(with)f(the)g(construction)h(of)g Fr(v)1336 5572 y Fu(0)1373 5560 y Fz(,)g Fy(\021)1457 5572 y Fu(0)1515 5560 y Fz(and)g Fy(p)1701 5572 y Fu(0)1738 5560 y Fz(.)31 b(F)o(or)21 b Fr(v)1977 5572 y Fu(0)2015 5560 y Fz(,)g(we)g(tak)o (e)g(the)g(one)h(de\256ned)g(by)g(\(3.19\).)33 b(By)21 b(Lemma)g(3.1,)-9 5660 y Fr(v)41 5672 y Fu(0)99 5660 y Fz(and)h Fy(@)287 5672 y Ft(y)327 5660 y Fr(v)377 5672 y Fu(0)436 5660 y Fz(are)f(continuous)i(in)e Fy(x;)14 b(y)s Fz(.)31 b(Let)20 b Fy(\021)1397 5672 y Fu(0)p Ft(;)p Fu(0)1508 5660 y Fz(be)i(de\256ned)g(by)g(\(3.18\).)33 b(By)21 b(Proposition)i(3.3,)g Fr(v)2976 5672 y Fu(0)3034 5660 y Fz(is)e(a)g(strong)h(solution)g(of)1838 5909 y(14)p eop %%Page: 15 15 bop -9 145 a Fz(\(3.12\)\261\(3.14\))25 b(with)c Fy(q)26 b Fv(=)d(0)d Fz(and)h Fy(\021)1033 157 y Fu(0)1093 145 y Fv(=)i Fy(\021)1222 157 y Fu(0)p Ft(;)p Fu(0)1312 145 y Fz(.)31 b(Let)20 b Fy(\021)1539 157 y Fu(0)1599 145 y Fv(=)j(0)d Fz(and)h Fy(p)1934 157 y Fu(0)1994 145 y Fv(=)i(0)p Fz(.)30 b(The)o(y)21 b(are)g(strong)h (solutions)f(of)g(\(3.15\)\261\(3.17\))26 b(with)-9 244 y Fy(h)39 256 y Ft(i)89 244 y Fv(=)d(0)d Fz(\(sic!\).)115 344 y(No)n(w)31 b(we)f(construct)g(inducti)n(v)o(ely)g(the)f(sequence)h(of)h(functions)g Fr(v)2158 356 y Ft(n)p Fu(+1)2287 344 y Fz(,)h Fy(\021)2381 356 y Ft(n)p Fu(+1)2541 344 y Fz(and)e Fy(p)2735 356 y Ft(n)p Fu(+1)2894 344 y Fz(for)h Fy(n)j Fs(\025)g Fv(0)p Fz(.)56 b(Suppose)32 b(that)-9 443 y Fr(v)41 455 y Ft(n)109 443 y Fs(2)24 b(B)e Fz(and)f(that)f Fr(v)606 455 y Ft(n)672 443 y Fz(and)h Fy(@)859 455 y Ft(y)899 443 y Fr(v)949 455 y Ft(n)1015 443 y Fz(are)f(continuous)i(in) f Fy(x;)14 b(y)s Fz(.)31 b(Evidently)-6 b(,)21 b Fy(q)2202 455 y Fu(0)2240 443 y Fv(\()p Fr(v)2322 455 y Ft(n)2367 443 y Fv(\))p Fz(,)g Fy(h)2489 455 y Fu(1)2527 443 y Fv(\()p Fr(v)2609 455 y Ft(n)2654 443 y Fv(\))g Fz(and)g Fy(h)2898 455 y Fu(2)2935 443 y Fv(\()p Fr(v)3017 455 y Ft(n)3063 443 y Fv(\))g Fz(are)f(also)g (continuous)-9 543 y(in)25 b Fy(x;)14 b(y)s Fz(.)45 b(Furthermore,)29 b(Proposition)f(4.4)f(ensures)f(that)g Fy(h)1781 555 y Fu(1)1818 543 y Fv(\()p Fr(v)1900 555 y Ft(n)1946 543 y Fv(\))p Fz(,)h Fy(h)2074 555 y Fu(2)2111 543 y Fv(\()p Fr(v)2193 555 y Ft(n)2239 543 y Fv(\))f Fz(satisfy)g(Condition)h(3.7.)46 b(By)25 b(Lemma)h(3.5,)i(this) -9 643 y(implies)d(the)h(e)o(xistence)f(of)h Fy(q)871 655 y Fu(1)909 643 y Fv(\()p Fr(v)991 655 y Ft(n)1037 643 y Fv(\))p Fz(,)h Fy(q)1154 655 y Fu(2)1192 643 y Fv(\()p Fr(v)1274 655 y Ft(n)1320 643 y Fv(\))p Fz(,)g Fy(q)1437 655 y Fu(1)p Ft(;p)1529 643 y Fv(\()p Fr(v)1611 655 y Ft(n)1656 643 y Fv(\))g Fz(and)f Fy(q)1900 655 y Fu(2)p Ft(;p)1992 643 y Fv(\()p Fr(v)2074 655 y Ft(n)2120 643 y Fv(\))g Fz(which)h(are)e(bounded)k(and)e(continuous,)i(as)d (well)f(as)-9 742 y(their)f(deri)n(v)n(ati)n(v)o(es)g(with)h(respect)f(to)h Fy(y)s Fz(.)41 b(By)25 b(Lemma)f(3.8)i(and)f(Proposition)i(4.4,)g Fy(q)s Fv(\()p Fr(v)2577 754 y Ft(n)2623 742 y Fv(\))h(=)f Fy(q)2812 754 y Fu(0)2849 742 y Fv(\()p Fr(v)2931 754 y Ft(n)2977 742 y Fv(\))21 b(+)f Fy(q)3152 754 y Fu(1)3189 742 y Fv(\()p Fr(v)3271 754 y Ft(n)3317 742 y Fv(\))g(+)g Fy(q)3491 754 y Fu(2)3529 742 y Fv(\()p Fr(v)3611 754 y Ft(n)3656 742 y Fv(\))25 b Fz(is)-9 842 y(bounded,)g(continuous)f(and)e(satis\256es)f(Condition)i (3.9.)35 b(By)22 b(Lemma)g(3.2,)h(this)f(implies)f(the)h(e)o(xistence)f(of)i Fy(N)3272 854 y Fu(1)3309 842 y Fv(\()p Fr(v)3391 854 y Ft(n)3437 842 y Fv(\))p Fy(;)14 b(N)3573 854 y Fu(2)3610 842 y Fv(\()p Fr(v)3692 854 y Ft(n)3738 842 y Fv(\))-9 942 y Fz(which)24 b(are)g(continuous)i(as)e(well)g(as)g(their)f(deri)n(v)n(ati)n(v)o(es)h(with) g(respect)g(to)g Fy(y)s Fz(,)h(and)g(the)f(e)o(xistence)f(of)i Fy(\021)3047 954 y Fu(0)p Ft(;q)3136 942 y Fv(\()p Fr(v)3218 954 y Ft(n)3264 942 y Fv(\))g Fz(which)f(is)g(also)-9 1041 y(continuous.)32 b(From)23 b(this,)e(we)g(conclude)h(with)g(Propositions)h (3.3)f(and)g(3.6)h(that)1375 1224 y Fy(u)1423 1236 y Ft(n)p Fu(+1)1575 1224 y Fv(=)g Fy(u)1711 1236 y Fu(0)1766 1224 y Fv(+)18 b Fy(N)1916 1236 y Fu(1)1953 1224 y Fv(\()p Fr(v)2035 1236 y Ft(n)2081 1224 y Fv(\))23 b Fz(,)1408 b(\(4.10\))1383 1348 y Fy(v)1423 1360 y Ft(n)p Fu(+1)1575 1348 y Fv(=)23 b Fy(v)1703 1360 y Fu(0)1759 1348 y Fv(+)18 b Fy(N)1909 1360 y Fu(2)1946 1348 y Fv(\()p Fr(v)2028 1360 y Ft(n)2074 1348 y Fv(\))23 b Fz(,)1415 b(\(4.11\))1382 1473 y Fy(\021)1423 1485 y Ft(n)p Fu(+1)1575 1473 y Fv(=)23 b Fy(q)1700 1485 y Fu(1)1737 1473 y Fv(\()p Fr(v)1819 1485 y Ft(n)1865 1473 y Fv(\))c(+)f Fy(q)2036 1485 y Fu(2)2073 1473 y Fv(\()p Fr(v)2155 1485 y Ft(n)2201 1473 y Fv(\))23 b Fz(,)1288 b(\(4.12\))1381 1597 y Fy(p)1423 1609 y Ft(n)p Fu(+1)1575 1597 y Fv(=)23 b Fy(q)1700 1609 y Fu(1)p Ft(;p)1791 1597 y Fv(\()p Fr(v)1873 1609 y Ft(n)1919 1597 y Fv(\))c(+)f Fy(q)2090 1609 y Fu(2)p Ft(;p)2181 1597 y Fv(\()p Fr(v)2263 1609 y Ft(n)2309 1597 y Fv(\))23 b Fz(,)1180 b(\(4.13\))-9 1780 y(are)21 b(strong)h(solutions)h(of)f (\(3.12\)\261\(3.14\))k(and)c(\(3.15\)\261\(3.17\))27 b(with)21 b Fy(\021)2076 1792 y Fu(0)2137 1780 y Fv(=)i Fy(\021)2266 1792 y Fu(0)p Ft(;)p Fu(0)2374 1780 y Fv(+)18 b Fy(\021)2498 1792 y Fu(0)p Ft(;q)2588 1780 y Fv(\()p Fr(v)2670 1792 y Ft(n)2716 1780 y Fv(\))p Fz(,)k Fy(q)k Fv(=)d Fy(q)s Fv(\()p Fr(v)3064 1792 y Ft(n)3110 1780 y Fv(\))p Fz(,)f Fy(h)3233 1792 y Fu(1)3293 1780 y Fv(=)h Fy(h)3429 1792 y Fu(1)3466 1780 y Fv(\()p Fr(v)3548 1792 y Ft(n)3594 1780 y Fv(\))e Fz(and)-9 1880 y Fy(h)39 1892 y Fu(2)99 1880 y Fv(=)h Fy(h)234 1892 y Fu(2)271 1880 y Fv(\()p Fr(v)353 1892 y Ft(n)399 1880 y Fv(\))p Fz(.)115 1979 y(By)c(Corollary)h (3.11,)i(the)d(tw)o(o)h(equations)f(\(4.10\))j(and)e(\(4.11\))h(de\256ning)g Fr(v)2310 1991 y Ft(n)p Fu(+1)2457 1979 y Fz(in)f(terms)f(of)h Fr(v)2888 1991 y Ft(n)2951 1979 y Fz(can)f(be)h(easily)e(interpreted)-9 2079 y(as)k(the)g(action)g(of)h(some)f(map)h Fs(F)31 b Fv(:)23 b Fs(B)i(!)e(B)s Fz(,)e(namely)-6 b(,)22 b(we)f(ha)n(v)o(e)h(the)f(follo)n (wing)h(lemma:)-9 2245 y FA(Lemma)e(4.5)43 b Fm(Ther)m(e)21 b(e)n(xist)g(constants)h Fy(C)1231 2257 y Fu(1)1268 2245 y Fy(;)14 b(C)1364 2257 y Fu(2)1423 2245 y Fm(suc)o(h)22 b(that)f(for)h(all)f Fr(v)2034 2257 y Fu(1)2071 2245 y Fy(;)14 b Fr(v)2158 2257 y Fu(2)2219 2245 y Fs(2)23 b(B)h Fm(we)c(have)1059 2428 y Fs(kF)8 b Fv(\()p Fr(v)1251 2440 y Ft(i)1279 2428 y Fv(\))19 b Fs(\000)f Fr(v)1463 2440 y Fu(0)1500 2428 y Fs(k)23 b(\024)g Fy(C)1712 2440 y Fu(1)1771 2428 y Fv(\()p Fy(")18 b Fv(+)g Fs(k)p Fr(v)2035 2440 y Ft(i)2063 2428 y Fs(k)p Fv(\))j Fs(k)p Fr(v)2250 2440 y Ft(i)2277 2428 y Fs(k)i Fm(,)1202 b Fz(\(4.14\))917 2552 y Fs(kF)8 b Fv(\()p Fr(v)1109 2564 y Fu(1)1147 2552 y Fv(\))18 b Fs(\000)g(F)8 b Fv(\()p Fr(v)1430 2564 y Fu(2)1468 2552 y Fv(\))p Fs(k)23 b(\024)g Fy(C)1712 2564 y Fu(1)1771 2552 y Fv(\()p Fy(")18 b Fv(+)g Fs(k)p Fr(v)2035 2564 y Fu(1)2072 2552 y Fs(k)g Fv(+)g Fs(k)p Fr(v)2307 2564 y Fu(2)2344 2552 y Fs(k)p Fv(\))j Fs(k)p Fr(v)2531 2564 y Fu(1)2587 2552 y Fs(\000)d Fr(v)2720 2564 y Fu(2)2758 2552 y Fs(k)k Fm(,)722 b Fz(\(4.15\))1371 2677 y Fs(k)p Fr(v)1463 2689 y Fu(0)1500 2677 y Fs(k)23 b(\024)g Fy(C)1712 2689 y Fu(2)1771 2677 y Fs(k)p Fr(v)1863 2689 y Fu(+)1918 2677 y Fs(k)1960 2689 y Ft(\013;\015)2058 2697 y Fo(1)2089 2689 y Ft(;\015)2144 2697 y Fo(2)2203 2677 y Fm(,)1341 b Fz(\(4.16\))-9 2859 y Fm(wher)m(e)20 b Fs(F)31 b Fv(:)23 b Fs(B)j(!)d(B)g Fm(is)e(given)g(by)h(the)f(abo)o(ve)h(construction.)-9 3025 y(Pr)l(oof)o(.)30 b Fz(Inequality)21 b(\(4.14\))g(is)f(a)f(direct)g (consequence)i(of)f(Lemma)f(3.10)j(and)e(Proposition)h(4.4,)h(inequality)d (\(4.16\))j(is)d(a)h(direct)-9 3125 y(consequence)k(of)g(Lemma)f(3.4.)38 b(This)23 b(implies)g(that)g Fs(F)31 b Fz(is)23 b(a)g(well)f(de\256ned)j(map) e Fs(B)28 b(!)e(B)s Fz(.)36 b(Finally)-6 b(,)24 b(for)h(all)d Fr(v)3300 3137 y Fu(1)3337 3125 y Fy(;)14 b Fr(v)3424 3137 y Fu(2)3488 3125 y Fs(2)26 b(B)s Fz(,)d(we)-9 3225 y(ha)n(v)o(e)e(the)g (estimates)f(v)n(alid)h(for)h(all)e Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)g Fv(\012)1404 3237 y Fu(+)475 3432 y Fs(j)p Fy(q)535 3444 y Fu(0)572 3432 y Fv(\()p Fr(v)654 3444 y Fu(1)692 3432 y Fv(\))19 b Fs(\000)f Fy(q)863 3444 y Fu(0)900 3432 y Fv(\()p Fr(v)982 3444 y Fu(2)1020 3432 y Fv(\))p Fs(j)23 b(\024)44 b Fy(C)28 b Fv(\()p Fy(")18 b Fv(+)g Fs(k)p Fr(v)1558 3444 y Fu(1)1595 3432 y Fs(k)g Fv(+)g Fs(k)p Fr(v)1830 3444 y Fu(2)1867 3432 y Fs(k)p Fv(\))j Fs(k)p Fr(v)2054 3444 y Fu(1)2110 3432 y Fs(\000)d Fr(v)2243 3444 y Fu(2)2281 3432 y Fs(k)34 b Fz(min)2503 3340 y Fn(n)2559 3337 y(\014)2559 3387 y(\014)2559 3436 y(\014)2586 3432 y Fy(K)6 b(W)2741 3463 y Ft(\013)p Fu(+1)p Ft(;)2898 3431 y Fo(1)p Fc(\000)p Fk(\015)3002 3443 y Fo(1)p 2898 3449 137 3 v 2952 3483 a(2)3049 3337 y Fn(\014)3049 3387 y(\014)3049 3436 y(\014)3113 3432 y Fz(,)24 b Fy(W)3236 3444 y Ft(\013;)p Fu(1)3336 3340 y Fn(o)3428 3432 y Fz(,)116 b(\(4.17\))332 3582 y Fs(j)p Fy(D)r(q)463 3594 y Fu(0)501 3582 y Fv(\()p Fr(v)583 3594 y Fu(1)621 3582 y Fv(\))19 b Fs(\000)f Fy(D)r(q)863 3594 y Fu(0)900 3582 y Fv(\()p Fr(v)982 3594 y Fu(2)1020 3582 y Fv(\))p Fs(j)23 b(\024)44 b Fy(C)28 b Fv(\()p Fy(")18 b Fv(+)g Fs(k)p Fr(v)1558 3594 y Fu(1)1595 3582 y Fs(k)g Fv(+)g Fs(k)p Fr(v)1830 3594 y Fu(2)1867 3582 y Fs(k)p Fv(\))j Fs(k)p Fr(v)2054 3594 y Fu(1)2110 3582 y Fs(\000)d Fr(v)2243 3594 y Fu(2)2281 3582 y Fs(k)i Fy(W)2421 3612 y Ft(\013;)2495 3581 y Fo(1)p Fc(\000)p Fk(\015)2599 3593 y Fo(1)p 2495 3599 V 2549 3632 a(2)2668 3582 y Fz(,)876 b(\(4.18\))453 3727 y Fs(j)p Fy(h)524 3739 y Fu(1)561 3727 y Fv(\()p Fr(v)643 3739 y Fu(1)681 3727 y Fv(\))19 b Fs(\000)f Fy(h)863 3739 y Fu(1)900 3727 y Fv(\()p Fr(v)982 3739 y Fu(2)1020 3727 y Fv(\))p Fs(j)23 b(\024)44 b Fy(C)28 b Fv(\()p Fy(")18 b Fv(+)g Fs(k)p Fr(v)1558 3739 y Fu(1)1595 3727 y Fs(k)g Fv(+)g Fs(k)p Fr(v)1830 3739 y Fu(2)1867 3727 y Fs(k)p Fv(\))j Fs(k)p Fr(v)2054 3739 y Fu(1)2110 3727 y Fs(\000)d Fr(v)2243 3739 y Fu(2)2281 3727 y Fs(k)i Fy(W)2421 3748 y Ft(\013)p Fq(\000)p Fu(1)p Ft(;)2580 3726 y Fo(3)p 2580 3735 29 3 v 2580 3768 a(2)2645 3727 y Fz(,)899 b(\(4.19\))311 3897 y Fs(j)p Fy(D)r(h)453 3909 y Fu(1)490 3897 y Fv(\()p Fr(v)572 3909 y Fu(1)610 3897 y Fv(\))19 b Fs(\000)f Fy(D)r(h)863 3909 y Fu(1)900 3897 y Fv(\()p Fr(v)982 3909 y Fu(2)1020 3897 y Fv(\))p Fs(j)23 b(\024)44 b Fy(C)28 b Fv(\()p Fy(")18 b Fv(+)g Fs(k)p Fr(v)1558 3909 y Fu(1)1595 3897 y Fs(k)g Fv(+)g Fs(k)p Fr(v)1830 3909 y Fu(2)1867 3897 y Fs(k)p Fv(\))j Fs(k)p Fr(v)2054 3909 y Fu(1)2110 3897 y Fs(\000)d Fr(v)2243 3909 y Fu(2)2281 3897 y Fs(k)2357 3801 y Fn(\014)2357 3851 y(\014)2357 3901 y(\014)2385 3897 y Fy(K)6 b(W)2540 3927 y Ft(\013;)2613 3896 y Fo(1)p Fc(\000)p Fk(\015)2717 3908 y Fo(1)p 2613 3914 137 3 v 2667 3947 a(2)2763 3801 y Fn(\014)2763 3851 y(\014)2763 3901 y(\014)2828 3897 y Fz(,)716 b(\(4.20\))453 4079 y Fs(j)p Fy(h)524 4091 y Fu(2)561 4079 y Fv(\()p Fr(v)643 4091 y Fu(1)681 4079 y Fv(\))19 b Fs(\000)f Fy(h)863 4091 y Fu(2)900 4079 y Fv(\()p Fr(v)982 4091 y Fu(2)1020 4079 y Fv(\))p Fs(j)23 b(\024)44 b Fy(C)28 b Fv(\()p Fy(")18 b Fv(+)g Fs(k)p Fr(v)1558 4091 y Fu(1)1595 4079 y Fs(k)g Fv(+)g Fs(k)p Fr(v)1830 4091 y Fu(2)1867 4079 y Fs(k)p Fv(\))j Fs(k)p Fr(v)2054 4091 y Fu(1)2110 4079 y Fs(\000)d Fr(v)2243 4091 y Fu(2)2281 4079 y Fs(k)2357 3984 y Fn(\014)2357 4034 y(\014)2357 4083 y(\014)2385 4079 y Fy(K)6 b(W)2540 4100 y Ft(\013;)2613 4077 y Fo(3)p 2613 4086 29 3 v 2613 4120 a(2)2655 3984 y Fn(\014)2655 4034 y(\014)2655 4083 y(\014)2720 4079 y Fz(,)824 b(\(4.21\))311 4229 y Fs(j)p Fy(D)r(h)453 4241 y Fu(2)490 4229 y Fv(\()p Fr(v)572 4241 y Fu(1)610 4229 y Fv(\))19 b Fs(\000)f Fy(D)r(h)863 4241 y Fu(2)900 4229 y Fv(\()p Fr(v)982 4241 y Fu(2)1020 4229 y Fv(\))p Fs(j)23 b(\024)44 b Fy(C)28 b Fv(\()p Fy(")18 b Fv(+)g Fs(k)p Fr(v)1558 4241 y Fu(1)1595 4229 y Fs(k)g Fv(+)g Fs(k)p Fr(v)1830 4241 y Fu(2)1867 4229 y Fs(k)p Fv(\))j Fs(k)p Fr(v)2054 4241 y Fu(1)2110 4229 y Fs(\000)d Fr(v)2243 4241 y Fu(2)2281 4229 y Fs(k)i Fy(W)2421 4249 y Ft(\013)p Fq(\000)p Fu(1)p Ft(;)2580 4227 y Fo(3)p 2580 4236 V 2580 4269 a(2)2645 4229 y Fy(:)897 b Fz(\(4.22\))-9 4423 y(This)20 b(follo)n(ws)g(easily)f(from)i(the)f(structure)g(of)h Fy(q)1406 4435 y Fu(0)1443 4423 y Fz(,)g Fy(h)1533 4435 y Fu(1)1590 4423 y Fz(and)f Fy(h)1780 4435 y Fu(2)1817 4423 y Fz(,)h(since,)f(for)h(e)o (xample)e Fy(u)2548 4435 y Fu(1)2585 4423 y Fy(@)2629 4435 y Ft(y)2669 4423 y Fy(v)2709 4435 y Fu(1)2762 4423 y Fs(\000)c Fy(u)2890 4435 y Fu(2)2927 4423 y Fy(@)2971 4435 y Ft(y)3011 4423 y Fy(v)3051 4435 y Fu(2)3111 4423 y Fv(=)23 b(\()p Fy(u)3279 4435 y Fu(1)3331 4423 y Fs(\000)15 b Fy(u)3459 4435 y Fu(2)3496 4423 y Fv(\))p Fy(@)3572 4435 y Ft(y)3613 4423 y Fy(v)3653 4435 y Fu(1)3706 4423 y Fv(+)-9 4522 y Fy(u)39 4534 y Fu(2)75 4522 y Fy(@)119 4534 y Ft(y)159 4522 y Fv(\()p Fy(v)231 4534 y Fu(1)288 4522 y Fs(\000)k Fy(v)412 4534 y Fu(2)449 4522 y Fv(\))p Fz(.)34 b(Inequalities)22 b(\(4.17\)\261\(4.22\))27 b(in)m(v)n(olv)o(e)c(the)f(same)g(functions)h(and)g(the)f(same)g(constants)h (on)g(the)f(right\261hand)-9 4622 y(side)f(as)g(the)g(one)h(of)g(Proposition) h(4.4,)f(so)g(that)f(inequality)g(\(4.15\))j(follo)n(ws)e(at)e(once.)p 2587 4622 43 43 v -9 4722 a(From)i(this)f(lemma,)g(we)g(get)g(that)g(the)g (map)g Fs(F)30 b Fz(is)20 b(a)h(contraction)h(in)f(an)h(appropriate)g(ball)f (of)h Fs(B)s Fz(.)-9 4888 y FA(Cor)n(ollary)g(4.6)43 b Fm(F)-9 b(or)20 b(all)g Fy(\033)27 b Fs(2)c Fv(\(0)p Fy(;)14 b Fv(1\))19 b Fm(and)j Fy(")h(>)f Fv(0)e Fm(satisfying)f Fy(")h(C)1938 4900 y Fu(1)1996 4888 y Fv(\(1)15 b(+)g(2)20 b Fy(C)2286 4900 y Fu(2)2323 4888 y Fv(\(1)c(+)f Fy(\033)s Fv(\)\))24 b Fy(<)f Fv(1)c Fm(the)h(map)h Fs(F)28 b Fm(has)21 b(a)f(unique)i(\256xed)-9 4987 y(point)f(denoted)i(by)e Fr(v)628 4999 y Ft(?)688 4987 y Fm(in)g(the)g(open)i(ball)e(of)h(r)o(adius)g Fy(\033)s Fs(k)p Fr(v)1707 4999 y Fu(0)1744 4987 y Fs(k)f Fm(center)m(ed)g(in)g Fr(v)2255 4999 y Fu(0)2316 4987 y Fs(2)i(B)g Fm(pr)l(o)o(vided)g Fs(k)p Fr(v)2883 4999 y Fu(+)2938 4987 y Fs(k)2980 4999 y Ft(\013;\015)3078 5007 y Fo(1)3110 4999 y Ft(;\015)3165 5007 y Fo(2)3224 4987 y Fy(<)g(")p Fm(.)-9 5153 y(Pr)l(oof)o(.)31 b Fz(By)20 b(inequalities)f (\(4.14\))j(and)f(\(4.15\),)i(the)d(map)h Fs(F)28 b Fz(is)19 b(a)h(contraction)h(in)f(the)g(open)h(ball)f(of)h(radius)g Fy(\033)s Fs(k)p Fr(v)3297 5165 y Fu(0)3334 5153 y Fs(k)f Fz(centered)g(in)-9 5253 y Fr(v)41 5265 y Fu(0)101 5253 y Fs(2)k(B)s Fz(,)c(pro)o(vided)j Fy(")e(C)720 5265 y Fu(1)779 5253 y Fv(\(1)d(+)g(2)j Fy(C)1076 5265 y Fu(2)1114 5253 y Fv(\(1)d(+)g Fy(\033)s Fv(\)\))24 b Fy(<)f Fv(1)p Fz(.)30 b(It)21 b(follo)n(ws)h(that)f(lim)2217 5265 y Ft(n)p Fq(!1)2409 5253 y Fr(v)2459 5265 y Ft(n)2527 5253 y Fv(=)i Fr(v)2665 5265 y Ft(?)2726 5253 y Fs(2)h(B)f Fz(is)e(a)g(\256x)o(ed)h(point)g(of)g Fs(F)8 b Fz(.)p 3639 5253 V -9 5353 a(This)21 b(corollary)h(implies)e(Lemma)h(1.5)i(by)f(the)f (follo)n(wing)h(observ)n(ation.)33 b(Since)21 b Fr(v)2458 5365 y Ft(?)2519 5353 y Fs(2)j(B)s Fz(,)c(we)i(can)f(de\256ne)1539 5535 y Fy(\021)1580 5547 y Ft(?)1642 5535 y Fv(=)h Fy(q)1766 5547 y Fu(1)1804 5535 y Fv(\()p Fr(v)1886 5547 y Ft(?)1924 5535 y Fv(\))d(+)f Fy(q)2095 5547 y Fu(2)2132 5535 y Fv(\()p Fr(v)2214 5547 y Ft(?)2253 5535 y Fv(\))24 b Fz(,)1431 5660 y Fy(p)1473 5672 y Ft(?)1534 5660 y Fv(=)e Fy(q)1658 5672 y Fu(1)p Ft(;p)1750 5660 y Fv(\()p Fr(v)1832 5672 y Ft(?)1870 5660 y Fv(\))d(+)f Fy(q)2041 5672 y Fu(2)p Ft(;p)2132 5660 y Fv(\()p Fr(v)2214 5672 y Ft(?)2253 5660 y Fv(\))24 b Fz(,)1838 5909 y(15)p eop %%Page: 16 16 bop 1788 145 a Fv(~)-48 b Fy(\021)1823 157 y Ft(?)1885 145 y Fv(=)23 b Fy(\021)2014 157 y Fu(0)p Ft(;?)2123 145 y Fv(+)18 b Fy(\021)2247 157 y Ft(?)2309 145 y Fz(,)1788 269 y Fv(~)-49 b Fy(p)1823 281 y Ft(?)1884 269 y Fv(=)23 b Fy(p)2014 281 y Fu(0)p Ft(;?)2123 269 y Fv(+)18 b Fy(\021)2247 281 y Ft(?)2309 269 y Fz(,)-9 421 y(where)1961 573 y Fy(\021)2002 585 y Fu(0)p Ft(;?)2116 573 y Fv(=)23 b Fy(\021)2245 585 y Fu(0)2282 573 y Fv(\()p Fr(v)2364 585 y Ft(?)2403 573 y Fv(\))g Fz(,)1288 697 y Fv(^)-49 b Fy(p)1323 709 y Fu(0)p Ft(;?)1414 697 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e Fs(\000)p Fy(i)34 b Fz(s)-5 b(i)g(g)g(n)r Fv(\()p Fy(k)s Fv(\))28 b(^)-48 b Fy(\021)2148 709 y Fu(0)p Ft(;?)2239 697 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)-9 849 y Fz(By)j(construction,)j(the)d(right\261hand)j(sides)d(of)h(equations)g (\(2.12\))h(and)f(\(2.13\),)j Fr(v)2449 861 y Ft(?)2488 849 y Fz(,)e Fy(@)2582 861 y Ft(y)2622 849 y Fr(v)2672 861 y Ft(?)2738 849 y Fz(and)f Fy(q)2925 861 y Fu(0)2962 849 y Fv(\()p Fr(v)3044 861 y Ft(?)3083 849 y Fv(\))22 b(+)27 b(~)-48 b Fy(\021)3264 861 y Ft(?)3330 849 y Fz(e)o(xist)26 b(and)i(are)-9 949 y(continuous)j(in)f Fy(x;)14 b(y)s Fz(,)31 b(so)f(that)f Fr(v)981 961 y Ft(?)1049 949 y Fz(is)h(a)f(strong)i(solution)f(of)g(\(2.12\))i(and)e(\(2.13\).)58 b(Furthermore)31 b Fr(v)3021 961 y Ft(?)3089 949 y Fz(obe)o(ys)g(the)e (boundary)-9 1048 y(conditions.)h(On)17 b(the)f(other)h(hand,)h(the)e (right\261hand)j(sides)d(of)h(equations)g(\(2.14\))h(and)f(\(2.15\),)j Fy(\021)2794 1060 y Ft(?)2832 1048 y Fz(,)d Fy(p)2912 1060 y Ft(?)2950 1048 y Fz(,)h Fy(@)3033 1060 y Ft(y)3073 1048 y Fy(\021)3114 1060 y Ft(?)3152 1048 y Fz(,)g Fy(@)3235 1060 y Ft(y)3275 1048 y Fy(p)3317 1060 y Ft(?)3355 1048 y Fz(,)f Fy(h)3441 1060 y Fu(1)3478 1048 y Fv(\()p Fr(v)3560 1060 y Ft(?)3599 1048 y Fv(\))f Fz(and)-9 1148 y Fy(h)39 1160 y Fu(2)76 1148 y Fv(\()p Fr(v)158 1160 y Ft(?)196 1148 y Fv(\))21 b Fz(e)o(xist)e(and)i (are)f(continuous,)i(so)f(that)e Fy(\021)1393 1160 y Ft(?)1432 1148 y Fz(,)h Fy(p)1515 1160 y Ft(?)1573 1148 y Fz(are)g(strong)h(solutions)g (of)g(\(2.14\))i(and)d(\(2.15\))j(and)d(satisfy)g(the)g(boundary)-9 1247 y(conditions.)39 b(It)24 b(only)g(remains)g(to)g(sho)n(w)h(that)e Fy(\021)1430 1259 y Fu(0)p Ft(;?)1545 1247 y Fz(and)h Fy(p)1733 1259 y Fu(0)p Ft(;?)1848 1247 y Fz(are)f(strong)j(solution)e(of)h(\(2.14\))h (and)e(\(2.15\))i(with)e Fy(h)3431 1259 y Fu(1)3468 1247 y Fy(;)14 b(h)3553 1259 y Fu(2)3616 1247 y Fv(=)26 b(0)p Fz(.)-9 1347 y(Since)17 b Fs(j)6 b Fv(^)-48 b Fy(\021)261 1359 y Fu(0)p Ft(;?)352 1347 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)c Fv(+)g Fs(j)d Fv(^)-49 b Fy(p)719 1359 y Fu(0)p Ft(;?)810 1347 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fv(\(1)10 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))1421 1317 y Fu(1)p Fq(\000)p Ft(\013)1553 1347 y Fy(e)1592 1317 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1915 1347 y Fz(with)18 b Fy(\013)23 b(>)g Fv(3)p Fz(,)17 b Fy(\021)2369 1359 y Fu(0)p Ft(;?)2478 1347 y Fz(and)h Fy(p)2660 1359 y Fu(0)p Ft(;?)2768 1347 y Fz(together)g(with)f (their)g(deri)n(v)n(ati)n(v)o(es)-9 1447 y(with)23 b(respect)g(to)g Fy(y)j Fz(are)d(bounded)j(and)e(continuous)h(in)e Fy(x;)14 b(y)26 b Fz(.)37 b(This)23 b(sho)n(ws)i(that)j Fv(~)-48 b Fy(\021)2467 1459 y Ft(?)2529 1447 y Fz(and)31 b Fv(~)-49 b Fy(p)2717 1459 y Ft(?)2778 1447 y Fz(are)23 b(strong)i(solutions)f(of)g(\(2.14\))-9 1546 y(and)19 b(\(2.15\).)32 b(Finally)-6 b(,)20 b(since)e (\(2.12\)\261\(2.15\))23 b(and)d(the)e(boundary)k(conditions)d(\(2.3\))i(and) e(\(2.4\))h(are)e(equi)n(v)n(alent)h(to)f(the)h(original)-9 1646 y(Na)n(vier)n(\261Stok)o(es)k(equations,)882 1645 y Fv(~)876 1646 y Fr(u)929 1658 y Ft(E)1004 1646 y Fv(+)18 b Fr(v)1137 1658 y Ft(?)1196 1646 y Fz(and)29 b Fv(~)-49 b Fy(p)1382 1658 y Ft(E)1457 1646 y Fv(+)25 b(~)-49 b Fy(p)21 b Fz(is)g(the)g(solution)h(to)f (\(1.1\)\261\(1.4\).)115 1746 y(W)-7 b(e)23 b(no)n(w)g(turn)g(to)g(the)f (proof)j(of)e(Theorem)g(1.7.)35 b(The)23 b(follo)n(wing)h(corollary)f(e)o (xpress)g(the)f(result)g(of)h(Theorem)g(1.7)h(in)e(the)-9 1845 y(v)n(ariables)e Fy(u;)14 b(v)s Fz(.)-9 1987 y FA(Cor)n(ollary)22 b(4.7)43 b Fm(Ther)m(e)21 b(e)n(xists)g(a)g(constant)i Fy(C)k Fm(suc)o(h)22 b(that)f(for)h(all)f Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(2)h Fv(\012)2313 1999 y Fu(+)2368 1987 y Fm(,)d(the)g(functions)h Fy(u)2916 1999 y Ft(?)2954 1987 y Fy(;)14 b(v)3031 1999 y Ft(?)3090 1987 y Fm(satisfy)994 2068 y Fn(\014)994 2118 y(\014)994 2167 y(\014)994 2217 y(\014)994 2267 y(\014)1021 2213 y Fy(u)1069 2225 y Ft(E)1125 2213 y Fv(\()p Fy(x;)g(y)s Fv(\))19 b(+)f Fy(u)1467 2225 y Ft(?)1505 2213 y Fv(\()p Fy(x;)c(y)s Fv(\))19 b Fs(\000)f Fy(c)1866 2157 y(e)1905 2127 y Fq(\000)p Ft(y)1993 2102 y Fo(2)2025 2127 y Ft(=)p Fu(4)p Ft(x)p 1866 2194 268 4 v 1885 2211 a Fs(p)p 1954 2211 161 4 v 68 x Fv(4)p Fy(\031)24 b(x)2144 2068 y Fn(\014)2144 2118 y(\014)2144 2167 y(\014)2144 2217 y(\014)2144 2267 y(\014)2194 2213 y Fs(\024)f Fy(C)28 b(")20 b(x)2475 2179 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)p Fq(\000)s Fu(~)-36 b Ft(\015)2746 2213 y Fm(,)798 b Fz(\(4.23\))1464 2422 y Fs(j)p Fy(v)1527 2434 y Ft(E)1583 2422 y Fv(\()p Fy(x;)14 b(y)s Fv(\))19 b(+)f Fy(v)1917 2434 y Ft(?)1956 2422 y Fv(\()p Fy(x;)c(y)s Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b(")20 b(x)2475 2388 y Fq(\000)p Fu(1)2588 2422 y Fm(,)956 b Fz(\(4.24\))-9 2574 y Fm(wher)m(e)23 b Fv(~)-45 b Fy(\015)28 b Fv(=)23 b Fz(min)521 2506 y Fn(\010)579 2541 y Fu(1)p 579 2555 34 4 v 579 2602 a(4)622 2574 y Fy(;)669 2537 y Fu(1)p Fq(\000)p Ft(\015)789 2545 y Fo(1)p 669 2554 153 4 v 729 2602 a Fu(2)831 2506 y Fn(\011)901 2574 y Fm(and)g Fy(c)g Fv(=)1196 2507 y Fn(R)1251 2574 y Fy(dy)h(u)1407 2586 y Fu(+)p Ft(;)p Fu(1)1515 2574 y Fv(\()p Fy(y)s Fv(\))18 b Fs(\000)g Fy(f)9 b Fv(\(0\))p Fm(.)-9 2715 y(Pr)l(oof)o(.)31 b Fz(W)-7 b(e)21 b(note)h(\256rst)f(that)g Fy(c)i Fv(=)28 b(^)-47 b Fy(u)1028 2727 y Fu(+)1082 2715 y Fv(\(0\))p Fz(,)22 b(so)g(that)e(we)i(ha) n(v)o(e)360 2796 y Fn(\014)360 2846 y(\014)360 2896 y(\014)360 2946 y(\014)360 2995 y(\014)387 2941 y Fy(u)435 2953 y Ft(E)491 2941 y Fv(\()p Fy(x;)14 b(y)s Fv(\))19 b(+)f Fy(u)833 2953 y Ft(?)871 2941 y Fv(\()p Fy(x;)c(y)s Fv(\))19 b Fs(\000)f Fy(c)1232 2885 y(e)1271 2855 y Fq(\000)p Ft(y)1359 2830 y Fo(2)1391 2855 y Ft(=)p Fu(4)p Ft(x)p 1232 2922 268 4 v 1251 2939 a Fs(p)p 1320 2939 161 4 v 69 x Fv(4)p Fy(\031)24 b(x)1509 2796 y Fn(\014)1509 2846 y(\014)1509 2896 y(\014)1509 2946 y(\014)1509 2995 y(\014)1560 2941 y Fs(\024)1648 2828 y Fn(Z)1731 2941 y Fy(dk)1855 2846 y Fn(\014)1855 2896 y(\014)1855 2946 y(\014)1888 2941 y Fv(^)-48 b Fy(u)1930 2953 y Ft(E)1986 2941 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)k(^)-47 b Fy(u)2330 2953 y Ft(?)2368 2941 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b Fs(\000)k Fv(^)-47 b Fy(u)2712 2953 y Fu(+)2767 2941 y Fv(\(0\))p Fy(e)2912 2907 y Fq(\000)p Ft(k)3000 2882 y Fo(2)3032 2907 y Ft(x)3074 2846 y Fn(\014)3074 2896 y(\014)3074 2946 y(\014)1560 3191 y Fs(\024)1648 3078 y Fn(Z)1731 3191 y Fy(dk)24 b Fs(j)5 b Fv(^)-47 b Fy(u)1912 3203 y Ft(E)1967 3191 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)20 b Fv(+)2287 3078 y Fn(Z)2369 3191 y Fy(dk)2494 3096 y Fn(\014)2494 3146 y(\014)2494 3196 y(\014)2527 3191 y Fv(^)-48 b Fy(u)2569 3203 y Ft(?)2607 3191 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b Fs(\000)k Fv(^)-47 b Fy(u)2951 3203 y Fu(+)3006 3191 y Fv(\(0\))p Fy(e)3151 3157 y Fq(\000)p Ft(k)3239 3132 y Fo(2)3272 3157 y Ft(x)3313 3096 y Fn(\014)3313 3146 y(\014)3313 3196 y(\014)3378 3191 y Fy(:)-9 3381 y Fz(Ne)o(xt,)21 b(we)g(ha)n(v)o(e)999 3453 y Fn(Z)1082 3566 y Fy(dk)j Fs(j)5 b Fv(^)-47 b Fy(u)1263 3578 y Ft(E)1319 3566 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)k(")1787 3453 y Fn(Z)1869 3566 y Fy(dk)e(e)2019 3532 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fu(+)p Ft(\016)r Fu(\))2346 3566 y Fs(\024)e Fy(C)k(")21 b(x)2627 3532 y Fq(\000)p Fu(1)2740 3566 y Fz(,)-9 3755 y(and)418 3811 y Fn(\014)418 3861 y(\014)418 3911 y(\014)451 3907 y Fv(^)-47 b Fy(u)494 3919 y Ft(?)532 3907 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b Fs(\000)k Fv(^)-47 b Fy(u)876 3919 y Fu(+)930 3907 y Fv(\(0\))p Fy(e)1075 3873 y Fq(\000)p Ft(k)1163 3847 y Fo(2)1196 3873 y Ft(x)1238 3811 y Fn(\014)1238 3861 y(\014)1238 3911 y(\014)1289 3907 y Fs(\024)1376 3811 y Fn(\014)1376 3861 y(\014)1376 3911 y(\014)1409 3907 y Fv(^)g Fy(u)1452 3919 y Ft(?)1490 3907 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b Fs(\000)k Fv(^)-47 b Fy(u)1834 3919 y Fu(+)1889 3907 y Fv(\(0\))p Fy(e)2034 3873 y Fu(\003)2079 3881 y Fc(\000)2128 3873 y Fu(\()p Ft(k)q Fu(\))p Ft(x)2258 3811 y Fn(\014)2258 3861 y(\014)2258 3911 y(\014)2304 3907 y Fv(+)2387 3811 y Fn(\014)2387 3861 y(\014)2387 3911 y(\014)2420 3907 y Fv(^)g Fy(u)2463 3919 y Fu(+)2518 3907 y Fv(\(0\)\()p Fy(e)2695 3873 y Fu(\003)2740 3881 y Fc(\000)2789 3873 y Fu(\()p Ft(k)q Fu(\))p Ft(x)2938 3907 y Fs(\000)18 b Fy(e)3060 3873 y Fq(\000)p Ft(k)3148 3847 y Fo(2)3181 3873 y Ft(x)3223 3907 y Fv(\))3255 3811 y Fn(\014)3255 3861 y(\014)3255 3911 y(\014)3319 3907 y Fy(:)-9 4078 y Fz(Using)k(\(3.41\),)i(we)d(get)559 4157 y Fn(Z)642 4270 y Fy(dk)766 4175 y Fn(\014)766 4225 y(\014)766 4275 y(\014)799 4270 y Fv(^)-47 b Fy(u)842 4282 y Ft(?)879 4270 y Fv(\()p Fy(x;)14 b(k)s Fv(\))20 b Fs(\000)j Fv(^)-47 b Fy(u)1224 4282 y Fu(+)1278 4270 y Fv(\(0\))p Fy(e)1423 4236 y Fu(\003)1468 4244 y Fc(\000)1517 4236 y Fu(\()p Ft(k)q Fu(\))p Ft(x)1648 4175 y Fn(\014)1648 4225 y(\014)1648 4275 y(\014)1699 4270 y Fs(\024)22 b Fy(C)28 b(")21 b(x)1980 4236 y Fq(\000)s Fu(~)-36 b Ft(\015)2110 4157 y Fn(Z)2192 4270 y Fy(dk)25 b(w)2362 4282 y Ft(\013)2409 4270 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(\024)f Fy(C)k(")21 b(x)2908 4236 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)p Fq(\000)s Fu(~)-36 b Ft(\015)3178 4270 y Fy(:)-9 4470 y Fz(Finally)-6 b(,)21 b(since)g Fv(\003)522 4482 y Fq(\000)578 4470 y Fv(\()p Fy(k)s Fv(\))e(+)f Fy(k)836 4440 y Fu(2)896 4470 y Fs(\025)k Fv(0)f Fz(and)h Fs(j)5 b Fv(^)-47 b Fy(u)1261 4482 y Fu(+)1316 4470 y Fv(\(0\))p Fs(j)23 b(\024)g Fy(")p Fz(,)e(we)g(get)771 4551 y Fn(Z)854 4664 y Fy(dk)978 4569 y Fn(\014)978 4618 y(\014)978 4668 y(\014)1011 4664 y Fv(^)-48 b Fy(u)1053 4676 y Fu(+)1108 4664 y Fv(\(0\)\()p Fy(e)1285 4630 y Fu(\003)1330 4638 y Fc(\000)1380 4630 y Fu(\()p Ft(k)q Fu(\))p Ft(x)1528 4664 y Fs(\000)18 b Fy(e)1650 4630 y Fq(\000)p Ft(k)1738 4605 y Fo(2)1771 4630 y Ft(x)1813 4664 y Fv(\))1845 4569 y Fn(\014)1845 4618 y(\014)1845 4668 y(\014)1896 4664 y Fs(\024)23 b Fy(")2057 4551 y Fn(Z)2140 4664 y Fy(dk)2264 4572 y Fn(\020)2314 4664 y Fy(e)2353 4630 y Fu(\003)2398 4638 y Fc(\000)2447 4630 y Fu(\()p Ft(k)q Fu(\))p Ft(x)2596 4664 y Fs(\000)18 b Fy(e)2718 4630 y Fq(\000)p Ft(k)2806 4605 y Fo(2)2838 4630 y Ft(x)2880 4572 y Fn(\021)2967 4664 y Fy(:)-9 4864 y Fz(Performing)23 b(the)e(change)h(of)g(v)n(ariables)f Fv(\003)1262 4876 y Fq(\000)1317 4864 y Fv(\()p Fy(k)s Fv(\))j(=)e Fs(\000)p Fy(\030)1643 4834 y Fu(2)1680 4864 y Fz(,)g(we)f(get)720 4977 y Fn(Z)802 5090 y Fy(dk)926 4998 y Fn(\020)976 5090 y Fy(e)1015 5056 y Fu(\003)1060 5064 y Fc(\000)1109 5056 y Fu(\()p Ft(k)q Fu(\))p Ft(x)1258 5090 y Fs(\000)d Fy(e)1380 5056 y Fq(\000)p Ft(k)1468 5031 y Fo(2)1501 5056 y Ft(x)1542 4998 y Fn(\021)1615 5090 y Fv(=)1703 4977 y Fn(Z)1786 4998 y Fq(1)1749 5166 y(\0001)1849 5090 y Fy(d\030)26 b(e)1993 5056 y Fq(\000)p Ft(\030)2077 5031 y Fo(2)2109 5056 y Ft(x)2165 4948 y Fn( )2241 5034 y Fv(1)17 b Fs(\000)2384 4961 y Fn(p)p 2467 4961 221 4 v 73 x Fv(1)g(+)i Fy(\030)2650 5010 y Fu(2)2705 5034 y Fv(+)f(2)p Fy(\030)2870 5004 y Fu(2)p 2241 5071 667 4 v 2422 5088 a Fn(p)p 2505 5088 221 4 v 73 x Fv(1)g(+)g Fy(\030)2688 5137 y Fu(2)2917 4948 y Fn(!)3020 5090 y Fz(,)1615 5345 y Fs(\024)23 b Fv(2)1759 5232 y Fn(Z)1841 5252 y Fq(1)1804 5420 y(1)1890 5345 y Fy(d\030)i(\030)2034 5310 y Fu(2)2093 5345 y Fy(e)2132 5310 y Fq(\000)p Ft(\030)2216 5285 y Fo(2)2248 5310 y Ft(x)2313 5345 y Fs(\024)e Fy(C)k(x)2534 5310 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2714 5345 y Fz(,)-9 5560 y(because)21 b Fv(1)d Fs(\000)425 5487 y Fn(p)p 508 5487 V 73 x Fv(1)g(+)g Fy(\030)691 5536 y Fu(2)747 5560 y Fv(+)g(2)p Fy(\030)912 5530 y Fu(2)973 5560 y Fs(\024)23 b Fv(2)p Fy(\030)1143 5530 y Fu(2)1180 5560 y Fz(.)32 b(This)22 b(achie)n(v)o(es)e(the)i(proof)i (of)e(\(4.23\).)35 b(The)21 b(proof)j(of)f(inequality)e(\(4.24\))j(is)d (similar)-5 b(.)-9 5660 y(W)e(e)21 b(omit)g(the)g(details.)p 720 5660 43 43 v 1838 5909 a(16)p eop %%Page: 17 17 bop -9 145 a Fd(Ackno)o(wledgements)-9 302 y Fz(The)23 b(author)i(w)o(ould)g (lik)o(e)e(to)h(e)o(xpress)g(his)g(gratitude)g(to)g(Peter)f(W)m(ittwer)g(for) h(proposing)j(the)c(problem)i(and)f(for)h(his)f(constant)-9 401 y(help)e(during)i(the)d(elaboration)i(of)f(the)g(results.)33 b(Jean\261Pierre)22 b(Eckmann')-5 b(s)24 b(suggestions)f(and)g(advices)e (during)j(the)e(redaction)-9 501 y(of)e(the)f(paper)h(were)g(in)m(v)n (aluable.)30 b(Finally)-6 b(,)20 b(the)f(author)i(w)o(ould)g(also)e(lik)o(e)g (to)h(thank)g(Martin)f(Hairer)h(and)g(Jacques)g(Rougemont)-9 600 y(for)i(helpful)g(discussions.)-9 884 y FB(A)m(ppendix)31 b(A:)g(Regularity)h(of)e(the)h(F)m(ourier)g(T)-9 b(ransf)m(orm)-9 1085 y Fz(In)24 b(this)f(section,)h(we)g(sho)n(w)h(that)e Fy(@)1036 1097 y Ft(k)1094 1063 y Fv(^)1077 1085 y Fy(f)8 b Fv(\()p Fy(k)s Fv(\))27 b Fs(\024)e Fy(C)6 b Fv(\(1)20 b(+)g Fs(j)p Fy(k)s Fs(j)1689 1055 y Fq(\000)p Ft(\015)1783 1085 y Fv(\))k Fz(with)f Fv(0)j Fy(<)g(\015)31 b(<)25 b Fv(1)e Fz(if)h(the)f(function)i Fy(f)32 b Fz(approaches)25 b(suf)n(\256ciently)-9 1184 y(f)o(ast)c Fs(j)p Fy(y)s Fs(j)224 1154 y Ft(\015)t Fq(\000)p Fu(2)372 1184 y Fz(as)g Fy(y)26 b Fs(!)d(1)p Fz(:)-9 1350 y FA(Lemma)d(A.1)43 b Fm(Let)21 b Fy(f)31 b Fv(:)23 b Fr(R)g Fs(!)g Fr(R)e Fm(be)g(an)i(odd/e)o (ven)e(function.)32 b(Assume)22 b(that)f(ther)m(e)g(e)n(xist)f Fv(0)j Fy(<)f(\015)28 b(<)22 b Fv(1)f Fm(and)i Fy(c)3224 1362 y Fu(1)3284 1350 y Fs(2)h Fr(R)c Fm(suc)o(h)i(that)1628 1533 y Fz(sup)1619 1605 y Ft(y)r Fq(2)p Fl(R)1769 1533 y Fs(j)p Fy(y)s Fs(j)1859 1499 y Fu(2)p Fq(\000)p Ft(\015)1987 1533 y Fs(j)p Fy(f)9 b Fv(\()p Fy(y)s 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b(a)g(lo)o(garythmic)g(diver)m(g)o(ence)f(at)h Fy(k)31 b Fv(=)d(0)d Fm(if)34 b Fz(sup)2733 5471 y Ft(y)r Fq(2)p Fl(R)2870 5399 y Fv(\(1)20 b(+)g Fs(j)p Fy(y)s Fs(j)p Fv(\))3171 5369 y Fu(2)3209 5399 y Fs(j)p Fy(f)9 b Fv(\()p Fy(y)s Fv(\))p Fs(j)28 b Fy(<)g Fs(1)e Fm(and)-9 5503 y Fn(R)46 5524 y Fq(1)30 5600 y Fu(0)116 5570 y Fy(dy)e(y)282 5500 y Fn(\014)282 5549 y(\014)310 5570 y Fy(f)9 b Fv(\()p Fy(y)s Fv(\))18 b Fs(\000)g Fy(c)605 5582 y Fu(1)642 5570 y Fv(\(1)g(+)g Fy(y)s Fv(\))893 5540 y Fq(\000)p Fu(2)982 5500 y Fn(\014)982 5549 y(\014)1033 5570 y Fy(<)23 b Fs(1)p Fm(.)1838 5909 y Fz(17)p eop %%Page: 18 18 bop -9 145 a FB(A)m(ppendix)31 b(B:)f(Existence)h(of)f(Euler')-5 b(s)31 b(Solution)-9 332 y Fz(In)22 b(this)f(section,)g(we)g(pro)o(v)o(e)h (the)f(follo)n(wing)i(lemma,)e(from)h(which)g(Lemma)f(1.3)h(follo)n(ws.)-9 496 y FA(Lemma)e(B.1)42 b Fm(Let)26 b Fy(f)40 b Fs(2)30 b(A)819 508 y Ft(\013;\015)917 516 y Fk(E)966 508 y Ft(;\016)1022 496 y Fm(.)47 b(Ther)m(e)27 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2628 y(\014)1680 2678 y(\014)1730 2674 y Fs(\024)1828 2618 y Fy(e)1867 2588 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p 1828 2655 V 1955 2731 a Fs(j)p Fy(k)s Fs(j)2197 2529 y Fn(\014)2197 2579 y(\014)2197 2628 y(\014)2197 2678 y(\014)2197 2728 y(\014)2225 2561 y(Z)2308 2582 y Ft(k)2271 2750 y Fu(0)2327 2674 y Fy(d\030)26 b(@)2476 2686 y Ft(\030)2526 2582 y Fn(\020)2575 2674 y Fy(f)9 b Fv(\()p Fy(\030)t Fv(\))p Fy(e)2768 2640 y Fq(\000j)p Ft(k)q Fq(\000)p Ft(\030)r Fq(j)p Fu(\(1+)p Ft(\016)r Fu(\))3153 2582 y Fn(\021)3202 2529 y(\014)3202 2579 y(\014)3202 2628 y(\014)3202 2678 y(\014)3202 2728 y(\014)1730 2889 y Fs(\024)23 b Fy(C)6 b(")21 b Fv(\(1)d(+)g Fs(j)p Fy(k)s Fs(j)2210 2855 y Fq(\000)p Ft(\015)2297 2863 y Fk(E)2349 2889 y Fv(\))k Fy(e)2442 2855 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fu(+)p Ft(\016)r Fu(\))-9 3069 y Fz(for)29 b(some)g(constant)g Fy(C)6 b Fz(.)52 b(Since)29 b Fy(\015)1037 3081 y Ft(E)1126 3069 y Fy(<)j Fv(1)p Fz(,)e(this)e(is)g(in)h Fy(L)1704 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Ft(x)p Fq(\000)p Fu(1\))2305 3197 y Fn(\014)2305 3247 y(\014)2356 3268 y Fs(\024)k(j)p Fy(\026)p Fs(j)e(j)p Fy(k)s Fs(j)p Fy(e)2690 3238 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2996 3268 y Fz(,)g(and)g(if)g Fy(\026)j Fs(2)g Fr(R)p Fz(,)d(one)g(has)-9 3379 y Fs(j)p Fy(e)53 3349 y Ft(ik)q Fu(\()p Ft(y)r Fu(+)p Ft(\026)p Fu(\))314 3379 y Fs(\000)e Fy(e)436 3349 y Ft(ik)q(y)536 3379 y Fs(j)23 b(\024)g(j)p Fy(\026)p Fs(j)e(j)p Fy(k)s Fs(j)p Fz(.)31 b(This)21 b(means)g(that)287 3601 y Fs(j)p Fv(\011)375 3613 y Ft(E)431 3601 y Fv(\()p Fy(x)e Fv(+)f Fy(\026;)c(y)s Fv(\))k Fs(\000)g Fv(\011)941 3613 y Ft(E)997 3601 y Fv(\()p Fy(x;)c(y)s Fv(\))p Fs(j)19 b Fv(+)f Fs(j)p Fv(\011)1402 3613 y Ft(E)1458 3601 y Fv(\()p Fy(x;)c(y)22 b Fv(+)c Fy(\026)p Fv(\))g Fs(\000)g Fv(\011)1968 3613 y Ft(E)2024 3601 y Fv(\()p Fy(x;)c(y)s Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b Fs(j)p Fy(\026)p Fs(j)21 b Fy(")2607 3488 y Fn(Z)2690 3601 y Fy(dk)j(e)2839 3567 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p 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4029 y Ft(k)1793 4016 y Fv(^)1787 4017 y Fr(u)1840 4029 y Ft(E)1937 4017 y Fs(2)42 b Fy(L)2091 3987 y Fu(1)2127 4017 y Fv(\()p Fr(R)p Fy(;)14 b(dk)s Fv(\))p Fz(,)39 b(so)c(that)f Fr(u)2774 4029 y Ft(E)2871 4017 y Fs(\030)41 b(j)p Fy(y)s Fs(j)3067 3987 y Fq(\000)p Fu(1)3191 4017 y Fz(as)34 b Fy(y)44 b Fs(!)d(1)p Fz(,)d(and)-9 4050 y Fn(R)46 4117 y Fy(dk)30 b Fs(j)191 4116 y Fv(^)185 4117 y Fr(u)238 4129 y Ft(E)294 4117 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)32 b(\024)f Fy(C)6 b Fv(\()p Fy(x)23 b Fv(+)e Fy(\016)s Fv(\))964 4087 y Fq(\000)p Fu(1)1054 4117 y Fz(,)29 b(so)e(that)g Fr(u)1414 4129 y Ft(E)1502 4117 y Fs(!)k Fv(0)c Fz(as)g Fy(x)1830 4087 y Fu(2)1889 4117 y Fv(+)21 b Fy(y)2019 4087 y Fu(2)2088 4117 y Fs(!)31 b(1)p Fz(.)49 b(Finally)-6 b(,)29 b Fs(r)p Fv(\011)2774 4129 y Ft(E)2857 4117 y Fz(satis\256es)e(also)g (the)g(boundary)-9 4217 y(condition)22 b(at)e Fy(x)k Fv(=)f(1)p Fz(,)e(since)f(it)h(is)g(continuous)i(in)e Fy(x)p Fz(.)p 1602 4217 43 43 v -9 4316 a(Lemma)f(1.3)j(is)e(a)g(consequence)h(of)g(this)f (lemma.)30 b(Namely)-6 b(,)22 b(de\256ne)1505 4495 y Fv(~)1499 4496 y Fr(u)1552 4508 y Ft(E)1631 4496 y Fv(=)h Fr(u)1772 4508 y Fq(1)1861 4496 y Fv(+)18 b Fs(r)p Fv(\011)2078 4508 y Ft(E)2157 4496 y Fz(,)1517 4651 y Fv(~)-49 b Fy(p)1552 4663 y Ft(E)1631 4651 y Fv(=)23 b Fs(\000)1794 4595 y Fv(1)p 1794 4632 42 4 v 1794 4708 a(2)1859 4651 y(\()1897 4650 y(~)1891 4651 y Fr(u)1944 4663 y Ft(E)2019 4651 y Fs(\001)2066 4650 y Fv(~)2060 4651 y Fr(u)2113 4663 y Ft(E)2169 4651 y Fv(\))37 b Fy(:)-9 4854 y Fz(By)21 b(Lemma)g(B.1,)560 4853 y Fv(~)555 4854 y Fr(u)608 4866 y Ft(E)685 4854 y Fz(satis\256es)f(\(continuously)k(in)d Fy(x;)14 b(y)s Fz(\))1750 5033 y Fs(4)1831 5032 y Fv(~)1825 5033 y Fr(u)1878 5045 y Ft(E)1956 5033 y Fv(=)22 b(0)h Fz(,)1244 5158 y Fs(\000)p Fv(\()1347 5157 y(~)1341 5158 y Fr(u)1394 5170 y Ft(E)1468 5158 y Fs(\001)c(r)p Fv(\))1617 5157 y(~)1611 5158 y Fr(u)1664 5170 y Ft(E)1739 5158 y Fs(\000)f(r)7 b Fv(~)-49 b Fy(p)23 b Fv(=)f(0)h Fz(,)1400 5283 y(lim)1296 5340 y Ft(x)1334 5323 y Fo(2)1366 5340 y Fu(+)p Ft(y)1453 5323 y Fo(2)1485 5340 y Fq(!1)1637 5282 y Fv(~)1631 5283 y Fr(u)1684 5295 y Ft(E)1740 5283 y Fv(\()p Fy(x;)14 b(y)s Fv(\))24 b(=)e Fr(u)2096 5295 y Fq(1)2190 5283 y Fz(,)1492 5455 y(lim)1480 5506 y Ft(x)p Fq(!)p Fu(1)1637 5454 y Fv(~)1631 5455 y Fr(u)1684 5467 y Ft(E)1740 5455 y Fv(\()p Fy(x;)14 b(y)s Fv(\))24 b(=)e Fr(u)2096 5467 y Fq(1)2185 5455 y Fv(+)c Fr(u)2321 5467 y Ft(f)2364 5455 y Fv(\()p Fy(y)s Fv(\))24 b Fz(,)-9 5660 y(so)d(that)242 5659 y Fv(~)236 5660 y Fr(u)289 5672 y Ft(E)345 5660 y Fy(;)g Fv(~)-49 b Fy(p)424 5672 y Ft(E)501 5660 y Fz(is)21 b(a)g(strong)i(solution)f(of)g (\(1.1\)\261\(1.3\))j(in)c Fv(\012)1795 5672 y Fu(+)1851 5660 y Fz(.)1838 5909 y(18)p eop %%Page: 19 19 bop -9 145 a FB(A)m(ppendix)31 b(C:)g(Pr)n(oofs)f(f)m(or)h(Section)g(3)-9 332 y Fz(In)23 b(this)f(section,)h(we)f(gi)n(v)o(e)g(the)h(proofs)h(for)g (the)e(lemmas)g(and)h(propositions)h(of)g(Sections)e(3.2)i(and)f(3.3.)35 b(The)23 b(proofs)i(in)m(v)n(olv)o(e)-9 432 y(elementary)20 b(properties)i(of)g(the)f(functions)i Fv(\003)1363 444 y Fq(\006)1419 432 y Fz(.)30 b(These)22 b(properties)g(are)f(gi)n(v)o(en)g(in)h(the)f(follo) n(wing)h(subsection.)-9 672 y Fd(C.1)102 b(Pr)n(eliminary)27 b(Bounds)-9 829 y FA(Lemma)20 b(C.1)43 b Fm(F)-9 b(or)22 b(all)f Fy(\015)27 b(>)c Fv(0)p Fm(,)e Fy(\013)i Fs(\025)g Fv(0)p Fm(,)e(ther)m(e)g (e)n(xists)f(a)i(constant)g Fy(C)27 b Fm(suc)o(h)22 b(that)g(for)f(all)g Fy(x)i Fs(\025)g Fv(1)e Fm(and)h Fv(0)h Fs(\024)g(j)p Fy(\030)t Fs(j)g(\024)g(j)p Fy(k)s Fs(j)p Fm(,)e(we)g(have)1803 1056 y Fv(0)h Fs(\024)h Fv(\003)2013 1068 y Fq(\000)2069 1056 y Fv(\()p Fy(k)s Fv(\))18 b(+)g Fs(j)p Fy(k)s Fs(j)23 b(\024)2493 1000 y(j)p Fv(\003)2574 1012 y Fq(\000)2630 1000 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p 2493 1037 271 4 v 2582 1113 a(j)p Fy(k)s Fs(j)2796 1056 y Fm(,)776 b Fz(\(C.1\))998 1175 y Fn(\022)1141 1236 y Fs(j)p Fy(k)s Fs(j)p 1069 1273 235 4 v 1069 1349 a Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1314 1175 y Fn(\023)1375 1192 y Ft(\015)1432 1292 y Fy(e)1471 1258 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(4)1867 1292 y Fs(\024)1977 1236 y Fy(C)p 1965 1273 91 4 v 1965 1349 a(x)2012 1325 y Ft(\015)2088 1292 y Fm(,)1484 b Fz(\(C.2\))943 1412 y Fn(\022)1086 1473 y Fs(j)p Fy(k)s Fs(j)p 1014 1510 235 4 v 1014 1586 a Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1259 1412 y Fn(\023)1320 1429 y Ft(\015)1377 1529 y Fy(e)1416 1495 y Fu(\003)1461 1503 y Fc(\000)1510 1495 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(4)1867 1529 y Fs(\024)2011 1473 y Fy(C)p 1965 1510 158 4 v 1965 1587 a(x)2012 1563 y Ft(\015)t(=)p Fu(2)2155 1529 y Fm(,)1417 b Fz(\(C.3\))1446 1773 y Fs(j)p Fy(k)s Fs(j)1538 1739 y Ft(\015)1581 1773 y Fy(e)1620 1739 y Fu(\003)1665 1747 y Fc(\000)1714 1739 y Fu(\()p Ft(k)q Fu(\))p Ft(x)1867 1773 y Fs(\024)23 b Fy(C)2034 1656 y Fn(\022)2105 1652 y Fs(p)p 2174 1652 191 4 v 65 x Fv(1)18 b(+)g Fy(x)p 2105 1754 260 4 v 2211 1830 a(x)2375 1656 y Fn(\023)2436 1666 y Ft(\015)2515 1773 y Fm(,)1057 b Fz(\(C.4\))1367 1959 y Fy(e)1406 1929 y Fu(\003)1451 1937 y Fc(\000)1500 1929 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(4)p 1367 1996 468 4 v 1427 2072 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))1726 2048 y Ft(\013)1867 2015 y Fs(\024)2164 1959 y Fy(C)p 1965 1996 464 4 v 1965 2073 a Fv(\(1)g(+)g Fs(j)p Fy(k)s Fs(j)2232 2013 y(p)p 2301 2013 48 4 v 60 x Fy(x)q Fv(\))2381 2049 y Ft(\013)2461 2015 y Fm(,)1111 b Fz(\(C.5\))1422 2200 y Fy(e)1461 2170 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(4)p 1422 2237 413 4 v 1455 2314 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))1754 2290 y Ft(\013)1867 2257 y Fs(\024)2164 2200 y Fy(C)p 1965 2237 464 4 v 1965 2314 a Fv(\(1)g(+)g Fs(j)p Fy(k)s Fs(j)2232 2254 y(p)p 2301 2254 48 4 v 60 x Fy(x)q Fv(\))2381 2290 y Ft(\013)2461 2257 y Fm(,)1111 b Fz(\(C.6\))1177 2430 y Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1477 2400 y Ft(\013)p 1177 2467 347 4 v 1180 2543 a Fv(\(1)g(+)g Fs(j)p Fy(\030)t Fs(j)p Fv(\))1473 2519 y Ft(\013)1534 2486 y Fy(e)1573 2452 y Fu(\003)1618 2460 y Fc(\000)1667 2452 y Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))1867 2486 y Fs(\024)23 b Fy(C)k(e)2080 2452 y Fu(\003)2125 2460 y Fc(\000)2175 2452 y Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))p Ft(=)p Fu(2)2442 2486 y Fy(:)1128 b Fz(\(C.7\))-9 2710 y Fm(Pr)l(oof)o(.)31 b Fz(Inequalities)21 b(\(C.1\))h(and)g(\(C.2\))g(are)f(elementary)-6 b(.)31 b(Inequality)22 b(\(C.3\))g(follo)n(ws)g(from)1276 2843 y Fn(\022)1418 2904 y Fs(j)p Fy(k)s Fs(j)p 1347 2941 235 4 v 1347 3017 a Fv(1)c(+)g Fs(j)p Fy(k)s Fs(j)1592 2843 y Fn(\023)1653 2860 y Ft(\015)1718 2960 y Fs(\024)1806 2843 y Fn(\022)1949 2904 y Fs(j)p Fv(\003)2030 2916 y Fq(\000)2085 2904 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p 1877 2941 414 4 v 1877 3017 a Fv(1)g(+)g Fs(j)p Fv(\003)2101 3029 y Fq(\000)2157 3017 y Fv(\()p Fy(k)s Fv(\))p Fs(j)2300 2843 y Fn(\023)2361 2860 y Ft(\015)t(=)p Fu(2)-9 3205 y Fz(and)j(\(C.2\))h(with)f Fs(j)p Fy(k)s Fs(j)g Fz(replaced)g(by)h Fs(j)p Fv(\003)1113 3217 y Fq(\000)1169 3205 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fz(.)31 b(F)o(or)22 b(inequality)f(\(C.4\),)h(since)f Fs(j)p Fy(k)s Fs(j)i(\024)2471 3134 y Fn(p)p 2554 3134 749 4 v 71 x Fs(j)p Fv(\003)2635 3217 y Fq(\000)2690 3205 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fv(\(1)c(+)f Fs(j)p Fv(\003)3080 3217 y Fq(\000)3136 3205 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fv(\))q Fz(,)j(we)g(ha)n(v)o(e)h(that) 642 3465 y Fs(j)p Fy(k)s Fs(j)734 3431 y Ft(\015)790 3348 y Fn(\022)967 3409 y Fy(x)p 861 3446 260 4 v 861 3463 a Fs(p)p 930 3463 191 4 v 65 x Fv(1)c(+)g Fy(x)1131 3348 y Fn(\023)1192 3365 y Ft(\015)1248 3465 y Fy(e)1287 3431 y Fu(\003)1332 3439 y Fc(\000)1381 3431 y Fu(\()p Ft(k)q Fu(\))p Ft(x)1535 3465 y Fs(\024)1622 3348 y Fn(\022)1746 3409 y Fy(x)1793 3379 y Fu(2)p 1694 3446 V 1694 3522 a Fv(1)f(+)h Fy(x)1894 3465 y Fs(j)p Fv(\003)1975 3477 y Fq(\000)2031 3465 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fv(\(1)g(+)g Fs(j)p Fv(\003)2420 3477 y Fq(\000)2476 3465 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fv(\))2641 3348 y Fn(\023)2703 3362 y Ft(\015)t(=)p Fu(2)2827 3465 y Fy(e)2866 3431 y Fu(\003)2911 3439 y Fc(\000)2960 3431 y Fu(\()p Ft(k)q Fu(\))p Ft(x)1535 3672 y Fs(\024)1622 3605 y Fn(\000)1660 3672 y Fs(j)p Fv(\003)1741 3684 y Fq(\000)1797 3672 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fy(x)i Fv(+)e Fs(j)p Fv(\003)2161 3684 y Fq(\000)2216 3672 y Fv(\()p Fy(k)s Fv(\))p Fs(j)2349 3638 y Fu(2)2387 3672 y Fy(x)2434 3638 y Fu(2)2472 3605 y Fn(\001)2510 3621 y Ft(\015)t(=)p Fu(2)2633 3672 y Fy(e)2672 3638 y Fu(\003)2717 3646 y Fc(\000)2767 3638 y Fu(\()p Ft(k)q Fu(\))p Ft(x)2920 3672 y Fs(\024)23 b Fy(C)29 b(:)-9 3851 y Fz(Inequalities)k(\(C.5\))i(and)g(\(C.6\))g(follo)n(w)g(from)g (\(C.2\))g(and)g(\(C.3\).)71 b(Finally)-6 b(,)38 b(inequality)c(\(C.7\))h (follo)n(ws)g(by)g(considering)-9 3951 y(separately)20 b(the)h(case)g Fv(0)h Fs(\024)h(j)p Fy(\030)t Fs(j)g(\024)g(j)p Fy(k)s Fs(j)p Fy(=)p Fv(2)d Fz(and)i Fs(j)p Fy(k)s Fs(j)p Fy(=)p Fv(2)g Fs(\024)h(j)p Fy(\030)t Fs(j)g(\024)g(j)p Fy(k)s Fs(j)p Fz(.)31 b(In)22 b(the)f(\256rst)g (case,)g(we)g(ha)n(v)o(e)645 4122 y Fv(\(1)d(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))944 4092 y Ft(\013)p 645 4159 347 4 v 648 4235 a Fv(\(1)g(+)g Fs(j)p Fy(\030)t Fs(j)p Fv(\))941 4211 y Ft(\013)1002 4178 y Fy(e)1041 4144 y Fu(\003)1086 4152 y Fc(\000)1135 4144 y Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))1335 4178 y Fs(\024)23 b Fy(e)1462 4144 y Fu(\003)1507 4152 y Fc(\000)1556 4144 y Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))p Ft(=)p Fu(2)1800 4178 y Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2099 4144 y Ft(\013)2147 4178 y Fy(e)2186 4144 y Fu(\003)2231 4152 y Fc(\000)2280 4144 y Fu(\()p Ft(k)q(=)p Fu(2\))p Ft(=)p Fu(2)2530 4178 y Fs(\024)k Fy(C)28 b(e)2743 4144 y Fu(\003)2788 4152 y Fc(\000)2837 4144 y Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))p Ft(=)p Fu(2)3104 4178 y Fz(,)-9 4402 y(and)21 b(in)h(the)f(second)h(case,)f (we)g(ha)n(v)o(e)773 4564 y Fv(\(1)d(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))1072 4534 y Ft(\013)p 773 4601 V 776 4677 a Fv(\(1)g(+)g Fs(j)p Fy(\030)t Fs(j)p Fv(\))1069 4653 y Ft(\013)1130 4620 y Fy(e)1169 4586 y Fu(\003)1214 4594 y Fc(\000)1263 4586 y Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))1463 4620 y Fs(\024)k Fy(e)1589 4586 y Fu(\003)1634 4594 y Fc(\000)1684 4586 y Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))p Ft(=)p Fu(2)1979 4564 y Fv(\(1)c(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2278 4534 y Ft(\013)p 1937 4601 430 4 v 1937 4677 a Fv(\(1)h(+)f Fs(j)p Fy(k)s Fs(j)p Fy(=)p Fv(2\))2321 4653 y Ft(\013)2400 4620 y Fs(\024)23 b Fy(C)k(e)2613 4586 y Fu(\003)2658 4594 y Fc(\000)2708 4586 y Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))p Ft(=)p Fu(2)2975 4620 y Fy(:)-9 4843 y Fz(This)21 b(completes)g(the)g(proof.)p 911 4843 43 43 v -9 5023 a FA(Lemma)f(C.2)43 b Fm(Let)22 b Fy(h)i Fv(:)h Fr(R)f Fs(!)h Fr(R)p Fm(.)34 b(Assume)23 b(that)g(ther)m(e)f(e) n(xist)f(constants)j Fy(H)31 b(>)24 b Fv(0)p Fm(,)f Fy(\013)i(>)f Fv(0)e Fm(and)i Fv(0)g Fs(\024)h Fy(\015)k(<)24 b Fv(1)e Fm(suc)o(h)h(that)g (for)f(all)-9 5122 y Fy(k)j Fs(2)f Fr(R)p Fm(,)1302 5334 y Fs(j)p Fy(h)p Fv(\()p Fy(k)s Fv(\))p Fs(j)g(\024)1763 5278 y Fy(H)p 1627 5315 347 4 v 1627 5391 a Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1927 5367 y Ft(\013)2007 5334 y Fm(,)1231 5581 y Fs(j)p Fy(D)r(h)p Fv(\()p Fy(k)s Fv(\))p Fs(j)24 b(\024)1763 5525 y Fy(H)p 1627 5562 V 1627 5638 a Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1927 5614 y Ft(\013)1998 5464 y Fn(\022)2140 5525 y Fs(j)p Fy(k)s Fs(j)p 2069 5562 235 4 v 2069 5638 a Fv(1)g(+)g Fs(j)p Fy(k)s Fs(j)2314 5464 y Fn(\023)2375 5481 y Fq(\000)p Ft(\015)2506 5581 y Fy(:)1838 5909 y Fz(19)p eop %%Page: 20 20 bop -9 145 a Fm(Then)22 b(ther)m(e)e(e)n(xists)h(a)g(constant)i Fy(C)k Fm(suc)o(h)22 b(that)f(for)h(all)f Fy(k)26 b Fs(2)d Fr(R)p Fm(,)924 286 y Fn(\014)924 336 y(\014)924 386 y(\014)951 382 y Fy(h)p Fv(\()p Fy(k)s Fv(\))c Fs(\000)f Fy(e)1250 348 y Fu(\003)1295 356 y Fc(\000)1344 348 y Fu(\()p Ft(k)q Fu(\))1437 382 y Fy(h)p Fv(\(0\))1591 286 y Fn(\014)1591 336 y(\014)1591 386 y(\014)1642 382 y Fs(\024)k Fy(C)28 b(H)2054 326 y Fv(1)p 1902 363 347 4 v 1902 439 a(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2201 415 y Ft(\013)2272 265 y Fn(\022)2415 326 y Fs(j)p Fy(k)s Fs(j)p 2343 363 235 4 v 2343 439 a Fv(1)g(+)g Fs(j)p Fy(k)s Fs(j)2588 265 y Fn(\023)2649 282 y Fu(1)p Fq(\000)p Ft(\015)2814 382 y Fy(:)756 b Fz(\(C.8\))-9 606 y Fm(Pr)l(oof)o(.)31 b Fz(Since)21 b Fv(\003)499 618 y Fq(\000)555 606 y Fv(\(0\))i(=)f(0)p Fz(,)f(we)h(ha)n(v)o(e)f(that)448 759 y Fn(\014)448 809 y(\014)448 858 y(\014)476 854 y Fy(h)p Fv(\()p Fy(k)s Fv(\))d Fs(\000)g Fy(e)774 820 y Fu(\003)819 828 y Fc(\000)868 820 y Fu(\()p Ft(k)q Fu(\))961 854 y Fy(h)p Fv(\(0\))1115 759 y Fn(\014)1115 809 y(\014)1115 858 y(\014)1166 854 y Fs(\024)1253 709 y Fn(\014)1253 759 y(\014)1253 809 y(\014)1253 858 y(\014)1253 908 y(\014)1281 741 y(Z)1364 762 y Ft(k)1327 930 y Fu(0)1383 854 y Fy(d\030)26 b(@)1532 866 y Ft(\030)1582 762 y Fn(\020)1632 854 y Fy(e)1671 820 y Fu(\003)1716 828 y Fc(\000)1765 820 y Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))1942 854 y Fy(h)p Fv(\()p Fy(\030)t Fv(\))2094 762 y Fn(\021)2144 709 y(\014)2144 759 y(\014)2144 809 y(\014)2144 858 y(\014)2144 908 y(\014)1166 1137 y Fs(\024)c Fy(C)1354 1024 y Fn(Z)1437 1044 y Fq(j)p Ft(k)q Fq(j)1400 1212 y Fu(0)1496 1137 y Fy(d\030)j(e)1639 1102 y Fu(\003)1684 1110 y Fc(\000)1733 1102 y Fu(\()p Fq(j)p Ft(k)q Fq(j\000)p Ft(\030)r Fu(\))1963 995 y Fn( )2171 1080 y Fv(2\()p Fs(j)p Fy(k)s Fs(j)18 b(\000)g Fy(\030)t Fv(\))p 2039 1117 603 4 v 2039 1134 a Fn(p)p 2122 1134 520 4 v 71 x Fv(1)g(+)g(4\()p Fs(j)p Fy(k)s Fs(j)g(\000)g Fy(\030)t Fv(\))2604 1181 y Fu(2)2652 1137 y Fs(j)p Fy(h)p Fv(\()p Fy(k)s Fv(\))p Fs(j)h Fv(+)f Fs(j)p Fy(D)r(h)p Fv(\()p Fy(k)s Fv(\))p Fs(j)3233 995 y Fn(!)1166 1419 y Fs(\024)1356 1363 y Fy(C)27 b(H)p 1263 1400 347 4 v 1263 1476 a Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1563 1452 y Ft(\013)1634 1306 y Fn(Z)1717 1326 y Fq(j)p Ft(k)q Fq(j)1680 1495 y Fu(0)1776 1419 y Fy(d\030)25 b(e)1919 1385 y Fu(\003)1964 1393 y Fc(\000)2014 1385 y Fu(\()p Fq(j)p Ft(k)q Fq(j\000)p Ft(\030)r Fu(\))p Ft(=)p Fu(2)2311 1277 y Fn( )2376 1419 y Fv(1)18 b(+)2519 1302 y Fn(\022)2662 1363 y Fs(j)p Fy(\030)t Fs(j)p 2591 1400 230 4 v 2591 1476 a Fv(1)g(+)g Fs(j)p Fy(\030)t Fs(j)2830 1302 y Fn(\023)2891 1319 y Fq(\000)p Ft(\015)2985 1277 y Fn(!)3088 1419 y Fy(:)-9 1668 y Fz(No)n(w)-6 b(,)23 b(for)f Fs(j)p Fy(k)s Fs(j)h(\024)g Fv(1)p Fz(,)e(we)g(ha)n(v)o(e)770 1803 y Fn(Z)853 1824 y Fq(j)p Ft(k)q Fq(j)816 1992 y Fu(0)912 1916 y Fy(d\030)26 b(e)1056 1882 y Fu(\003)1101 1890 y Fc(\000)1150 1882 y Fu(\()p Fq(j)p Ft(k)q Fq(j\000)p Ft(\030)r Fu(\))p Ft(=)p Fu(2)1447 1774 y Fn( )1513 1916 y Fv(1)18 b(+)1656 1799 y Fn(\022)1798 1860 y Fs(j)p Fy(\030)t Fs(j)p 1727 1897 V 1727 1973 a Fv(1)g(+)g Fs(j)p Fy(\030)t Fs(j)1966 1799 y Fn(\023)2027 1816 y Fq(\000)p Ft(\015)2122 1774 y Fn(!)2210 1916 y Fs(\024)23 b Fy(C)k Fs(j)p Fy(k)s Fs(j)2476 1882 y Fu(1)p Fq(\000)p Ft(\015)2604 1916 y Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2871 1882 y Ft(\015)2914 1916 y Fv(\))23 b Fz(,)-9 2165 y(and)e(for)i Fs(j)p Fy(k)s Fs(j)g(\025)f Fv(1)p Fz(,)f(we)h(ha)n(v)o(e)1062 2300 y Fn(Z)1145 2321 y Fq(j)p Ft(k)q Fq(j)1108 2489 y Fu(0)1203 2413 y Fy(d\030)k(e)1347 2379 y Fu(\003)1392 2387 y Fc(\000)1441 2379 y Fu(\()p Fq(j)p Ft(k)q Fq(j\000)p Ft(\030)r Fu(\))p Ft(=)p Fu(2)1738 2272 y Fn( )1804 2413 y Fv(1)18 b(+)1947 2296 y Fn(\022)2090 2357 y Fs(j)p Fy(\030)t Fs(j)p 2018 2394 V 2018 2470 a Fv(1)g(+)g Fs(j)p Fy(\030)t Fs(j)2257 2296 y Fn(\023)2318 2314 y Fq(\000)p Ft(\015)2413 2272 y Fn(!)2502 2413 y Fs(\024)k Fy(C)30 b Fz(,)-9 2667 y(because)d(the)h(inte)o(gral)f(is)g(con)m(v)o(er)n(gent.)51 b(If)29 b Fs(j)p Fy(k)s Fs(j)i(\024)h Fv(1)p Fz(,)d Fs(j)p Fy(k)s Fs(j)1720 2637 y Fu(1)p Fq(\000)p Ft(\015)1847 2667 y Fv(\(1)22 b(+)g Fs(j)p Fy(k)s Fs(j)2122 2637 y Ft(\015)2164 2667 y Fv(\))32 b Fs(\024)g Fy(C)6 b Fs(j)p Fy(k)s Fs(j)2482 2637 y Fu(1)p Fq(\000)p Ft(\015)2609 2667 y Fv(\(1)22 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2916 2637 y Ft(\015)t Fq(\000)p Fu(1)3071 2667 y Fz(and)29 b(the)e(bound)j(\(C.8\))-9 2767 y(follo)n(ws)21 b(for)h(this)f(case.)30 b(If)22 b Fs(j)p Fy(k)s Fs(j)h(\025)f Fv(1)p Fz(,)f(then)g Fv(1)i Fs(\024)f(j)p Fy(k)s Fs(j)1501 2737 y Fu(1)p Fq(\000)p Ft(\015)1629 2767 y Fv(\(1)c(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1927 2737 y Ft(\015)t Fq(\000)p Fu(1)2076 2767 y Fz(and)k(the)g(bound)j(\(C.8\))e(follo)n(ws)f(for)h(this)f (other)h(case.)p 3728 2767 43 43 v -9 2867 a(The)f(follo)n(wing)h(functions)h (enter)e(the)g(de\256nition)h(of)27 b Fv(^)-47 b Fy(u)1634 2879 y Fu(0)1692 2867 y Fz(and)25 b Fv(^)-45 b Fy(v)1876 2879 y Fu(0)1914 2867 y Fz(,)704 3086 y Fy(P)757 3098 y Fu(1)p Ft(;)p Fu(1)847 3086 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e Fy(e)1191 3052 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1516 3086 y Fv(+)1772 3030 y Fs(j)p Fy(k)s Fs(j)p 1609 3067 418 4 v 1609 3143 a Fv(\003)1667 3155 y Fq(\000)1723 3143 y Fv(\()p Fy(k)s Fv(\))d(+)f Fs(j)p Fy(k)s Fs(j)2051 2994 y Fn(\020)2100 3086 y Fy(e)2139 3052 y Fu(\003)2184 3060 y Fc(\000)2233 3052 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2519 3086 y Fs(\000)g Fy(e)2641 3052 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2947 2994 y Fn(\021)3034 3086 y Fz(,)704 3316 y Fy(P)757 3328 y Fu(1)p Ft(;)p Fu(2)847 3316 y Fv(\()p Fy(x;)c(k)s Fv(\))24 b(=)1325 3259 y Fs(j)p Fy(k)s Fs(j)p 1162 3297 V 1162 3373 a Fv(\003)1220 3385 y Fq(\000)1276 3373 y Fv(\()p Fy(k)s Fv(\))19 b(+)f Fs(j)p Fy(k)s Fs(j)1604 3223 y Fn(\020)1653 3316 y Fy(e)1692 3281 y Fu(\003)1737 3289 y Fc(\000)1787 3281 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2072 3316 y Fs(\000)g Fy(e)2194 3281 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2501 3223 y Fn(\021)2587 3316 y Fz(,)704 3545 y Fy(P)757 3557 y Fu(2)p Ft(;)p Fu(1)847 3545 y Fv(\()p Fy(x;)c(k)s Fv(\))24 b(=)1259 3489 y(\003)1317 3501 y Fq(\000)1373 3489 y Fv(\()p Fy(k)s Fv(\))p 1162 3526 V 1162 3602 a(\003)1220 3614 y Fq(\000)1276 3602 y Fv(\()p Fy(k)s Fv(\))19 b(+)f Fs(j)p Fy(k)s Fs(j)1604 3453 y Fn(\020)1653 3545 y Fy(e)1692 3511 y Fu(\003)1737 3519 y Fc(\000)1787 3511 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2072 3545 y Fs(\000)g Fy(e)2194 3511 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2501 3453 y Fn(\021)2587 3545 y Fz(,)704 3774 y Fy(P)757 3786 y Fu(2)p Ft(;)p Fu(2)847 3774 y Fv(\()p Fy(x;)c(k)s Fv(\))24 b(=)e Fy(e)1191 3740 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1516 3774 y Fv(+)1706 3718 y(\003)1764 3730 y Fq(\000)1820 3718 y Fv(\()p Fy(k)s Fv(\))p 1609 3755 V 1609 3831 a(\003)1667 3843 y Fq(\000)1723 3831 y Fv(\()p Fy(k)s Fv(\))d(+)f Fs(j)p Fy(k)s Fs(j)2051 3682 y Fn(\020)2100 3774 y Fy(e)2139 3740 y Fu(\003)2184 3748 y Fc(\000)2233 3740 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2519 3774 y Fs(\000)g Fy(e)2641 3740 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2947 3682 y Fn(\021)3034 3774 y Fy(:)-9 3992 y Fz(W)-7 b(e)21 b(summarize)g(their)g(properties)h(in)f (the)g(follo)n(wing)i(lemma.)-9 4151 y FA(Lemma)d(C.3)43 b Fm(F)-9 b(or)24 b(all)g Fv(0)i Fs(\024)h Fy(\026)g Fs(\024)f Fv(1)e Fm(and)h(for)f(all)g Fy(i;)14 b(j)31 b Fv(=)c(1)p Fy(;)14 b Fv(2)p Fm(,)24 b(ther)m(e)f(e)n(xists)g(a)i(constant)g Fy(C)30 b Fm(suc)o(h)25 b(that)f(for)g(all)g Fv(\()p Fy(x;)14 b(k)s Fv(\))28 b Fs(2)f Fv(\012)3694 4163 y Fu(+)3749 4151 y Fm(,)-9 4251 y Fs(j)p Fy(\033)s Fs(j)c(\024)g Fv(1)e Fm(with)f Fy(x)f Fv(+)f Fy(\033)27 b Fs(\025)22 b Fv(1)p Fm(,)f(one)h(has)628 4424 y Fs(j)p Fy(P)704 4436 y Fu(1)p Ft(;)p Fu(1)795 4424 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fs(j)p Fy(P)1190 4436 y Fu(1)p Ft(;)p Fu(2)1280 4424 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)6 b(e)1713 4390 y Fu(\003)1758 4398 y Fc(\000)1807 4390 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(2)2165 4424 y Fm(,)1114 4600 y Fs(j)p Fy(P)1190 4612 y Fu(2)p Ft(;)p Fu(1)1280 4600 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)1756 4544 y Fs(j)p Fy(k)s Fs(j)p 1684 4581 235 4 v 1684 4657 a Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1929 4600 y Fy(e)1968 4566 y Fu(\003)2013 4574 y Fc(\000)2062 4566 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(2)2420 4600 y Fm(,)1114 4829 y Fs(j)p Fy(P)1190 4841 y Fu(2)p Ft(;)p Fu(2)1280 4829 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)6 b(e)1713 4795 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2038 4829 y Fv(+)18 b Fy(C)2268 4773 y Fs(j)p Fy(k)s Fs(j)p 2196 4810 V 2196 4886 a Fv(1)g(+)g Fs(j)p Fy(k)s Fs(j)2441 4829 y Fy(e)2480 4795 y Fu(\003)2525 4803 y Fc(\000)2574 4795 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(2)2932 4829 y Fm(,)551 5002 y Fs(j)p Fy(P)627 5014 y Ft(i;j)705 5002 y Fv(\()p Fy(x)i Fv(+)e Fy(\033)n(;)c(k)s Fv(\))19 b Fs(\000)f Fy(P)1202 5014 y Ft(i;j)1280 5002 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)6 b Fs(j)p Fy(\033)s Fs(j)1770 4968 y Ft(\026)1836 5002 y Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))2136 4968 y Ft(\026)2204 5002 y Fm(,)1043 5137 y Fs(j)p Fy(D)r(P)1190 5149 y Fu(1)p Ft(;)p Fu(1)1280 5137 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)1674 5073 y Fs(p)p 1743 5073 48 4 v 64 x Fy(x)q(e)1830 5103 y Fu(\003)1875 5111 y Fc(\000)1924 5103 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(2)2281 5137 y Fm(,)1043 5272 y Fs(j)p Fy(D)r(P)1190 5284 y Fu(1)p Ft(;)p Fu(2)1280 5272 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)6 b Fv(\()p Fy(x)19 b Fs(\000)f Fv(1\))p Fy(e)1968 5238 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2293 5272 y Fv(+)g Fy(C)2441 5208 y Fs(p)p 2510 5208 V 64 x Fy(x)q(e)2597 5238 y Fu(\003)2642 5246 y Fc(\000)2691 5238 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(2)3049 5272 y Fm(,)1043 5407 y Fs(j)p Fy(D)r(P)1190 5419 y Fu(2)p Ft(;)p Fu(1)1280 5407 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)6 b(e)1713 5373 y Fu(\003)1758 5381 y Fc(\000)1807 5373 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(2)2165 5407 y Fm(,)1043 5582 y Fs(j)p Fy(D)r(P)1190 5594 y Fu(2)p Ft(;)p Fu(2)1280 5582 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)6 b Fv(\()p Fy(x)19 b Fs(\000)f Fv(1\))p Fy(e)1968 5548 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2293 5582 y Fv(+)g Fy(C)2523 5526 y Fs(j)p Fy(k)s Fs(j)p 2451 5563 235 4 v 2451 5639 a Fv(1)g(+)g Fs(j)p Fy(k)s Fs(j)2696 5582 y Fy(e)2735 5548 y Fu(\003)2780 5556 y Fc(\000)2829 5548 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(2)3187 5582 y Fy(:)1838 5909 y Fz(20)p eop %%Page: 21 21 bop -9 145 a Fm(Pr)l(oof)o(.)31 b Fz(All)21 b(these)g(properties)h(are)f (elementary)-6 b(.)32 b(As)21 b(an)h(e)o(xample,)f(we)g(sho)n(w)i(the)e (inequalities)f(in)m(v)n(olving)j Fy(P)3266 157 y Fu(1)p Ft(;)p Fu(2)3357 145 y Fz(.)31 b(The)21 b(others)-9 244 y(follo)n(w)27 b(by)g(using)g(v)o(ery)g(similar)e(ar)n(guments.)48 b(By)26 b(Lemma)g(C.1,)i(we)e(ha)n(v)o(e)h Fv(0)i Fs(\024)h Fv(\003)2529 256 y Fq(\000)2584 244 y Fv(\()p Fy(k)s Fv(\))22 b(+)f Fs(j)p Fy(k)s Fs(j)30 b(\024)f(j)p Fy(k)s Fs(j)3110 214 y Fq(\000)p Fu(1)3199 244 y Fs(j)p Fv(\003)3280 256 y Fq(\000)3336 244 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fz(.)46 b(On)27 b(the)-9 344 y(other)21 b(hand,)i Fv(\003)452 356 y Fq(\000)508 344 y Fv(\()p Fy(k)s Fv(\))g Fs(\030)g(\000)p Fy(k)840 314 y Fu(2)898 344 y Fz(as)e Fy(k)26 b Fs(!)d Fv(0)d Fz(and)i Fv(\003)1429 356 y Fq(\000)1485 344 y Fv(\()p Fy(k)s Fv(\))i Fs(\030)e(\000j)p Fy(k)s Fs(j)f Fz(as)g Fy(k)26 b Fs(!)d(1)p Fz(.)31 b(Using)22 b(this,)f(we)h(get)914 551 y Fs(j)p Fy(P)990 563 y Fu(1)p Ft(;)p Fu(2)1080 551 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)k Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1809 456 y Fn(\014)1809 506 y(\014)1809 555 y(\014)1836 551 y Fy(e)1875 517 y Fu(\003)1920 525 y Fc(\000)1970 517 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2255 551 y Fs(\000)g Fy(e)2377 517 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2684 456 y Fn(\014)2684 506 y(\014)2684 555 y(\014)1321 734 y Fs(\024)23 b Fy(C)k(e)1534 700 y Fu(\003)1579 708 y Fc(\000)1629 700 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))1914 734 y Fv(+)18 b Fy(C)28 b Fs(j)p Fy(k)s Fs(j)p Fy(e)2215 700 y Fu(\003)2260 708 y Fc(\000)2309 700 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2590 639 y Fn(\014)2590 688 y(\014)2590 738 y(\014)2618 734 y Fv(1)18 b Fs(\000)g Fy(e)2800 700 y Fq(\000)p Fu(\(\003)2923 708 y Fc(\000)2972 700 y Fu(\()p Ft(k)q Fu(\)+)p Fq(j)p Ft(k)q Fq(j)p Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))3392 639 y Fn(\014)3392 688 y(\014)3392 738 y(\014)1321 899 y Fs(\024)23 b Fy(C)k(e)1534 865 y Fu(\003)1579 873 y Fc(\000)1629 865 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))1914 899 y Fv(+)18 b Fy(C)28 b Fs(j)p Fy(k)s Fs(j)p Fv(\(\003)2266 911 y Fq(\000)2322 899 y Fv(\()p Fy(k)s Fv(\))19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))j(\()p Fy(x)e Fs(\000)f Fv(1\))j Fy(e)2994 865 y Fu(\003)3039 873 y Fc(\000)3089 865 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))1321 1024 y Fs(\024)i Fy(C)k(e)1534 989 y Fu(\003)1579 997 y Fc(\000)1629 989 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))1914 1024 y Fv(+)18 b Fy(C)28 b Fs(j)p Fv(\003)2165 1036 y Fq(\000)2221 1024 y Fv(\()p Fy(k)s Fv(\))p Fs(j)21 b Fv(\()p Fy(x)e Fs(\000)f Fv(1\))k Fy(e)2691 989 y Fu(\003)2736 997 y Fc(\000)2785 989 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))1321 1148 y Fs(\024)h Fy(C)k(e)1534 1114 y Fu(\003)1579 1122 y Fc(\000)1629 1114 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(2)1986 1148 y Fz(,)327 1289 y Fs(j)p Fy(P)403 1301 y Fu(1)p Ft(;)p Fu(2)493 1289 y Fv(\()p Fy(x)20 b Fv(+)e Fy(\033)n(;)c(k)s Fv(\))19 b Fs(\000)f Fy(P)990 1301 y Fu(1)p Ft(;)p Fu(2)1080 1289 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)24 b(\024)1409 1196 y Fn(\020)1458 1289 y Fv(1)18 b Fs(\000)g Fy(e)1640 1254 y Fu(\003)1685 1262 y Fc(\000)1735 1254 y Fu(\()p Ft(k)q Fu(\))p Fq(j)p Ft(\033)r Fq(j)1907 1196 y Fn(\021)2054 1289 y Fz(sup)1971 1364 y Fu(\()p Ft(x;k)q Fu(\))p Fq(2)p Fu(\012)2209 1372 y Fo(+)2269 1289 y Fs(j)p Fy(P)2345 1301 y Fu(1)p Ft(;)p Fu(2)2436 1289 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)1381 1540 y Fv(+)1637 1483 y Fs(j)p Fy(k)s Fs(j)p 1474 1520 418 4 v 1474 1597 a Fv(\003)1532 1609 y Fq(\000)1588 1597 y Fv(\()p Fy(k)s Fv(\))19 b(+)f Fs(j)p Fy(k)s Fs(j)1902 1540 y Fy(e)1941 1505 y Fu(\003)1986 1513 y Fc(\000)2035 1505 y Fu(\()p Ft(k)q Fu(\))p Fq(j)p Ft(\033)r Fq(j)2221 1447 y Fn(\020)2271 1540 y Fv(1)g Fs(\000)g Fy(e)2453 1505 y Fq(\000)p Fu(\(\003)2576 1513 y Fc(\000)2625 1505 y Fu(\()p Ft(k)q Fu(\)+)p Fq(j)p Ft(k)q Fq(j)p Fu(\))p Fq(j)p Ft(\033)r Fq(j)2951 1447 y Fn(\021)1321 1728 y Fs(\024)23 b Fy(C)k Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1795 1694 y Ft(\026)1839 1728 y Fs(j)p Fy(\033)s Fs(j)1935 1694 y Ft(\026)1999 1728 y Fv(+)g Fy(C)27 b Fs(j)p Fy(k)s Fs(j)22 b(j)p Fy(\033)s Fs(j)p Fy(e)2417 1694 y Fu(\003)2462 1702 y Fc(\000)2511 1694 y Fu(\()p Ft(k)q Fu(\))p Fq(j)p Ft(\033)r Fq(j)1321 1853 y Fs(\024)h Fy(C)k Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1795 1819 y Ft(\026)1839 1853 y Fs(j)p Fy(\033)s Fs(j)1935 1819 y Ft(\026)1999 1853 y Fv(+)g Fy(C)27 b Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))2468 1819 y Ft(\026)2534 1853 y Fs(j)p Fy(\033)s Fs(j)j(j)p Fy(k)s Fs(j)2743 1819 y Fu(1)p Fq(\000)p Ft(\026)2873 1853 y Fy(e)2912 1819 y Fu(\003)2957 1827 y Fc(\000)3006 1819 y Fu(\()p Ft(k)q Fu(\))p Fq(j)p Ft(\033)r Fq(j)1321 2002 y Fs(\024)i Fy(C)k Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1795 1968 y Ft(\026)1853 1910 y Fn(\020)1903 2002 y Fs(j)p Fy(\033)s Fs(j)1999 1968 y Ft(\026)2062 2002 y Fv(+)g Fy(C)28 b Fs(j)p Fy(\033)s Fs(j)2338 1941 y Fo(1+)p Fk(\026)p 2338 1955 107 3 v 2377 1988 a Fo(2)2459 1910 y Fn(\021)1321 2152 y Fs(\024)23 b Fy(C)k Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1795 2117 y Ft(\026)1839 2152 y Fs(j)p Fy(\033)s Fs(j)1935 2117 y Ft(\026)2004 2152 y Fy(:)-9 2334 y Fz(The)j(bound)j(on)e(the)f(deri)n (v)n(ati)n(v)o(e)f(of)i Fy(P)1107 2346 y Fu(1)p Ft(;)p Fu(2)1219 2334 y Fz(is)f(similar)-5 b(.)30 b(W)-7 b(e)21 b(omit)g(the)h(details.)p 2310 2334 43 43 v -9 2434 a(The)f(follo)n(wing)h(functions)h(enter)e(the)g (de\256nition)h(of)g Fy(N)1653 2446 y Fu(1)1711 2434 y Fz(and)g Fy(N)1922 2446 y Fu(2)1959 2434 y Fz(.)414 2664 y Fy(P)467 2676 y Fu(1)505 2664 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e Fs(\000)p Fv(\(\003)965 2676 y Fu(+)1020 2664 y Fv(\()p Fy(k)s Fv(\))d(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1370 2547 y Fn(\024)1717 2608 y Fy(ik)p 1424 2645 662 4 v 1424 2721 a Fv(\001\()p Fy(k)s Fv(\)\(\003)1693 2733 y Fq(\000)1749 2721 y Fv(\()p Fy(k)s Fv(\))h(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))2095 2664 y Fy(e)2134 2630 y Fu(\003)2179 2638 y Fc(\000)2229 2630 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2514 2664 y Fs(\000)g Fy(i)35 b Fz(s)-5 b(i)g(g)g(n)r Fv(\()p Fy(k)s Fv(\))p Fy(e)2938 2630 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))3244 2547 y Fn(\025)3325 2664 y Fz(,)414 2897 y Fy(P)467 2909 y Fu(2)505 2897 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e(\(\003)900 2909 y Fu(+)955 2897 y Fv(\()p Fy(k)s Fv(\))d(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1305 2780 y Fn(\024)1578 2841 y Fv(\003)1636 2853 y Fq(\000)1692 2841 y Fv(\()p Fy(k)s Fv(\))p 1359 2878 V 1359 2954 a(\001\()p Fy(k)s Fv(\)\(\003)1628 2966 y Fq(\000)1685 2954 y Fv(\()p Fy(k)s Fv(\))h(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))2031 2897 y Fy(e)2070 2862 y Fu(\003)2115 2870 y Fc(\000)2164 2862 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2450 2897 y Fv(+)g Fy(e)2572 2862 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2878 2780 y Fn(\025)2959 2897 y Fy(:)-9 3124 y Fz(W)-7 b(e)21 b(summarize)g(their)g(properties)h(in)f(the)g(follo)n (wing)i(lemma.)-9 3290 y FA(Lemma)d(C.4)43 b Fm(F)-9 b(or)27 b(all)f Fv(0)j Fs(\024)g Fy(\026)h Fs(\024)g Fv(1)p Fm(,)d(ther)m(e)f(e)n (xists)f(a)i(constant)g Fy(C)33 b Fm(suc)o(h)26 b(that)h(for)f(all)g Fv(\()p Fy(x;)14 b(k)s Fv(\))31 b Fs(2)f Fv(\012)2984 3302 y Fu(+)3065 3290 y Fm(and)e(all)e Fs(j)p Fy(\033)s Fs(j)k(\024)g Fv(1)c Fm(with)-9 3390 y Fy(x)18 b Fv(+)g Fy(\033)27 b Fs(\025)c Fv(1)p Fm(,)e(we)f(have)1657 3572 y Fs(j)p Fy(P)1733 3584 y Fu(1)1771 3572 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fs(j)p Fy(P)2166 3584 y Fu(2)2204 3572 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b(e)2658 3538 y Fu(\003)2703 3546 y Fc(\000)2752 3538 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(2)3110 3572 y Fm(,)589 3697 y Fs(j)p Fy(P)665 3709 y Fu(1)703 3697 y Fv(\()p Fy(x)19 b Fv(+)f Fy(\033)n(;)c(k)s Fv(\))19 b Fs(\000)f Fy(P)1199 3709 y Fu(1)1237 3697 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)19 b Fv(+)f Fs(j)p Fy(P)1632 3709 y Fu(2)1670 3697 y Fv(\()p Fy(x)h Fv(+)f Fy(\033)n(;)c(k)s Fv(\))19 b Fs(\000)f Fy(P)2166 3709 y Fu(2)2204 3697 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b Fs(j)p Fy(\033)s Fs(j)2715 3663 y Ft(\026)2781 3697 y Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))3080 3663 y Ft(\026)3148 3697 y Fy(:)-9 3880 y Fm(Pr)l(oof)o(.)31 b Fz(The)21 b(proof)j(is)c(v)o(ery)i(similar)f(to)g (the)g(one)h(of)g(Lemma)f(C.3,)h(we)f(omit)g(the)g(details.)p 2705 3880 43 43 v -9 4120 a Fd(C.2)102 b(Pr)n(oof)26 b(of)f(Lemma)h(3.1)-9 4277 y FA(Lemma)20 b(C.5)j(\(Lemma)d(3.1\))44 b Fm(Assume)21 b(that)h Fr(v)1425 4289 y Fu(+)1504 4277 y Fv(=)h(\()p Fy(u)1672 4289 y Fu(+)1727 4277 y Fy(;)14 b(v)1804 4289 y Fu(+)1859 4277 y Fv(\))23 b Fs(2)h(W)2075 4289 y Ft(\013;\015)2173 4297 y Fo(1)2205 4289 y Ft(;\015)2260 4297 y Fo(2)2318 4277 y Fm(for)d(some)h Fy(\013)i(>)f Fv(3)p Fm(,)e Fy(\015)2926 4289 y Fu(1)2986 4277 y Fs(2)j Fv([0)p Fy(;)14 b Fv(1\))21 b Fm(and)h Fy(\015)3453 4289 y Fu(2)3514 4277 y Fs(2)i Fv([0)p Fy(;)3704 4244 y Fu(1)p 3704 4258 34 4 v 3704 4306 a(2)3747 4277 y Fv(])-9 4377 y Fm(with)d Fy(\015)202 4389 y Fu(2)262 4377 y Fs(\024)j Fy(\015)394 4389 y Fu(1)431 4377 y Fm(.)32 b(Assume)22 b(that)g Fy(u)967 4389 y Fu(+)1043 4377 y Fm(and)h Fy(v)1232 4389 y Fu(+)1309 4377 y Fm(ar)m(e)f(e)o(ven)f(and)i(odd)g(in)f Fy(y)j Fm(r)m(espectively)-5 b(.)30 b(De\256ne)22 b Fy(\021)2804 4389 y Fu(0)p Ft(;)p Fu(0)2894 4377 y Fm(,)g Fy(u)2985 4389 y Fu(0)3044 4377 y Fm(and)h Fy(v)3233 4389 y Fu(0)3292 4377 y Fm(as)f(the)g(F)-9 b(ourier)-9 4476 y(tr)o(ansforms)22 b(of)1166 4659 y Fv(^)-48 b Fy(\021)1201 4671 y Fu(0)p Ft(;)p Fu(0)1291 4659 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)k(^)-48 b Fy(\021)1637 4671 y Fu(0)p Ft(;)p Fu(+)1746 4659 y Fv(\()p Fy(k)s Fv(\))p Fy(e)1895 4625 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2224 4659 y Fm(,)1084 4719 y Fn(\022)1150 4780 y Fv(^)h Fy(u)1193 4792 y Fu(0)1152 4893 y Fv(^)i Fy(v)1189 4905 y Fu(0)1230 4719 y Fn(\023)1291 4836 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1596 4719 y Fn(\022)1658 4786 y Fy(P)1711 4798 y Fu(1)p Ft(;)p Fu(1)1884 4786 y Fy(P)1937 4798 y Fu(1)p Ft(;)p Fu(2)1658 4885 y Fy(P)1711 4897 y Fu(2)p Ft(;)p Fu(1)1884 4885 y Fy(P)1937 4897 y Fu(2)p Ft(;)p Fu(2)2027 4719 y Fn(\023)2102 4836 y Fv(\()p Fy(x;)14 b(k)s Fv(\))2296 4719 y Fn(\022)2363 4780 y Fv(^)-47 b Fy(u)2406 4792 y Fu(+)2365 4893 y Fv(^)i Fy(v)2402 4905 y Fu(+)2461 4719 y Fn(\023)2522 4836 y Fv(\()p Fy(k)s Fv(\))23 b Fm(,)-9 5064 y(wher)m(e)d Fy(P)269 5076 y Ft(i;j)369 5064 y Fm(ar)m(e)h(de\256ned)i(in)e(the)g(pr)m(e)o (vious)h(section)f(and)890 5297 y Fv(^)-48 b Fy(\021)925 5309 y Fu(0)p Ft(;)p Fu(+)1033 5297 y Fv(\()p Fy(k)s Fv(\))23 b(=)g Fs(\000)p Fy(ik)1404 5241 y Fv(\003)1462 5253 y Fu(+)1516 5241 y Fv(\()p Fy(k)s Fv(\))18 b(+)g Fs(j)p Fy(k)s Fs(j)p 1404 5278 417 4 v 1500 5354 a Fv(\003)1558 5366 y Fu(+)1612 5354 y Fv(\()p Fy(k)s Fv(\))1835 5297 y(^)-48 b Fy(u)1877 5309 y Fu(+)1932 5297 y Fv(\()p Fy(k)s Fv(\))19 b Fs(\000)f Fv(\(\003)2234 5309 y Fu(+)2289 5297 y Fv(\()p Fy(k)s Fv(\))h(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))s(^)-45 b Fy(v)2665 5309 y Fu(+)2720 5297 y Fv(\()p Fy(k)s Fv(\))24 b Fy(:)-9 5524 y Fm(Then)e(the)f(functions)h Fy(\021)681 5536 y Fu(0)p Ft(;)p Fu(0)772 5524 y Fm(,)f Fy(u)862 5536 y Fu(0)899 5524 y Fm(,)h Fy(v)982 5536 y Fu(0)1019 5524 y Fm(,)g Fy(@)1106 5536 y Ft(y)1146 5524 y Fy(u)1194 5536 y Fu(0)1231 5524 y Fm(,)g Fy(@)1318 5536 y Ft(y)1358 5524 y Fy(v)1398 5536 y Fu(0)1457 5524 y Fm(ar)m(e)f(bounded)j(and)f(continuous)h(in)d Fy(x;)14 b(y)s Fm(,)22 b(the)g(function)g Fy(u)3178 5536 y Fu(0)3236 5524 y Fm(is)f(e)o(ven)g(in)g Fy(y)k Fm(and)-9 5624 y(the)c(functions)g Fy(v)488 5636 y Fu(0)526 5624 y Fy(;)14 b(\021)604 5636 y Fu(0)p Ft(;)p Fu(0)715 5624 y Fm(ar)m(e)22 b(odd)g(in)g Fy(y)s Fm(.)1838 5909 y Fz(21)p eop %%Page: 22 22 bop -9 145 a Fm(Pr)l(oof)o(.)34 b Fz(First)22 b(of)h(all,)f(the)g(parity)h (of)g(the)f(functions)h(is)f(tri)n(vial.)33 b(From)24 b(Lemma)e(C.3,)h(it)e (follo)n(ws)i(that)f(there)g(e)o(xists)g(a)g(constant)-9 244 y Fy(C)27 b Fz(such)22 b(that)f(for)h(all)e Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)985 256 y Fu(+)1041 244 y Fz(,)1182 463 y Fs(j)5 b Fv(^)-47 b Fy(u)1253 475 y Fu(0)1289 463 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)20 b Fv(+)e Fs(j)s Fv(^)-45 b Fy(v)1672 475 y Fu(0)1709 463 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)2113 406 y Fs(k)p Fr(v)2205 418 y Fu(+)2260 406 y Fs(k)2302 418 y Ft(\013;\015)2400 426 y Fo(1)2432 418 y Ft(;\015)2487 426 y Fo(2)p 2113 444 410 4 v 2145 520 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2444 496 y Ft(\013)2556 463 y Fy(:)1014 b Fz(\(C.9\))-9 677 y(Since)21 b Fy(\013)j(>)f Fv(3)p Fz(,)f(this)f(implies)g (that)g(the)g(F)o(ourier)i(transforms)g(of)28 b Fv(^)-48 b Fy(u)1937 689 y Fu(0)1974 677 y Fz(,)22 b Fy(K)11 b Fv(^)-47 b Fy(u)2142 689 y Fu(0)2178 677 y Fz(,)25 b Fv(^)-45 b Fy(v)2261 689 y Fu(0)2320 677 y Fz(and)23 b Fy(K)9 b Fv(^)-45 b Fy(v)2582 689 y Fu(0)2640 677 y Fz(are)22 b(bounded.)34 b(T)-7 b(o)22 b(see)g(that)f(the)o(y)g(are)-9 777 y(continuous)e(in)f Fy(x)p Fz(,)i(use)e(Lemma)f(C.3)i(to)f(get)f(that)h(for)h(all)e Fs(j)p Fy(\033)s Fs(j)23 b(\024)g Fv(1)17 b Fz(with)h Fy(x)11 b Fv(+)g Fy(\033)27 b Fs(\025)c Fv(1)p Fz(,)18 b(and)h(for)g(all)e Fy(\026)23 b Fs(2)g Fr(R)18 b Fz(with)g Fv(0)k Fy(<)h(\026)g(<)g(\013)12 b Fs(\000)f Fv(2)p Fz(,)227 882 y Fn(Z)309 995 y Fy(dk)25 b Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))35 b Fs(j)6 b Fv(^)-48 b Fy(u)825 1007 y Fu(0)862 995 y Fv(\()p Fy(x)19 b Fv(+)f Fy(\033)n(;)c(k)s Fv(\))20 b Fs(\000)j Fv(^)-47 b Fy(u)1354 1007 y Fu(0)1390 995 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f(j)p Fy(\033)s Fs(j)1815 961 y Ft(\026)1881 995 y Fy(C)1982 882 y Fn(Z)2064 995 y Fy(dk)2242 939 y Fs(k)p Fr(v)2334 951 y Fu(+)2389 939 y Fs(k)2431 951 y Ft(\013;\015)2529 959 y Fo(1)2560 951 y Ft(;\015)2615 959 y Fo(2)p 2184 976 525 4 v 2184 1052 a Fv(\(1)c(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))2484 1028 y Ft(\013)p Fq(\000)p Fu(1)p Fq(\000)p Ft(\026)2742 995 y Fs(\024)k(j)p Fy(\033)s Fs(j)2925 961 y Ft(\026)2991 995 y Fy(C)28 b Fs(k)p Fr(v)3170 1007 y Fu(+)3225 995 y Fs(k)3267 1007 y Ft(\013;\015)3365 1015 y Fo(1)3397 1007 y Ft(;\015)3452 1015 y Fo(2)3511 995 y Fy(:)-9 1210 y Fz(This)h(implies)g(that)g Fy(u)656 1222 y Fu(0)723 1210 y Fz(and)h Fy(@)919 1222 y Ft(y)959 1210 y Fy(u)1007 1222 y Fu(0)1074 1210 y Fz(are)f(Ho)-35 b(\310)7 b(lder)n(\261)p Fy(\026)32 b Fz(in)d Fy(x)i Fz(for)f(all)f Fv(0)34 b Fy(<)g(\026)h(<)f(\013)23 b Fs(\000)g Fv(2)p Fz(.)56 b(T)-7 b(o)30 b(see)f(that)g Fy(u)3082 1222 y Fu(0)3149 1210 y Fz(and)h Fy(@)3345 1222 y Ft(y)3385 1210 y Fy(u)3433 1222 y Fu(0)3500 1210 y Fz(are)f(also)-9 1310 y(Ho)-35 b(\310)7 b(lder)n(\261)p Fy(\026)23 b Fz(in)e Fy(y)s Fz(,)g(use)647 1239 y Fn(\014)647 1289 y(\014)674 1310 y Fy(e)713 1279 y Ft(ik)q Fu(\()p Ft(y)r Fu(+)p Ft(\033)r Fu(\))975 1310 y Fs(\000)d Fy(e)1097 1279 y Ft(ik)q(y)1197 1239 y Fn(\014)1197 1289 y(\014)1247 1310 y Fs(\024)23 b(j)p Fy(\033)s Fs(j)1431 1279 y Ft(\026)1476 1310 y Fs(j)p Fy(k)s Fs(j)1568 1279 y Ft(\026)1634 1310 y Fz(to)e(get)g(that)336 1536 y Fs(j)p Fy(@)403 1548 y Ft(y)443 1536 y Fy(u)491 1548 y Fu(0)528 1536 y Fv(\()p Fy(x;)14 b(y)22 b Fv(+)c Fy(\033)s Fv(\))h Fs(\000)f Fy(@)1018 1548 y Ft(y)1058 1536 y Fy(u)1106 1548 y Fu(0)1143 1536 y Fv(\()p Fy(x;)c(y)s Fv(\))p Fs(j)19 b Fv(+)f Fs(j)p Fy(u)1531 1548 y Fu(0)1568 1536 y Fv(\()p Fy(x;)c(y)21 b Fv(+)d Fy(\033)s Fv(\))i Fs(\000)e Fy(u)2062 1548 y Fu(0)2098 1536 y Fv(\()p Fy(x;)c(y)s Fv(\))p Fs(j)24 b(\024)f(j)p Fy(\033)s Fs(j)2521 1502 y Ft(\026)2587 1536 y Fy(C)2687 1423 y Fn(Z)2770 1536 y Fy(dk)2948 1480 y Fs(k)p Fr(v)3040 1492 y Fu(+)3095 1480 y Fs(k)3137 1492 y Ft(\013;\015)3235 1500 y Fo(1)3266 1492 y Ft(;\015)3321 1500 y Fo(2)p 2890 1517 V 2890 1593 a Fv(\(1)c(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))3190 1569 y Ft(\013)p Fq(\000)p Fu(1)p Fq(\000)p Ft(\026)2337 1709 y Fs(\024)23 b(j)p Fy(\033)s Fs(j)2521 1675 y Ft(\026)2587 1709 y Fy(C)28 b Fs(k)p Fr(v)2766 1721 y Fu(+)2821 1709 y Fs(k)2863 1721 y Ft(\013;\015)2961 1729 y Fo(1)2992 1721 y Ft(;\015)3047 1729 y Fo(2)3107 1709 y Fy(:)-9 1880 y Fz(F)o(or)22 b(the)f(continuity)g(of)h Fy(v)751 1892 y Fu(0)789 1880 y Fz(,)f(the)h(proof)h(is)e(done)h(along)g(the)f(same)g (lines.)30 b(Finally)-6 b(,)22 b(by)g(\(C.9\),)h(we)e(ha)n(v)o(e)1134 1983 y Fn(Z)1217 2004 y Fq(1)1180 2172 y(\0001)1281 2096 y Fy(dk)j Fs(j)6 b Fv(^)-48 b Fy(\021)1455 2108 y Fu(0)p Ft(;)p Fu(0)1545 2096 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)1874 1983 y Fn(Z)1957 2004 y Fq(1)1920 2172 y(\0001)2020 2096 y Fy(dk)2152 2040 y Fs(k)p Fr(v)2244 2052 y Fu(+)2299 2040 y Fs(k)2341 2052 y Ft(\013;\015)2439 2060 y Fo(1)2470 2052 y Ft(;\015)2525 2060 y Fo(2)p 2141 2077 432 4 v 2141 2153 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2440 2129 y Ft(\013)p Fq(\000)p Fu(1)2606 2096 y Fz(,)-9 2320 y(so)j(that)g Fy(\021)277 2332 y Fu(0)p Ft(;)p Fu(0)367 2320 y Fv(\()p Fy(x;)14 b(y)s Fv(\))22 b Fz(is)f(bounded.)34 b(The)21 b(continuity)h(of)g Fy(\021)1656 2332 y Fu(0)p Ft(;)p Fu(0)1767 2320 y Fz(follo)n(ws)g(along)g(the)f(same)g (lines)g(as)g(the)g(one)h(for)g Fy(u)3283 2332 y Fu(0)3320 2320 y Fz(.)p 3393 2320 43 43 v -9 2559 a Fd(C.3)102 b(Pr)n(oof)26 b(of)f(Lemma)h(3.2)-9 2716 y FA(Lemma)20 b(C.6)j(\(Lemma)d(3.2\))44 b Fm(Let)37 b Fv(^)-49 b Fy(q)40 b Fv(:)d(\012)1277 2728 y Fu(+)1369 2716 y Fs(!)f Fr(R)31 b Fm(be)h(bounded.)64 b(Assume)31 b(that)38 b Fv(^)-48 b Fy(q)34 b Fm(is)d(an)i(odd)f(function)g(of)g Fy(k)s Fm(,)h(that)38 b Fv(^)-48 b Fy(q)40 b Fs(2)-9 2815 y Fy(L)48 2785 y Fu(1)84 2815 y Fv(\(\012)176 2827 y Fu(+)232 2815 y Fy(;)14 b(k)34 b(dk)g(dx)p Fv(\))e Fm(and)h(that)e Fz(sup)1060 2838 y Ft(x)p Fq(\025)p Fu(1)1201 2748 y Fn(R)1256 2815 y Fy(dk)j Fv(\(1)24 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))p Fs(j)6 b Fv(^)-48 b Fy(q)t Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)37 b Fy(<)f Fs(1)p Fm(.)61 b(De\256ne)32 b Fy(N)2590 2827 y Fu(1)2658 2815 y Fm(and)h Fy(N)2884 2827 y Fu(2)2952 2815 y Fm(and)g Fy(\021)3152 2827 y Fu(0)p Ft(;q)3273 2815 y Fm(as)f(the)f(F)-9 b(ourier)-9 2915 y(tr)o(ansforms)22 b(of)1007 3064 y Fv(^)983 3085 y Fy(N)1050 3097 y Fu(1)1086 3085 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1416 3064 y(^)1392 3085 y Fy(N)1459 3097 y Fu(1)p Ft(;)p Fu(1)1549 3085 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)1869 3064 y(^)1845 3085 y Fy(N)1912 3097 y Fu(1)p Ft(;)p Fu(2)2002 3085 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(P)2351 3097 y Fu(1)2388 3085 y Fv(\()p Fy(x;)c(k)s Fv(\))22 b Fy(\034)9 b Fv(\()p Fy(k)s Fv(\))25 b Fm(,)1007 3200 y Fv(^)983 3221 y Fy(N)1050 3233 y Fu(2)1086 3221 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1416 3200 y(^)1392 3221 y Fy(N)1459 3233 y Fu(2)p Ft(;)p Fu(1)1549 3221 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)1869 3200 y(^)1845 3221 y Fy(N)1912 3233 y Fu(2)p Ft(;)p Fu(2)2002 3221 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(P)2351 3233 y Fu(2)2388 3221 y Fv(\()p Fy(x;)c(k)s Fv(\))22 b Fy(\034)9 b Fv(\()p Fy(k)s Fv(\))25 b Fm(,)962 3356 y Fv(^)-48 b Fy(\021)997 3368 y Fu(0)p Ft(;q)1086 3356 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)29 b(^)-48 b Fy(\021)1433 3368 y Ft(q)r(;)p Fu(+)1541 3356 y Fv(\()p Fy(k)s Fv(\))p Fy(e)1690 3322 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2019 3356 y Fm(,)-9 3526 y(wher)m(e)20 b Fy(P)269 3538 y Fu(1)307 3526 y Fy(;)14 b(P)397 3538 y Fu(2)456 3526 y Fm(ar)m(e)21 b(de\256ned)i(in)e(Section)g(C.1)h(and)996 3720 y Fv(^)972 3741 y Fy(N)1039 3753 y Fu(1)p Ft(;)p Fu(1)1129 3741 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1497 3684 y Fy(ik)p 1444 3721 180 4 v 1444 3798 a Fv(\001\()p Fy(k)s Fv(\))1648 3628 y Fn(Z)1731 3648 y Ft(x)1694 3816 y Fu(1)1751 3741 y Fy(dx)1841 3753 y Fu(0)1900 3741 y Fy(e)1939 3706 y Fu(\003)1984 3714 y Fc(\000)2033 3706 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)2275 3714 y Fo(0)2308 3706 y Fu(\))2344 3741 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2457 3753 y Fu(0)2495 3741 y Fy(;)14 b(k)s Fv(\))23 b Fm(,)996 3948 y Fv(^)972 3969 y Fy(N)1039 3981 y Fu(1)p Ft(;)p Fu(2)1129 3969 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1497 3913 y Fy(ik)p 1444 3950 V 1444 4026 a Fv(\001\()p Fy(k)s Fv(\))1648 3856 y Fn(Z)1731 3876 y Fq(1)1694 4045 y Ft(x)1780 3969 y Fy(dx)1870 3981 y Fu(0)1929 3969 y Fy(e)1968 3935 y Fu(\003)2013 3943 y Fo(+)2060 3935 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)2302 3943 y Fo(0)2334 3935 y Fu(\))2370 3969 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2483 3981 y Fu(0)2521 3969 y Fy(;)14 b(k)s Fv(\))23 b Fm(,)996 4177 y Fv(^)972 4198 y Fy(N)1039 4210 y Fu(2)p Ft(;)p Fu(1)1129 4198 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e Fs(\000)1509 4142 y Fv(\003)1567 4154 y Fq(\000)1623 4142 y Fv(\()p Fy(k)s Fv(\))p 1509 4179 225 4 v 1531 4255 a(\001\()p Fy(k)s Fv(\))1757 4085 y Fn(Z)1840 4106 y Ft(x)1803 4274 y Fu(1)1860 4198 y Fy(dx)1950 4210 y Fu(0)2009 4198 y Fy(e)2048 4164 y Fu(\003)2093 4172 y Fc(\000)2142 4164 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)2384 4172 y Fo(0)2417 4164 y Fu(\))2453 4198 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2566 4210 y Fu(0)2604 4198 y Fy(;)14 b(k)s Fv(\))23 b Fm(,)996 4407 y Fv(^)972 4428 y Fy(N)1039 4440 y Fu(2)p Ft(;)p Fu(2)1129 4428 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e Fs(\000)1509 4372 y Fv(\003)1567 4384 y Fu(+)1622 4372 y Fv(\()p Fy(k)s Fv(\))p 1509 4409 224 4 v 1531 4485 a(\001\()p Fy(k)s Fv(\))1756 4315 y Fn(Z)1839 4335 y Fq(1)1802 4503 y Ft(x)1888 4428 y Fy(dx)1978 4440 y Fu(0)2037 4428 y Fy(e)2076 4393 y Fu(\003)2121 4401 y Fo(+)2168 4393 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)2410 4401 y Fo(0)2442 4393 y Fu(\))2478 4428 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2591 4440 y Fu(0)2629 4428 y Fy(;)14 b(k)s Fv(\))23 b Fm(,)1168 4656 y Fy(\034)9 b Fv(\()p Fy(k)s Fv(\))24 b(=)1434 4543 y Fn(Z)1517 4564 y Fq(1)1481 4732 y Fu(1)1566 4656 y Fy(dx)1656 4668 y Fu(0)1715 4656 y Fy(e)1754 4622 y Fu(\003)1799 4630 y Fo(+)1846 4622 y Fu(\()p Ft(k)q Fu(\)\(1)p Fq(\000)p Ft(x)2083 4630 y Fo(0)2116 4622 y Fu(\))2152 4656 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2265 4668 y Fu(0)2303 4656 y Fy(;)14 b(k)s Fv(\))23 b Fm(,)1071 4827 y Fv(^)-48 b Fy(\021)1106 4839 y Ft(q)r(;)p Fu(+)1213 4827 y Fv(\()p Fy(k)s Fv(\))24 b(=)e Fs(\000)p Fv(\(\003)1589 4839 y Fu(+)1644 4827 y Fv(\()p Fy(k)s Fv(\))d(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))p Fy(\034)9 b Fv(\()p Fy(k)s Fv(\))24 b Fy(:)-9 4998 y Fm(Then)d(the)f(functions)i Fy(N)705 5010 y Fu(1)741 4998 y Fm(,)f Fy(N)850 5010 y Fu(2)887 4998 y Fm(,)g Fy(@)973 5010 y Ft(y)1013 4998 y Fy(N)1080 5010 y Fu(1)1117 4998 y Fy(;)14 b(@)1198 5010 y Ft(y)1238 4998 y Fy(N)1305 5010 y Fu(2)1363 4998 y Fm(and)22 b Fy(\021)1552 5010 y Fu(0)p Ft(;q)1662 4998 y Fm(ar)m(e)f(bounded)i(and)f(continuous)g(in)f Fy(x;)14 b(y)s Fm(,)21 b(the)f(function)h Fy(N)3394 5010 y Fu(1)3452 4998 y Fm(is)f(e)o(ven)g(in)-9 5097 y Fy(y)s Fm(,)h(and)h(the)f(functions)h Fy(N)750 5109 y Fu(2)787 5097 y Fy(;)14 b(\021)865 5109 y Fu(0)p Ft(;q)976 5097 y Fm(ar)m(e)21 b(odd)i(in)e Fy(y)s Fm(.)-9 5279 y(Pr)l(oof)o(.)36 b Fz(First)23 b(of)g(all,)g(the)f(parity)h(of)h(the)f (functions)h(is)e(tri)n(vial.)35 b(W)-7 b(e)23 b(be)o(gin)h(the)e(proof)j(by) f(sho)n(wing)h(that)3151 5258 y Fv(^)3127 5279 y Fy(N)3194 5291 y Ft(i;j)3297 5279 y Fs(2)h Fy(L)3435 5249 y Fu(1)3472 5279 y Fv(\()p Fr(R)p Fy(;)14 b Fv(\(1)19 b(+)-9 5390 y Fs(j)p Fy(k)s Fs(j)p Fv(\))115 5360 y Fu(1+)p Ft(\026)243 5390 y Fy(dk)s Fv(\))25 b Fz(for)g(all)e Fv(0)k Fs(\024)f Fy(\026)h(<)g Fv(1)p Fz(.)39 b(W)-7 b(e)24 b(be)o(gin)g(with)1598 5369 y Fv(^)1574 5390 y Fy(N)1641 5402 y Ft(i;)p Fu(1)1721 5390 y Fz(.)39 b(W)-7 b(e)24 b(use)h(\256rst)f(that)g(there)g(e)o(xists)f(a)h(constant)g Fy(C)31 b Fz(such)24 b(that)g(for)h(all)-9 5489 y Fy(k)g Fs(2)f Fr(R)p Fz(,)1430 5660 y Fs(j)p Fv(\003)1511 5672 y Fq(\000)1566 5660 y Fv(\()p Fy(k)s Fv(\))p Fs(j)g(\024)f(j)p Fy(k)s Fs(j)g Fz(,)1838 5909 y(22)p eop %%Page: 23 23 bop 1272 145 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))1571 110 y Fu(1+)p Ft(\026)1723 145 y Fs(\024)23 b Fv(\001\()p Fy(k)s Fv(\))e(\()p Fy(C)k Fv(+)18 b Fy(C)6 b Fs(j)p Fy(k)s Fs(j)2367 110 y Ft(\026)2412 145 y Fv(\))24 b Fz(,)-9 324 y(to)d(get)323 431 y Fn(Z)405 544 y Fy(dk)j Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))815 510 y Fu(1+)p Ft(\026)944 544 y Fs(j)991 523 y Fv(^)967 544 y Fy(N)1034 556 y Ft(i;)p Fu(1)1114 544 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)p Fv(\))24 b Fs(\024)f Fy(C)1554 431 y Fn(Z)1636 544 y Fy(dk)i Fv(\001\()p Fy(k)s Fv(\))p Fs(j)1973 523 y Fv(^)1949 544 y Fy(N)2016 556 y Ft(i;)p Fu(1)2097 544 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fy(C)2495 431 y Fn(Z)2578 544 y Fy(dk)24 b Fv(\001\()p Fy(k)s Fv(\))p Fs(j)p Fy(k)s Fs(j)2959 510 y Ft(\026)3004 544 y Fs(j)3051 523 y Fv(^)3027 544 y Fy(N)3094 556 y Ft(i;)p Fu(1)3174 544 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b Fy(:)-9 767 y Fz(Since)d Fv(\003)259 779 y Fq(\000)314 767 y Fv(\()p Fy(k)s Fv(\))j Fs(\024)e Fv(0)p Fz(,)f(we)h(ha)n(v)o(e)888 881 y Fn(Z)971 994 y Fy(dk)i Fv(\001\()p Fy(k)s Fv(\))p Fs(j)1308 973 y Fv(^)1283 994 y Fy(N)1350 1006 y Ft(i;)p Fu(1)1431 994 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)1760 881 y Fn(Z)1842 994 y Fy(dk)1966 881 y Fn(Z)2049 901 y Fq(1)2012 1069 y Fu(1)2098 994 y Fy(dx)2188 1006 y Fu(0)2247 994 y Fs(j)p Fy(k)31 b Fv(^)-49 b Fy(q)s Fv(\()p Fy(x)2456 1006 y Fu(0)2494 994 y Fy(;)14 b(k)s Fv(\))p Fs(j)24 b Fy(<)e Fs(1)h Fy(:)-9 1215 y Fz(By)e(Lemma)g(C.1,)h(we)f(ha)n(v)o(e)1323 1455 y Fs(j)p Fy(k)s Fs(j)1415 1420 y Ft(\026)1459 1455 y Fy(e)1498 1420 y Fu(\003)1543 1428 y Fc(\000)1592 1420 y Fu(\()p Ft(k)q Fu(\))p Ft(x)1746 1455 y Fs(\024)h Fy(C)1934 1337 y Fn(\022)2005 1333 y Fs(p)p 2074 1333 191 4 v 65 x Fv(1)c(+)g Fy(x)p 2005 1435 260 4 v 2111 1511 a(x)2274 1337 y Fn(\023)2335 1347 y Ft(\026)2417 1455 y Fz(,)-9 1697 y(and)j(since)g(for)i(all)d(for)i Fv(1)h Fs(\024)f Fy(x)876 1709 y Fu(0)937 1697 y Fs(\024)h Fy(x)e Fz(we)h(ha)n(v)o(e)1392 1635 y Fs(p)p 1462 1635 377 4 v 1462 1697 a Fv(1)c(+)g Fy(x)g Fs(\000)h Fy(x)1801 1709 y Fu(0)1861 1697 y Fs(\024)1949 1629 y(p)p 2018 1629 89 4 v 68 x Fv(3)p Fy(x)p Fz(,)j(we)f(get)712 1821 y Fn(Z)795 1934 y Fy(dk)j Fv(\001\()p Fy(k)s Fv(\))e Fs(j)p Fy(k)s Fs(j)1198 1900 y Ft(\026)1264 1934 y Fs(j)1311 1913 y Fv(^)1287 1934 y Fy(N)1354 1946 y Ft(i;)p Fu(1)1434 1934 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)1841 1821 y Fn(Z)1924 1934 y Fy(dk)2048 1821 y Fn(Z)2131 1842 y Ft(x)2094 2010 y Fu(1)2152 1934 y Fy(dx)2242 1946 y Fu(0)2402 1878 y Fy(x)2449 1848 y Ft(\026=)p Fu(2)p 2311 1915 343 4 v 2311 1991 a Fv(\()p Fy(x)d Fs(\000)f Fy(x)2539 2003 y Fu(0)2577 1991 y Fv(\))2609 1967 y Ft(\026)2663 1934 y Fs(j)p Fy(k)31 b Fv(^)-49 b Fy(q)s Fv(\()p Fy(x)2872 1946 y Fu(0)2910 1934 y Fy(;)14 b(k)s Fv(\))p Fs(j)1675 2158 y(\024)22 b Fy(C)6 b(x)1874 2124 y Fu(1)p Fq(\000)p Ft(\026=)p Fu(2)2104 2158 y Fz(sup)2086 2230 y Ft(x)2124 2238 y Fo(0)2156 2230 y Fq(\025)p Fu(1)2254 2045 y Fn(Z)2337 2158 y Fy(dk)24 b Fs(j)p Fy(k)31 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2657 2170 y Fu(0)2695 2158 y Fy(;)14 b(k)s Fv(\))p Fs(j)23 b Fy(:)-9 2395 y Fz(This)e(is)f(bounded)k(for)e Fy(x)f Fz(bounded.)34 b(T)-7 b(o)21 b(sho)n(w)h(that)f(it)f(is)h(also)f(bounded)k(as)d Fy(x)i Fs(!)g(1)p Fz(,)f(suppose)g(that)f Fy(x)i Fs(\025)g Fv(4)p Fz(.)30 b(In)22 b(this)f(case,)f(we)-9 2495 y(split)g(the)h(inte)o (gration)g(interv)n(al,)g(which)h(gi)n(v)o(es)258 2619 y Fn(Z)341 2732 y Fy(dk)465 2619 y Fn(Z)548 2639 y Ft(x)p Fq(\000)p Fu(2)511 2807 y(1)653 2732 y Fy(dx)743 2744 y Fu(0)812 2675 y Fv(\(1)d(+)f Fy(x)h Fs(\000)f Fy(x)1184 2687 y Fu(0)1221 2675 y Fv(\))1253 2645 y Ft(\026=)p Fu(2)p 812 2713 553 4 v 917 2789 a Fv(\()p Fy(x)i Fs(\000)e Fy(x)1146 2801 y Fu(0)1183 2789 y Fv(\))1215 2765 y Ft(\026)1375 2732 y Fs(j)p Fy(k)31 b Fv(^)-49 b Fy(q)s Fv(\()p Fy(x)1584 2744 y Fu(0)1622 2732 y Fy(;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)1950 2619 y Fn(Z)2033 2732 y Fy(dk)2157 2619 y Fn(Z)2240 2639 y Fq(1)2203 2807 y Fu(1)2289 2732 y Fy(dx)2379 2744 y Fu(0)2438 2732 y Fs(j)p Fy(k)30 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2647 2744 y Fu(0)2685 2732 y Fy(;)14 b(k)s Fv(\))p Fs(j)23 b Fy(<)g Fs(1)g Fz(,)295 2859 y Fn(Z)378 2972 y Fy(dk)502 2859 y Fn(Z)585 2880 y Ft(x)548 3048 y(x)p Fq(\000)p Fu(2)653 2972 y Fy(dx)743 2984 y Fu(0)812 2916 y Fv(\(1)c(+)f Fy(x)h Fs(\000)f Fy(x)1184 2928 y Fu(0)1221 2916 y Fv(\))1253 2886 y Ft(\026=)p Fu(2)p 812 2953 V 917 3029 a Fv(\()p Fy(x)i Fs(\000)e Fy(x)1146 3041 y Fu(0)1183 3029 y Fv(\))1215 3005 y Ft(\026)1375 2972 y Fs(j)p Fy(k)31 b Fv(^)-49 b Fy(q)s Fv(\()p Fy(x)1584 2984 y Fu(0)1622 2972 y Fy(;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)39 b Fz(sup)1950 3044 y Ft(x)1988 3052 y Fo(0)2020 3044 y Fq(\025)p Fu(1)2119 2859 y Fn(Z)2202 2972 y Fy(dk)24 b Fs(j)p Fy(k)30 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2521 2984 y Fu(0)2559 2972 y Fy(;)14 b(k)s Fv(\))p Fs(j)2711 2859 y Fn(Z)2794 2880 y Ft(x)2757 3048 y(x)p Fq(\000)p Fu(2)2862 2972 y Fy(dy)2981 2916 y Fv(\(1)k(+)g Fy(x)h Fs(\000)f Fy(y)s Fv(\))3381 2886 y Ft(\026=)p Fu(2)p 2981 2953 512 4 v 3086 3029 a Fv(\()p Fy(x)h Fs(\000)f Fy(y)s Fv(\))3343 3005 y Ft(\026)1783 3214 y Fs(\024)23 b Fy(C)39 b Fz(sup)1950 3286 y Ft(x)1988 3294 y Fo(0)2020 3286 y Fq(\025)p Fu(1)2119 3101 y Fn(Z)2202 3214 y Fy(dk)24 b Fs(j)p Fy(k)30 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2521 3226 y Fu(0)2559 3214 y Fy(;)14 b(k)s Fv(\))p Fs(j)2711 3101 y Fn(Z)2794 3122 y Ft(x)2757 3290 y(x)p Fq(\000)p Fu(2)2862 3214 y Fy(dy)3111 3158 y Fv(1)p 2981 3195 302 4 v 2981 3271 a(\()p Fy(x)19 b Fs(\000)f Fy(y)s Fv(\))3238 3247 y Ft(\026)1783 3452 y Fs(\024)23 b Fy(C)39 b Fz(sup)1950 3524 y Ft(x)1988 3532 y Fo(0)2020 3524 y Fq(\025)p Fu(1)2119 3339 y Fn(Z)2202 3452 y Fy(dk)24 b Fs(j)p Fy(k)30 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2521 3464 y Fu(0)2559 3452 y Fy(;)14 b(k)s Fv(\))p Fs(j)23 b Fy(<)g Fs(1)g Fy(:)-9 3710 y Fz(This)14 b(sho)n(ws)g(that)545 3689 y Fv(^)521 3710 y Fy(N)588 3722 y Ft(i;)p Fu(1)692 3710 y Fs(2)23 b Fy(L)827 3680 y Fu(1)864 3710 y Fv(\()p Fr(R)p Fy(;)14 b Fv(\(1)s(+)s Fs(j)p Fy(k)s Fs(j)p Fv(\))1274 3680 y Fu(1+)p Ft(\026)1401 3710 y Fy(dk)s Fv(\))h Fz(for)f(all)g Fv(0)22 b Fs(\024)h Fy(\026)g(<)g Fv(1)p Fz(.)28 b(W)-7 b(e)14 b(no)n(w)g(sho)n(w)g(that)2793 3689 y Fv(^)2769 3710 y Fy(N)2836 3722 y Ft(i;)p Fu(2)2939 3710 y Fs(2)23 b Fy(L)3074 3680 y Fu(1)3111 3710 y Fv(\()p Fr(R)p Fy(;)14 b Fv(\(1)s(+)s Fs(j)p Fy(k)s Fs(j)p Fv(\))3521 3680 y Fu(1+)p Ft(\026)3649 3710 y Fy(dk)s Fv(\))-9 3809 y Fz(also)21 b(for)h(all)e Fv(0)j Fs(\024)f Fy(\026)h(<)g Fv(1)p Fz(.)31 b(W)-7 b(e)21 b(use)g(\256rst)h(that)f(there)g(e)o(xists)f(a)h (constant)h Fy(C)27 b Fz(such)22 b(that)f(for)h(all)e Fy(k)26 b Fs(2)e Fr(R)p Fz(,)1529 3976 y Fs(j)p Fy(k)s Fs(j)p 1485 4013 180 4 v 1485 4090 a Fv(\001\()p Fy(k)s Fv(\))1693 4033 y(+)1786 3976 y Fs(j)p Fv(\003)1867 3988 y Fu(+)1922 3976 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p 1786 4013 270 4 v 1831 4090 a Fv(\001\()p Fy(k)s Fv(\))2088 4033 y Fs(\024)f Fy(C)29 b Fz(,)-9 4261 y(and)21 b(then,)h(since)f Fv(\003)577 4273 y Fu(+)632 4261 y Fv(\()p Fy(k)s Fv(\))i Fs(\025)g Fv(1)p Fz(,)e(we)g(ha)n(v)o(e)970 4370 y Fn(Z)1053 4483 y Fy(dk)j Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1430 4449 y Fu(1+)p Ft(\026)1559 4483 y Fv(\))p Fs(j)1638 4462 y Fv(^)1614 4483 y Fy(N)1681 4495 y Ft(i;)p Fu(2)1761 4483 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)6 b(B)2218 4495 y Fu(1)2255 4483 y Fv(\()p Fy(x)p Fv(\))20 b(+)e Fy(C)6 b(B)2597 4495 y Fu(2)2634 4483 y Fv(\()p Fy(x)p Fv(\))24 b Fz(,)-9 4701 y(where)951 4908 y Fy(B)1014 4920 y Fu(1)1051 4908 y Fv(\()p Fy(x)p Fv(\))g(=)1274 4795 y Fn(Z)1357 4815 y Fq(1)1320 4983 y Ft(x)1406 4908 y Fy(dx)1496 4920 y Fu(0)1555 4908 y Fy(e)1594 4873 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)1748 4881 y Fo(0)1779 4873 y Fu(\))1823 4795 y Fn(Z)1906 4908 y Fy(dk)g Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2158 4920 y Fu(0)2196 4908 y Fy(;)14 b(k)s Fv(\))p Fs(j)23 b Fz(,)951 5134 y Fy(B)1014 5146 y Fu(2)1051 5134 y Fv(\()p Fy(x)p Fv(\))h(=)1274 5021 y Fn(Z)1357 5134 y Fy(dk)g Fs(j)p Fy(k)s Fs(j)1559 5100 y Fu(1+)p Ft(\026)1701 5021 y Fn(Z)1784 5042 y Fq(1)1748 5210 y Ft(x)1833 5134 y Fy(dx)1923 5146 y Fu(0)1982 5134 y Fy(e)2021 5100 y Fu(\003)2066 5108 y Fo(+)2113 5100 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)2355 5108 y Fo(0)2387 5100 y Fu(\))2431 5134 y Fs(j)7 b Fv(^)-49 b Fy(q)s Fv(\()p Fy(x)2573 5146 y Fu(0)2611 5134 y Fy(;)14 b(k)s Fv(\))p Fs(j)37 b Fy(:)-9 5355 y Fz(No)n(w)-6 b(,)23 b(we)e(ha)n(v)o(e)1259 5569 y Fy(B)1322 5581 y Fu(1)1359 5569 y Fv(\()p Fy(x)p Fv(\))j Fs(\024)41 b Fz(sup)1582 5641 y Ft(x)1620 5649 y Fo(0)1652 5641 y Fq(\025)p Fu(1)1751 5456 y Fn(Z)1833 5569 y Fy(dk)25 b Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2086 5581 y Fu(0)2124 5569 y Fy(;)14 b(k)s Fv(\))p Fs(j)23 b Fy(<)g Fs(1)g Fy(:)1838 5909 y Fz(23)p eop %%Page: 24 24 bop -9 145 a Fz(T)-7 b(o)25 b(estimate)d Fy(B)473 157 y Fu(2)510 145 y Fz(,)k(we)e(\256rst)h(use)f(that)g(for)h(all)f Fv(0)j Fs(\024)f Fy(\026)i Fs(\024)f Fv(1)p Fz(,)d(there)h(e)o(xists)e(a)h(constant) h Fy(C)30 b Fz(such)c(that)d(for)j(all)d Fy(x)3258 157 y Fu(0)3323 145 y Fs(\025)k Fy(x)h Fs(\025)f Fv(1)c Fz(and)-9 244 y Fy(k)i Fs(2)f Fr(R)p Fz(,)d(we)g(ha)n(v)o(e)856 423 y Fs(j)p Fy(k)s Fs(j)948 389 y Ft(\026)993 423 y Fy(e)1032 389 y Fu(\003)1077 397 y Fo(+)1124 389 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)1366 397 y Fo(0)1398 389 y Fu(\))1451 423 y Fs(\024)i(j)p Fv(\003)1620 435 y Fu(+)1674 423 y Fv(\()p Fy(k)s Fv(\))p Fs(j)1807 389 y Ft(\026)1852 423 y Fy(e)1891 389 y Fu(\003)1936 397 y Fo(+)1983 389 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)2225 397 y Fo(0)2257 389 y Fu(\))2311 423 y Fs(\024)f Fy(C)6 b Fv(\()p Fy(x)2542 435 y Fu(0)2599 423 y Fs(\000)18 b Fy(x)p Fv(\))2761 389 y Fq(\000)p Ft(\026)2881 423 y Fy(:)-9 603 y Fz(Splitting)j(the)g(inte)o (gration)g(interv)n(al)f(in)i(tw)o(o)g(parts)f(gi)n(v)o(es)381 726 y Fn(Z)464 839 y Fy(dk)588 726 y Fn(Z)671 747 y Ft(x)p Fu(+2)634 915 y Ft(x)775 839 y Fy(dx)865 851 y Fu(0)925 839 y Fy(e)964 805 y Fu(\003)1009 813 y Fo(+)1055 805 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)1297 813 y Fo(0)1330 805 y Fu(\))1360 839 y Fs(j)p Fy(k)s Fs(j)1452 805 y Fu(1+)p Ft(\026)1580 839 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)1722 851 y Fu(0)1760 839 y Fy(;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)2088 726 y Fn(Z)2171 839 y Fy(dk)2295 726 y Fn(Z)2378 747 y Ft(x)p Fu(+2)2341 915 y Ft(x)2482 839 y Fy(dx)2572 851 y Fu(0)2792 783 y Fv(1)p 2641 820 343 4 v 2641 896 a(\()p Fy(x)2720 908 y Fu(0)2777 896 y Fs(\000)18 b Fy(x)p Fv(\))2939 872 y Ft(\026)2994 839 y Fs(j)p Fy(k)30 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)3203 851 y Fu(0)3241 839 y Fy(;)14 b(k)s Fv(\))p Fs(j)1921 1063 y(\024)23 b Fy(C)44 b Fz(sup)2109 1135 y Ft(x)p Fq(\025)p Fu(1)2246 950 y Fn(Z)2329 1063 y Fy(dk)24 b Fs(j)p Fy(k)30 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)418 1192 y Fn(Z)501 1305 y Fy(dk)625 1192 y Fn(Z)708 1212 y Fq(1)671 1380 y Ft(x)p Fu(+2)775 1305 y Fy(dx)865 1317 y Fu(0)925 1305 y Fy(e)964 1271 y Fu(\003)1009 1279 y Fo(+)1055 1271 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)1297 1279 y Fo(0)1330 1271 y Fu(\))1360 1305 y Fs(j)p Fy(k)s Fs(j)1452 1271 y Fu(1+)p Ft(\026)1580 1305 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)1722 1317 y Fu(0)1760 1305 y Fy(;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)2088 1192 y Fn(Z)2171 1305 y Fy(dk)2295 1192 y Fn(Z)2378 1212 y Fq(1)2341 1380 y Ft(x)p Fu(+2)2445 1305 y Fy(dx)2535 1317 y Fu(0)2755 1249 y Fv(1)p 2605 1286 V 2605 1362 a(\()p Fy(x)2684 1374 y Fu(0)2740 1362 y Fs(\000)18 b Fy(x)p Fv(\))2902 1338 y Ft(\026)2957 1305 y Fs(j)p Fy(k)9 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)3145 1317 y Fu(0)3183 1305 y Fy(;)14 b(k)s Fv(\))p Fs(j)1921 1538 y(\024)23 b Fy(C)2109 1425 y Fn(Z)2192 1446 y Fq(1)2156 1614 y Fu(1)2241 1538 y Fy(dx)2331 1550 y Fu(0)2383 1425 y Fn(Z)2466 1538 y Fy(dk)h Fs(j)p Fy(k)30 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b Fy(:)-9 1786 y Fz(Since)c Fs(j)p Fy(\034)9 b Fv(\()p Fy(k)s Fv(\))p Fs(j)24 b Fv(=)541 1745 y Fu(\001\()p Ft(k)q Fu(\))p 523 1766 181 4 v 523 1814 a(\003)568 1822 y Fo(+)615 1814 y Fu(\()p Ft(k)q Fu(\))713 1786 y Fs(j)760 1765 y Fv(^)736 1786 y Fy(N)803 1798 y Fu(2)p Ft(;)p Fu(2)893 1786 y Fv(\(1)p Fy(;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g(j)1263 1765 y Fv(^)1239 1786 y Fy(N)1306 1798 y Fu(2)p Ft(;)p Fu(2)1396 1786 y Fv(\(1)p Fy(;)14 b(k)s Fv(\))p Fs(j)p Fz(,)20 b(we)h(get)f(that)g Fy(\034)32 b Fs(2)24 b Fy(L)2245 1755 y Fu(1)2282 1786 y Fv(\()p Fr(R)p Fy(;)14 b Fv(\(1)i(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2718 1755 y Fu(1+)p Ft(\026)2847 1786 y Fy(dk)s Fv(\))k Fz(for)i(all)d Fv(0)k Fs(\024)f Fy(\026)h(<)g Fv(1)p Fz(.)30 b(This)-9 1916 y(implies)16 b(that)426 1895 y Fv(^)402 1916 y Fy(N)469 1928 y Fu(1)523 1916 y Fz(and)687 1895 y Fv(^)663 1916 y Fy(N)730 1928 y Fu(2)784 1916 y Fz(are)h(both)h(in)f Fy(L)1213 1886 y Fu(1)1250 1916 y Fv(\()p Fr(R)p Fy(;)d Fv(\(1)9 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))1672 1886 y Fu(1+)p Ft(\026)1801 1916 y Fy(dk)s Fv(\))17 b Fz(for)h(all)e Fv(0)23 b Fs(\024)f Fy(\026)h(<)g Fv(1)p Fz(,)18 b(and)f(that)23 b Fv(^)-48 b Fy(\021)2876 1928 y Fu(0)p Ft(;q)2989 1916 y Fs(2)23 b Fy(L)3124 1886 y Fu(1)3161 1916 y Fv(\()p Fr(R)p Fy(;)14 b Fv(\(1)9 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))3583 1886 y Ft(\026)3628 1916 y Fy(dk)s Fv(\))p Fz(.)-9 2016 y(From)21 b(this)f(we)g(conclude)g(that)g Fy(N)998 2028 y Fu(1)1035 2016 y Fz(,)g Fy(N)1143 2028 y Fu(2)1180 2016 y Fz(,)h Fy(@)1266 2028 y Ft(y)1306 2016 y Fy(N)1373 2028 y Fu(1)1410 2016 y Fz(,)f Fy(@)1495 2028 y Ft(y)1535 2016 y Fy(N)1602 2028 y Fu(2)1659 2016 y Fz(and)h Fy(\021)1843 2028 y Fu(0)p Ft(;q)1953 2016 y Fz(are)f(bounded.)33 b(W)-7 b(e)20 b(no)n(w)h(turn)g(to)f(the)g(continuity)-6 b(.)32 b(First,)20 b(we)-9 2116 y(sho)n(w)k(that)g(the)f(F)o(ourier)i(transform)g(of)f Fy(P)1244 2128 y Fu(1)1281 2116 y Fy(\034)9 b Fz(,)26 b Fy(P)1426 2128 y Fu(2)1463 2116 y Fy(\034)34 b Fz(and)24 b(their)f(deri)n(v)n(ati)n(v)o (es)g(with)h(respect)f(to)h Fy(y)i Fz(are)e(Ho)-35 b(\310)7 b(lder)n(\261)p Fy(\026)25 b Fz(in)f Fy(x;)14 b(y)26 b Fz(for)f(all)-9 2215 y Fv(0)e Fy(<)g(\026)h(<)f Fv(1)p Fz(,)f(and)g(thus)h(continuous)g(in)f Fy(x;)14 b(y)s Fz(.)33 b(T)-7 b(o)22 b(do)h(this,)f(let)e Fs(j)p Fy(\033)s Fs(j)25 b(\024)e Fv(1)e Fz(with)h Fy(x)d Fv(+)f Fy(\033)28 b Fs(\025)23 b Fv(1)e Fz(and)i Fv(0)g Fy(<)g(\026)h(<)f Fv(1)p Fz(.)32 b(By)22 b(Lemma)f(C.4,)-9 2315 y(we)g(ha)n(v)o(e)287 2404 y Fn(Z)370 2517 y Fy(dk)j Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))j Fs(j)p Fy(P)877 2529 y Ft(i)905 2517 y Fv(\()p Fy(x)e Fv(+)f Fy(\033)n(;)c(k)s Fv(\))20 b Fs(\000)e Fy(P)1402 2529 y Ft(i)1430 2517 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)22 b(j)p Fy(\034)9 b Fv(\()p Fy(k)s Fv(\))p Fs(j)24 b(\024)f(j)p Fy(\033)s Fs(j)2078 2483 y Ft(\026)2144 2517 y Fy(C)2244 2404 y Fn(Z)2327 2517 y Fy(dk)h Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2736 2483 y Fu(1+)p Ft(\026)2887 2517 y Fs(j)p Fy(\034)9 b Fv(\()p Fy(k)s Fv(\))p Fs(j)24 b(\024)e(j)p Fy(\033)s Fs(j)3295 2483 y Ft(\026)3362 2517 y Fy(C)29 b(:)-9 2755 y Fz(Ne)o(xt)35 b(we)g(sho)n(w)i(that)e(the)g(same)g(holds)i(for)1427 2734 y Fv(^)1403 2755 y Fy(N)1470 2767 y Ft(i;)p Fu(1)1586 2755 y Fz(and)1768 2734 y Fv(^)1744 2755 y Fy(N)1811 2767 y Ft(i;)p Fu(2)1891 2755 y Fz(.)73 b(Namely)-6 b(,)40 b(let)35 b Fs(j)p Fy(\033)s Fs(j)42 b(\024)g Fv(1)35 b Fz(with)h Fy(x)26 b Fv(+)g Fy(\033)46 b Fs(\025)c Fv(1)p Fz(,)c(and)e(de\256ne)-4 2855 y Fv(\026)-47 b Fy(x)23 b Fv(=)g Fz(min)p Fs(f)p Fy(x;)14 b(x)19 b Fv(+)f Fy(\033)s Fs(g)p Fz(.)31 b(W)-7 b(e)21 b(ha)n(v)o(e)295 2996 y Fn(\014)295 3046 y(\014)295 3096 y(\014)347 3071 y Fv(^)323 3092 y Fy(N)390 3104 y Ft(i;)p Fu(1)470 3092 y Fv(\()p Fy(x)e Fv(+)f Fy(\033)n(;)c(k)s Fv(\))19 b Fs(\000)938 3071 y Fv(^)913 3092 y Fy(N)980 3104 y Ft(i;)p Fu(1)1061 3092 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1255 2996 y Fn(\014)1255 3046 y(\014)1255 3096 y(\014)1306 3092 y Fs(\024)23 b Fz(s)-5 b(i)g(g)g(n)q Fv(\()p Fy(\033)s Fv(\))1649 2979 y Fn(Z)1733 2999 y Ft(x)p Fu(+)p Ft(\033)1696 3167 y(x)1845 3092 y Fy(dx)1935 3104 y Fu(0)1994 3092 y Fs(j)p Fy(q)s Fv(\()p Fy(x)2136 3104 y Fu(0)2174 3092 y Fy(;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f(\(1)g Fs(\000)g Fy(e)2628 3057 y Fq(\000j)p Fu(\003)2745 3065 y Fc(\000)2794 3057 y Fu(\()p Ft(k)q Fu(\))p Ft(\033)r Fq(j)2947 3092 y Fv(\))p Fs(j)3026 3071 y Fv(^)3002 3092 y Fy(N)3069 3104 y Ft(i;)p Fu(1)3149 3092 y Fv(\()5 b(\026)-47 b Fy(x)q(;)14 b(k)s Fv(\))p Fs(j)1306 3329 y(\024)23 b Fz(s)-5 b(i)g(g)g(n)q Fv(\()p Fy(\033)s Fv(\))1649 3216 y Fn(Z)1733 3237 y Ft(x)p Fu(+)p Ft(\033)1696 3405 y(x)1845 3329 y Fy(dx)1935 3341 y Fu(0)1994 3329 y Fs(j)p Fy(q)s Fv(\()p Fy(x)2136 3341 y Fu(0)2174 3329 y Fy(;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fy(C)27 b Fs(j)p Fy(\033)s Fs(j)2596 3295 y Ft(\026)2641 3329 y Fs(j)p Fv(\003)2722 3341 y Fq(\000)2778 3329 y Fv(\()p Fy(k)s Fv(\))p Fs(j)2911 3295 y Ft(\026)2956 3329 y Fs(j)3003 3308 y Fv(^)2979 3329 y Fy(N)3046 3341 y Ft(i;)p Fu(1)3126 3329 y Fv(\()5 b(\026)-47 b Fy(x)q(;)14 b(k)s Fv(\))p Fs(j)1306 3566 y(\024)23 b Fz(s)-5 b(i)g(g)g(n)q Fv(\()p Fy(\033)s Fv(\))1649 3453 y Fn(Z)1733 3474 y Ft(x)p Fu(+)p Ft(\033)1696 3642 y(x)1845 3566 y Fy(dx)1935 3578 y Fu(0)1994 3566 y Fs(j)p Fy(q)s Fv(\()p Fy(x)2136 3578 y Fu(0)2174 3566 y Fy(;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fy(C)27 b Fs(j)p Fy(\033)s Fs(j)2596 3532 y Ft(\026)2641 3566 y Fs(j)p Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))p Fs(j)2986 3532 y Ft(\026)3031 3566 y Fs(j)3078 3545 y Fv(^)3054 3566 y Fy(N)3121 3578 y Ft(i;)p Fu(1)3202 3566 y Fv(\()5 b(\026)-47 b Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b Fy(:)-9 3787 y Fz(From)f(this,)f(we)g (conclude)h(easily)f(that)810 3894 y Fn(Z)892 4007 y Fy(dk)k Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))1316 3911 y Fn(\014)1316 3961 y(\014)1316 4011 y(\014)1368 3986 y Fv(^)1344 4007 y Fy(N)1411 4019 y Ft(i;)p Fu(1)1491 4007 y Fv(\()p Fy(x)h Fv(+)f Fy(\033)n(;)c(k)s Fv(\))19 b Fs(\000)1958 3986 y Fv(^)1934 4007 y Fy(N)2001 4019 y Ft(i;)p Fu(1)2082 4007 y Fv(\()p Fy(x;)14 b(k)s Fv(\))2276 3911 y Fn(\014)2276 3961 y(\014)2276 4011 y(\014)2327 4007 y Fs(\024)23 b Fv(\()p Fs(j)p Fy(\033)s Fs(j)c Fv(+)f Fs(j)p Fy(\033)s Fs(j)2741 3972 y Ft(\026)2786 4007 y Fv(\))k Fy(C)29 b(:)-9 4245 y Fz(F)o(or)153 4224 y Fv(^)129 4245 y Fy(N)196 4257 y Ft(i;)p Fu(2)276 4245 y Fz(,)21 b(let)f Fs(j)p Fy(\033)s Fs(j)k(\024)f Fv(1)d Fz(with)i Fy(x)d Fv(+)f Fy(\033)26 b Fs(\025)d Fv(1)p Fz(,)e(and)h(de\256ne)27 b Fv(\026)-47 b Fy(x)23 b Fv(=)g Fz(max)p Fs(f)p Fy(x;)14 b(x)19 b Fv(+)f Fy(\033)s Fs(g)p Fz(.)31 b(W)-7 b(e)21 b(ha)n(v)o(e)322 4389 y Fn(\014)322 4439 y(\014)322 4489 y(\014)374 4464 y Fv(^)350 4485 y Fy(N)417 4497 y Ft(i;)p Fu(2)497 4485 y Fv(\()p Fy(x)e Fv(+)g Fy(\033)n(;)14 b(k)s Fv(\))19 b Fs(\000)965 4464 y Fv(^)941 4485 y Fy(N)1008 4497 y Ft(i;)p Fu(2)1088 4485 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1282 4389 y Fn(\014)1282 4439 y(\014)1282 4489 y(\014)1333 4485 y Fs(\024)23 b Fz(s)-5 b(i)g(g)g(n)r Fv(\()p Fy(\033)s Fv(\))1677 4372 y Fn(Z)1760 4392 y Ft(x)p Fu(+)p Ft(\033)1724 4560 y(x)1872 4485 y Fy(dx)1962 4497 y Fu(0)2021 4485 y Fs(j)p Fy(q)s Fv(\()p Fy(x)2163 4497 y Fu(0)2201 4485 y Fy(;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f(\(1)g Fs(\000)g Fy(e)2655 4451 y Fq(\000j)p Fu(\003)2772 4459 y Fo(+)2819 4451 y Fu(\()p Ft(k)q Fu(\))p Ft(\033)r Fq(j)2972 4485 y Fv(\))p Fs(j)3051 4464 y Fv(^)3027 4485 y Fy(N)3094 4497 y Ft(i;)p Fu(2)3174 4485 y Fv(\()5 b(\026)-47 b Fy(x)q(;)14 b(k)s Fv(\))p Fs(j)23 b Fy(:)-9 4706 y Fz(By)e(the)g(same)g(ar)n(guments)i (as)e(abo)o(v)o(e,)g(we)h(conclude)g(that)810 4812 y Fn(Z)892 4925 y Fy(dk)j Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))1316 4830 y Fn(\014)1316 4879 y(\014)1316 4929 y(\014)1368 4904 y Fv(^)1344 4925 y Fy(N)1411 4937 y Ft(i;)p Fu(2)1491 4925 y Fv(\()p Fy(x)h Fv(+)f Fy(\033)n(;)c(k)s Fv(\))19 b Fs(\000)1958 4904 y Fv(^)1934 4925 y Fy(N)2001 4937 y Ft(i;)p Fu(2)2082 4925 y Fv(\()p Fy(x;)14 b(k)s Fv(\))2276 4830 y Fn(\014)2276 4879 y(\014)2276 4929 y(\014)2327 4925 y Fs(\024)23 b Fv(\()p Fs(j)p Fy(\033)s Fs(j)c Fv(+)f Fs(j)p Fy(\033)s Fs(j)2741 4891 y Ft(\026)2786 4925 y Fv(\))k Fy(C)29 b(:)-9 5142 y Fz(W)-7 b(e)21 b(ha)n(v)o(e)h(thus)f(sho)n(wn)i(that)863 5231 y Fn(Z)946 5344 y Fy(dk)h Fv(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1370 5248 y Fn(\014)1370 5298 y(\014)1370 5348 y(\014)1421 5323 y Fv(^)1397 5344 y Fy(N)1464 5356 y Ft(i)1492 5344 y Fv(\()p Fy(x)h Fv(+)f Fy(\033)n(;)c(k)s Fv(\))19 b Fs(\000)1959 5323 y Fv(^)1935 5344 y Fy(N)2002 5356 y Ft(i)2030 5344 y Fv(\()p Fy(x;)14 b(k)s Fv(\))2224 5248 y Fn(\014)2224 5298 y(\014)2224 5348 y(\014)2275 5344 y Fs(\024)23 b Fy(C)k Fv(\()p Fs(j)p Fy(\033)s Fs(j)20 b Fv(+)e Fs(j)p Fy(\033)s Fs(j)2776 5309 y Ft(\026)2821 5344 y Fv(\))23 b Fz(,)-9 5560 y(so)e(that)553 5660 y Fs(j)p Fy(N)643 5672 y Ft(i)671 5660 y Fv(\()p Fy(x)e Fv(+)f Fy(\033)n(;)c(y)s Fv(\))19 b Fs(\000)f Fy(N)1179 5672 y Ft(i)1206 5660 y Fv(\()p Fy(x;)c(y)s Fv(\))p Fs(j)20 b Fv(+)e Fs(j)p Fy(@)1591 5672 y Ft(y)1631 5660 y Fy(N)1698 5672 y Ft(i)1725 5660 y Fv(\()p Fy(x)h Fv(+)f Fy(\033)n(;)c(y)s Fv(\))19 b Fs(\000)f Fy(@)2210 5672 y Ft(y)2250 5660 y Fy(N)2317 5672 y Ft(i)2345 5660 y Fv(\()p Fy(x;)c(y)s Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fv(\()p Fs(j)p Fy(\033)s Fs(j)19 b Fv(+)f Fs(j)p Fy(\033)s Fs(j)3084 5626 y Ft(\026)3129 5660 y Fv(\))23 b Fy(:)1838 5909 y Fz(24)p eop %%Page: 25 25 bop -9 145 a Fz(This)24 b(implies)f(that)h Fy(N)659 157 y Ft(i)710 145 y Fz(and)h Fy(@)901 157 y Ft(y)941 145 y Fy(N)1008 157 y Ft(i)1060 145 y Fz(are)f(Ho)-35 b(\310)7 b(lder)n(\261)p Fy(\026)25 b Fz(in)g Fy(x)f Fz(for)i(all)d Fv(0)j Fy(<)h(\026)g(<)f Fv(1)p Fz(.)40 b(T)-7 b(o)24 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y(\014)2985 482 y(\014)3037 457 y Fv(^)3013 478 y Fy(N)3080 490 y Ft(i)3107 478 y Fv(\()p Fy(x;)c(k)s Fv(\))3301 382 y Fn(\014)3301 432 y(\014)3301 482 y(\014)3352 478 y Fs(\024)23 b Fy(C)6 b Fs(j)p Fy(\033)s Fs(j)3601 443 y Ft(\026)3669 478 y Fy(:)-9 708 y Fz(Finally)-6 b(,)21 b(to)h(see)e(that)h Fy(\021)675 720 y Fu(0)p Ft(;q)786 708 y Fz(is)g(continuous)i(in)e Fy(x;)14 b(y)s Fz(,)22 b(use)f(that)27 b Fv(^)-48 b Fy(\021)1840 720 y Fu(+)p Ft(;q)1971 708 y Fs(2)23 b Fy(L)2106 678 y Fu(1)2143 708 y Fv(\()p Fr(R)p Fy(;)14 b Fv(\(1)k(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2583 678 y Ft(\026)2628 708 y Fy(dk)s Fv(\))p Fz(,)k(and)f(that)1080 823 y Fn(\014)1080 873 y(\014)1080 923 y(\014)1108 918 y Fy(e)1147 884 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fu(+)p Ft(\033)r Fq(\000)p Fu(1\))1563 918 y Fs(\000)d Fy(e)1685 884 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1991 823 y Fn(\014)1991 873 y(\014)1991 923 y(\014)2042 918 y Fs(\024)23 b(j)p Fy(\033)s Fs(j)2226 884 y Ft(\026)2292 918 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Fs(j)p Fy(k)s Fs(j)p Fv(\))2951 1492 y Fu(1+)p Ft(\026)3080 1526 y Fs(j)p Fy(\034)9 b Fv(\()p Fy(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)6 b Fs(j)p Fy(\033)s Fs(j)3554 1492 y Ft(\026)3599 1526 y Fy(:)-9 1746 y Fz(This)21 b(completes)g(the)g(proof.)p 911 1746 43 43 v -9 1987 a Fd(C.4)102 b(Pr)n(oof)26 b(of)f(Lemma)h(3.5)-9 2144 y FA(Lemma)20 b(C.7)j(\(Lemma)d(3.5\))44 b Fm(Let)1072 2122 y Fv(^)1072 2144 y Fy(h)1120 2156 y Fu(1)1157 2144 y Fy(;)1194 2122 y Fv(^)1194 2144 y Fy(h)1242 2156 y Fu(2)1304 2144 y Fv(:)24 b(\012)1411 2156 y Fu(+)1491 2144 y Fs(!)h Fr(R)d Fm(be)h(bounded,)j(and)e (assume)g(that)2711 2122 y Fv(^)2710 2144 y Fy(h)2758 2156 y Ft(i)2811 2144 y Fs(2)h Fy(L)2948 2114 y Fu(1)2985 2144 y Fv(\(\012)3077 2156 y Fu(+)3132 2144 y Fy(;)14 b Fv(\(1)19 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))3470 2114 y Ft(\026)3514 2144 y Fy(dk)26 b(dx)p Fv(\))p Fm(,)-9 2253 y Fz(sup)109 2275 y Ft(x)p Fq(\025)p Fu(1)249 2186 y Fn(R)304 2253 y Fy(dk)e Fv(\(1)16 b(+)h Fs(j)p Fy(k)s Fs(j)p Fv(\))710 2222 y Ft(\026)755 2253 y Fs(j)779 2231 y Fv(^)778 2253 y Fy(h)826 2265 y Ft(i)853 2253 y Fv(\()p Fy(x;)d(k)s Fv(\))p Fs(j)24 b Fy(<)f Fs(1)d Fm(for)h(some)g Fy(\026)i(>)g Fv(0)p Fm(.)30 b(Assume)21 b(furthermor)m(e)g (that)2716 2231 y Fv(^)2715 2253 y Fy(h)2763 2265 y Fu(1)2820 2253 y Fm(is)f(e)o(ven)g(in)h Fy(k)i Fm(and)3374 2231 y Fv(^)3373 2253 y Fy(h)3421 2265 y Fu(2)3479 2253 y Fm(is)d(odd)i(in)-9 2352 y Fy(k)s Fm(.)30 b(De\256ne)22 b Fy(q)368 2364 y Fu(1)406 2352 y Fm(,)f Fy(q)485 2364 y Fu(2)522 2352 y Fm(,)h Fy(q)602 2364 y Fu(1)p Ft(;p)714 2352 y Fm(and)h Fy(q)900 2364 y Fu(2)p Ft(;p)1013 2352 y Fm(as)e(the)g(F)-9 b(ourier)22 b(tr)o(ansforms)g(of)526 2588 y Fv(^)-48 b Fy(q)557 2600 y Fu(1)594 2588 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)900 2475 y Fn(Z)983 2495 y Fq(1)946 2663 y Fu(1)1031 2588 y Fy(dx)1121 2600 y Fu(0)1194 2471 y Fn(\024)1238 2588 y Fs(\000)1313 2531 y Fv(1)p 1313 2569 42 4 v 1313 2645 a(2)1364 2588 y Fy(i)35 b Fz(s)-5 b(i)g(g)g(n)q Fv(\()p Fy(k)s Fv(\))1679 2495 y Fn(\020)1729 2588 y Fy(e)1768 2553 y Fq(\000j)p Ft(x)p Fq(\000)p Ft(x)1968 2561 y Fo(0)1999 2553 y Fq(jj)p Ft(k)q Fq(j)2117 2588 y Fs(\000)19 b Fy(e)2240 2553 y Fq(\000)p Fu(\()p Ft(x)p Fu(+)p Ft(x)2445 2561 y Fo(0)2476 2553 y Fq(\000)p Fu(2\))p Fq(j)p Ft(k)q Fq(j)2667 2495 y Fn(\021)2716 2471 y(\025)2796 2566 y Fv(^)2795 2588 y Fy(h)2843 2600 y Fu(1)2880 2588 y Fv(\()p Fy(x)2959 2600 y Fu(0)2997 2588 y Fy(;)14 b(k)s Fv(\))24 b Fm(,)526 2820 y Fv(^)-48 b Fy(q)557 2832 y Fu(2)594 2820 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)900 2707 y Fn(Z)983 2728 y Fq(1)946 2896 y Fu(1)1031 2820 y Fy(dx)1121 2832 y Fu(0)1194 2703 y Fn(\024)1248 2764 y Fv(1)p 1248 2801 V 1248 2877 a(2)1313 2728 y Fn(\020)1363 2820 y Fy(e)1402 2786 y Fq(\000j)p Ft(x)p Fq(\000)p Ft(x)1602 2794 y Fo(0)1633 2786 y Fq(jj)p Ft(k)q Fq(j)1747 2820 y Fz(s)-5 b(i)g(g)g(n)q Fv(\()p Fy(x)19 b Fs(\000)f Fy(x)2102 2832 y Fu(0)2140 2820 y Fv(\))h(+)f Fy(e)2313 2786 y Fq(\000)p Fu(\()p Ft(x)p Fu(+)p Ft(x)2518 2794 y Fo(0)2549 2786 y Fq(\000)p Fu(2\))p Fq(j)p Ft(k)q Fq(j)2740 2728 y Fn(\021)2790 2703 y(\025)2870 2798 y Fv(^)2869 2820 y Fy(h)2917 2832 y Fu(2)2954 2820 y Fv(\()p Fy(x)3033 2832 y Fu(0)3071 2820 y Fy(;)c(k)s Fv(\))23 b Fm(,)472 3053 y Fv(^)-48 b Fy(q)503 3065 y Fu(1)p Ft(;p)594 3053 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)900 2940 y Fn(Z)983 2960 y Fq(1)946 3128 y Fu(1)1031 3053 y Fy(dx)1121 3065 y Fu(0)1194 2936 y Fn(\024)1238 3053 y Fs(\000)1313 2996 y Fv(1)p 1313 3033 V 1313 3110 a(2)1378 2960 y Fn(\020)1428 3053 y Fy(e)1467 3018 y Fq(\000j)p Ft(x)p Fq(\000)p Ft(x)1667 3026 y Fo(0)1697 3018 y Fq(jj)p Ft(k)q Fq(j)1811 3053 y Fz(s)-5 b(i)g(g)g(n)r Fv(\()p Fy(x)19 b Fs(\000)f Fy(x)2167 3065 y Fu(0)2205 3053 y Fv(\))g Fs(\000)g Fy(e)2377 3018 y Fq(\000)p Fu(\()p Ft(x)p Fu(+)p Ft(x)2582 3026 y Fo(0)2614 3018 y Fq(\000)p Fu(2\))p Fq(j)p Ft(k)q Fq(j)2805 2960 y Fn(\021)2854 2936 y(\025)2934 3031 y Fv(^)2933 3053 y Fy(h)2981 3065 y Fu(1)3018 3053 y Fv(\()p Fy(x)3097 3065 y Fu(0)3135 3053 y Fy(;)c(k)s Fv(\))23 b Fm(,)472 3285 y Fv(^)-48 b Fy(q)503 3297 y Fu(2)p Ft(;p)594 3285 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)900 3172 y Fn(Z)983 3193 y Fq(1)946 3361 y Fu(1)1031 3285 y Fy(dx)1121 3297 y Fu(0)1194 3168 y Fn(\024)1238 3285 y Fs(\000)1313 3229 y Fv(1)p 1313 3266 V 1313 3342 a(2)1364 3285 y Fy(i)35 b Fz(s)-5 b(i)g(g)g(n)q Fv(\()p Fy(k)s Fv(\))1679 3193 y Fn(\020)1729 3285 y Fy(e)1768 3251 y Fq(\000j)p Ft(x)p Fq(\000)p Ft(x)1968 3259 y Fo(0)1999 3251 y Fq(jj)p Ft(k)q Fq(j)2117 3285 y Fv(+)19 b Fy(e)2240 3251 y Fq(\000)p Fu(\()p Ft(x)p Fu(+)p Ft(x)2445 3259 y Fo(0)2476 3251 y Fq(\000)p Fu(2\))p Fq(j)p Ft(k)q Fq(j)2667 3193 y Fn(\021)2716 3168 y(\025)2796 3263 y Fv(^)2795 3285 y Fy(h)2843 3297 y Fu(2)2880 3285 y Fv(\()p Fy(x)2959 3297 y Fu(0)2997 3285 y Fy(;)14 b(k)s Fv(\))24 b Fy(:)-9 3513 y Fm(Then)i(the)g(functions)g Fy(q)690 3525 y Ft(i)718 3513 y Fm(,)h Fy(q)803 3525 y Ft(i;p)884 3513 y Fm(,)g Fy(@)976 3525 y Ft(y)1016 3513 y Fy(q)1053 3525 y Ft(i)1107 3513 y Fm(and)g Fy(@)1304 3525 y Ft(y)1344 3513 y Fy(q)1381 3525 y Ft(i;p)1488 3513 y Fm(ar)m(e)f(bounded)i(and)f(continuous)h (in)e Fy(x;)14 b(y)s Fm(,)27 b(the)e(functions)h Fy(q)3265 3525 y Ft(i)3319 3513 y Fm(ar)m(e)g(e)o(ven)f(in)g Fy(y)-9 3613 y Fm(and)d(the)f(functions)h Fy(q)634 3625 y Ft(i;p)737 3613 y Fm(ar)m(e)f(odd)i(in)e Fy(y)s Fm(.)-9 3795 y(Pr)l(oof)o(.)36 b Fz(First)23 b(of)h(all,)e(the)h(parity)g(of)h(the)f(functions)h(is)f(tri)n (vial.)35 b(The)23 b(proof)i(is)e(v)o(ery)h(similar)e(to)h(the)g(one)g(of)h (Lemma)f(3.2.)37 b(W)-7 b(e)-9 3895 y(sho)n(w)22 b(that)28 b Fv(^)-48 b Fy(q)376 3907 y Ft(i)427 3895 y Fs(2)24 b Fy(L)563 3865 y Fu(1)600 3895 y Fv(\()p Fr(R)p Fy(;)14 b Fv(\(1)k(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))1040 3865 y Fu(1+)p Ft(\027)1166 3895 y Fy(dk)s Fv(\))p Fz(,)28 b Fv(^)-48 b Fy(q)1367 3907 y Ft(i;p)1472 3895 y Fs(2)24 b Fy(L)1608 3865 y Fu(1)1645 3895 y Fv(\()p Fr(R)p Fy(;)14 b Fv(\(1)k(+)h Fs(j)p Fy(k)s Fs(j)p Fv(\))2086 3865 y Fu(1+)p Ft(\027)2211 3895 y Fy(dk)s Fv(\))j Fz(for)h(all)e Fv(0)i Fs(\024)g Fy(\027)29 b(<)23 b(\026)p Fz(.)32 b(W)-7 b(e)22 b(\256rst)g(note)g(that)f(there)-9 3994 y(e)o(xists)f(a)h(constant)g Fy(C)28 b Fz(such)22 b(that)f(for)h(all)e Fy(k)26 b Fs(2)d Fr(R)p Fz(,)f Fy(x;)14 b(x)1599 4006 y Fu(0)1660 3994 y Fs(\025)22 b Fv(1)p Fz(,)1489 4177 y Fv(\(1)c(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))1788 4143 y Fu(1+)p Ft(\027)1937 4177 y Fs(\024)k Fy(C)6 b Fv(\(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))2421 4143 y Ft(\026)2484 4177 y Fv(+)g Fs(j)p Fy(k)s Fs(j)2659 4143 y Fu(1+)p Ft(\027)2785 4177 y Fv(\))23 b Fz(,)899 4231 y Fn(\014)899 4281 y(\014)899 4331 y(\014)927 4326 y Fy(e)966 4292 y Fq(\000j)p Ft(k)q Fq(j)e(j)p Ft(x)p Fq(\000)p Ft(x)1263 4300 y Fo(0)1294 4292 y Fq(j)1337 4326 y Fs(\006)d Fy(e)1459 4292 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fu(+)p Ft(x)1740 4300 y Fo(0)1771 4292 y Fq(\000)p Fu(2\))1886 4231 y Fn(\014)1886 4281 y(\014)1886 4331 y(\014)1937 4326 y Fs(\024)k Fv(2)f Fy(e)2126 4292 y Fq(\000j)p Ft(k)q Fq(j)g(j)p Ft(x)p Fq(\000)p Ft(x)2423 4300 y Fo(0)2454 4292 y Fq(j)2501 4326 y Fy(:)-9 4529 y Fz(This)d(last)g(inequality)g(sho)n (ws)i(that)e(the)g(proof)j(will)d(be)g(v)o(ery)i(similar)d(for)j Fy(q)2182 4541 y Ft(i)2228 4529 y Fz(and)f(for)h Fy(q)2524 4541 y Ft(i;p)2605 4529 y Fz(.)30 b(F)o(or)20 b(this)e(reason,)i(we)f (consider)g(only)-9 4629 y Fy(q)28 4641 y Ft(i)76 4629 y Fz(in)j(details.)30 b(W)-7 b(e)21 b(ha)n(v)o(e)272 4741 y Fn(Z)355 4854 y Fy(dk)j Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))764 4819 y Fu(1+)p Ft(\027)890 4854 y Fs(j)6 b Fv(^)-48 b Fy(q)950 4866 y Ft(i)978 4854 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)1385 4741 y Fn(Z)1468 4854 y Fy(dk)1592 4741 y Fn(Z)1675 4761 y Fq(1)1638 4929 y Fu(1)1724 4854 y Fy(dx)1814 4866 y Fu(0)1873 4854 y Fv(\(1)c(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2172 4819 y Ft(\026)2217 4854 y Fs(j)p Fy(h)2288 4866 y Ft(i)2316 4854 y Fv(\()p Fy(x)2395 4866 y Fu(0)2433 4854 y Fy(;)c(k)s Fv(\))p Fs(j)k Fv(+)g Fy(C)2751 4741 y Fn(Z)2834 4854 y Fy(dk)24 b Fs(j)p Fy(k)s Fs(j)3036 4819 y Fu(1+)p Ft(\027)3183 4854 y Fs(j)6 b Fv(^)-48 b Fy(q)3243 4866 y Ft(i)3271 4854 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)1219 5075 y(\024)22 b Fy(C)j Fv(+)18 b Fy(C)1552 4962 y Fn(Z)1635 5075 y Fy(dk)24 b Fs(j)p Fy(k)s Fs(j)1837 5041 y Fu(1+)p Ft(\027)1984 5075 y Fs(j)6 b Fv(^)-48 b Fy(q)2044 5087 y Ft(i)2071 5075 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b Fy(:)-9 5295 y Fz(Ne)o(xt,)d(we)g(ha)n(v)o(e)987 5399 y Fn(Z)1070 5512 y Fy(dk)j Fs(j)p Fy(k)s Fs(j)1272 5477 y Fu(1+)p Ft(\027)1397 5512 y Fs(j)6 b Fv(^)-48 b Fy(q)1457 5524 y Ft(i)1485 5512 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fv(2)d Fy(H)1945 5524 y Ft(i;)p Fu(1)2026 5512 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f(2)j Fy(H)2454 5524 y Ft(i;)p Fu(2)2534 5512 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fz(,)1838 5909 y(25)p eop %%Page: 26 26 bop -9 145 a Fz(where)882 367 y Fy(H)951 379 y Ft(i;)p Fu(1)1032 367 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1337 254 y Fn(Z)1420 367 y Fy(dk)1544 254 y Fn(Z)1627 274 y Ft(x)p Fu(+2)1590 442 y(1)1731 367 y Fy(dx)1821 379 y Fu(0)1881 367 y Fs(j)p Fy(k)s Fs(j)1973 333 y Fu(1+)p Ft(\027)2098 367 y Fy(e)2137 333 y Fq(\000j)p Ft(k)q Fu(\()p Ft(x)p Fq(\000)p Ft(x)2399 341 y Fo(0)2430 333 y Fu(\))p Fq(j)2480 367 y Fs(j)p Fy(h)2551 379 y Ft(i)2579 367 y Fv(\()p Fy(x)2658 379 y Fu(0)2696 367 y Fy(;)14 b(k)s Fv(\))p Fs(j)23 b Fz(,)882 593 y Fy(H)951 605 y Ft(i;)p Fu(2)1032 593 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1337 480 y Fn(Z)1420 593 y Fy(dk)1544 480 y Fn(Z)1627 501 y Fq(1)1590 669 y Ft(x)p Fu(+2)1695 593 y Fy(dx)1785 605 y Fu(0)1844 593 y Fs(j)p Fy(k)s Fs(j)p Fy(e)1975 559 y Fq(\000j)p Ft(k)q Fu(\()p Ft(x)p Fq(\000)p Ft(x)2237 567 y Fo(0)2268 559 y Fu(\))p Fq(j)2318 593 y Fs(j)p Fy(k)s Fs(j)2410 559 y Ft(\026)2454 593 y Fs(j)p Fy(h)2525 605 y Ft(i)2553 593 y Fv(\()p Fy(x)2632 605 y Fu(0)2670 593 y Fy(;)14 b(k)s Fv(\))p Fs(j)23 b Fy(:)-9 841 y Fz(Since)e Fs(j)p Fy(k)s Fs(j)293 811 y Ft(\033)337 841 y Fy(e)376 811 y Fq(\000j)p Ft(k)q Fq(jj)p Ft(x)p Fq(\000)p Ft(x)652 819 y Fo(0)683 811 y Fq(j)730 841 y Fs(\024)i Fy(C)6 b Fs(j)p Fy(x)19 b Fs(\000)f Fy(x)1102 853 y Fu(0)1140 841 y Fs(j)1163 811 y Fq(\000)p Ft(\033)1259 841 y Fz(,)k(we)f(ha)n(v)o(e)752 1070 y Fy(H)821 1082 y Ft(i;)p Fu(2)902 1070 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(\024)e Fy(C)6 b Fs(j)p Fy(x)20 b Fs(\000)e Fv(2)p Fs(j)1510 1036 y Fq(\000)p Fu(1)1612 957 y Fn(Z)1695 978 y Fq(1)1658 1146 y Fu(1)1744 1070 y Fy(dx)1870 957 y Fn(Z)1952 1070 y Fy(dk)25 b Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2362 1036 y Ft(\026)2407 1070 y Fs(j)p Fy(h)2478 1082 y Ft(i)2505 1070 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)24 b Fy(<)f Fs(1)g Fz(,)752 1309 y Fy(H)821 1321 y Ft(i;)p Fu(1)902 1309 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(\024)e Fy(C)39 b Fz(sup)1286 1381 y Ft(x)1324 1389 y Fo(0)1357 1381 y Fq(\025)p Fu(1)1455 1196 y Fn(Z)1538 1309 y Fy(dk)24 b Fs(j)p Fy(k)s Fs(j)1740 1275 y Ft(\026)1806 1309 y Fs(j)p Fy(h)1877 1321 y Ft(i)1905 1309 y Fv(\()p Fy(x)1984 1321 y Fu(0)2022 1309 y Fy(;)14 b(k)s Fv(\))p Fs(j)2174 1196 y Fn(Z)2256 1217 y Ft(x)p Fu(+2)2220 1385 y(1)2361 1309 y Fy(dx)2451 1321 y Fu(0)2510 1309 y Fs(j)p Fy(x)19 b Fs(\000)f Fy(x)2729 1321 y Fu(0)2767 1309 y Fs(j)2790 1275 y Ft(\026)p Fq(\000)p Ft(\027)t Fq(\000)p Fu(1)1120 1546 y Fs(\024)k Fy(C)6 b Fv(\(1)19 b(+)f Fy(x)1495 1512 y Ft(\026)p Fq(\000)p Ft(\027)1629 1546 y Fv(\))33 b Fz(sup)1675 1619 y Ft(x)1713 1627 y Fo(0)1745 1619 y Fq(\025)p Fu(1)1844 1433 y Fn(Z)1927 1546 y Fy(dk)24 b Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2336 1512 y Ft(\026)2402 1546 y Fs(j)p Fy(h)2473 1558 y Ft(i)2501 1546 y Fv(\()p Fy(x)2580 1558 y Fu(0)2618 1546 y Fy(;)c(k)s Fv(\))p Fs(j)23 b Fy(:)-9 1786 y Fz(This)e(is)g(bounded)j (for)e Fy(x)g Fz(bounded.)33 b(T)-7 b(o)22 b(see)f(that)f(it)h(is)g(also)g (bounded)j(as)d Fy(x)i Fs(!)g(1)p Fz(,)f(let)e Fy(x)k Fs(\025)e Fv(4)p Fz(,)f(then)1236 1969 y Fy(H)1305 1981 y Ft(i;)p Fu(1)1385 1969 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fy(H)1760 1981 y Ft(i;)p Fu(3)1840 1969 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(H)2205 1981 y Ft(i;)p Fu(4)2286 1969 y Fv(\()p Fy(x;)c(k)s Fv(\))24 b Fz(,)-9 2151 y(where)882 2374 y Fy(H)951 2386 y Ft(i;)p Fu(3)1031 2374 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1336 2261 y Fn(Z)1419 2374 y Fy(dk)1543 2261 y Fn(Z)1626 2281 y Ft(x)p Fq(\000)p Fu(2)1589 2449 y(1)1731 2374 y Fy(dx)1821 2386 y Fu(0)1881 2374 y Fs(j)p Fy(k)s Fs(j)1973 2339 y Fu(1+)p Ft(\027)2098 2374 y Fy(e)2137 2339 y Fq(\000j)p Ft(k)q Fu(\()p Ft(x)p Fq(\000)p Ft(x)2399 2347 y Fo(0)2430 2339 y Fu(\))p Fq(j)2480 2374 y Fs(j)p Fy(h)2551 2386 y Ft(i)2579 2374 y Fv(\()p Fy(x)2658 2386 y Fu(0)2696 2374 y Fy(;)14 b(k)s Fv(\))p Fs(j)23 b Fz(,)882 2612 y Fy(H)951 2624 y Ft(i;)p Fu(4)1031 2612 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1336 2499 y Fn(Z)1419 2612 y Fy(dk)1543 2499 y Fn(Z)1626 2520 y Ft(x)p Fu(+2)1589 2688 y Ft(x)p Fq(\000)p Fu(2)1731 2612 y Fy(dx)1821 2624 y Fu(0)1880 2612 y Fs(j)p Fy(k)s Fs(j)1972 2578 y Fu(1+)p Ft(\027)2097 2612 y Fy(e)2136 2578 y Fq(\000j)p Ft(k)q Fu(\()p Ft(x)p Fq(\000)p Ft(x)2398 2586 y Fo(0)2430 2578 y Fu(\))p Fq(j)2479 2612 y Fs(j)p Fy(h)2550 2624 y Ft(i)2578 2612 y Fv(\()p Fy(x)2657 2624 y Fu(0)2695 2612 y Fy(;)14 b(k)s Fv(\))p Fs(j)23 b Fy(:)-9 2844 y Fz(As)e(abo)o(v)o(e,)h(we)f(ha)n(v)o(e)819 3064 y Fy(H)888 3076 y Ft(i;)p Fu(3)969 3064 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(\024)e(j)p Fy(x)d Fs(\000)f Fv(2)p Fs(j)1511 3030 y Fq(\000)p Fu(1)1614 2951 y Fn(Z)1697 2972 y Fq(1)1660 3140 y Fu(1)1745 3064 y Fy(dx)1849 2951 y Fn(Z)1933 3064 y Fy(dk)24 b Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2342 3030 y Ft(\026)2387 3064 y Fs(j)p Fy(h)2458 3076 y Ft(i)2486 3064 y Fv(\()p Fy(x;)c(k)s Fv(\))p Fs(j)23 b Fy(<)g Fs(1)g Fz(,)819 3286 y Fy(H)888 3298 y Ft(i;)p Fu(4)969 3286 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(\024)e Fy(C)6 b Fv(2)1381 3252 y Ft(\026)p Fq(\000)p Ft(\027)1531 3286 y Fz(sup)1528 3358 y Ft(x)p Fq(\025)p Fu(1)1665 3173 y Fn(Z)1748 3286 y Fy(dk)24 b Fs(j)p Fy(k)s Fs(j)1950 3252 y Ft(\026)2016 3286 y Fs(j)p Fy(h)2087 3298 y Ft(i)2114 3286 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b Fy(<)f Fs(1)g Fy(:)-9 3526 y Fz(This)e(sho)n(ws)h(that)f(for)h(all)f Fv(0)h Fs(\024)h Fy(\027)28 b(<)23 b(\026)p Fz(,)e(we)h(ha)n(v)o(e)1327 3633 y Fn(Z)1409 3746 y Fy(dk)j Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))1819 3712 y Fu(1+)p Ft(\027)1944 3746 y Fs(j)6 b Fv(^)-48 b Fy(q)2004 3758 y Ft(i)2032 3746 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b Fy(<)f Fs(1)g Fz(,)1273 3851 y Fn(Z)1355 3964 y Fy(dk)i Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))1765 3930 y Fu(1+)p Ft(\027)1890 3964 y Fs(j)6 b Fv(^)-48 b Fy(q)1950 3976 y Ft(i;p)2032 3964 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b Fy(<)f Fs(1)g Fz(,)-9 4184 y(so)32 b(that)g Fy(q)295 4196 y Ft(i)323 4184 y Fz(,)j Fy(q)416 4196 y Ft(i;p)498 4184 y Fz(,)g Fy(@)598 4196 y Ft(y)638 4184 y Fy(q)675 4196 y Ft(i)735 4184 y Fz(and)f Fy(@)935 4196 y Ft(y)975 4184 y Fy(q)1012 4196 y Ft(i;p)1126 4184 y Fz(are)e(bounded.)67 b(T)-7 b(o)33 b(pro)o(v)o(e)g(that) f(the)o(y)g(are)g(also)g(continuous)j(in)d Fy(x;)14 b(y)s Fz(,)36 b(use)c(the)g(same)-9 4284 y(ar)n(guments)26 b(as)g(in)f(the)g(proof)j(of)e (Lemma)f(3.2)h(to)g(get)f(that)g(the)o(y)g(are)g(Ho)-35 b(\310)7 b(lder)n(\261)p Fy(\027)33 b Fz(in)25 b Fv(\()p Fy(x;)14 b(y)s Fv(\))26 b Fz(for)h(all)d Fv(0)k Fs(\024)h Fy(\027)34 b(<)28 b(\026)p Fz(,)f(and)f(hence,)-9 4384 y(continuous.)32 b(Since)21 b(the)g(proof)j(is)d(v)o(ery)h(similar)m(,)e(we)h(omit)g(the)g(details.)p 2197 4384 43 43 v -9 4624 a Fd(C.5)102 b(Pr)n(oof)26 b(of)f(Lemma)h(3.4)-9 4781 y FA(Lemma)20 b(C.8)j(\(Lemma)d(3.4\))44 b Fm(Let)27 b Fr(v)1127 4793 y Fu(+)1214 4781 y Fv(=)k(\()p Fy(u)1390 4793 y Fu(+)1445 4781 y Fy(;)14 b(v)1522 4793 y Fu(+)1578 4781 y Fv(\))32 b Fs(2)g(W)1811 4793 y Ft(\013;\015)1909 4801 y Fo(1)1941 4793 y Ft(;\015)1996 4801 y Fo(2)2060 4781 y Fm(for)27 b(some)i Fy(\013)j(>)f Fv(3)p Fm(,)e Fy(\015)2705 4793 y Fu(1)2774 4781 y Fs(2)j Fv([0)p Fy(;)14 b Fv(1\))27 b Fm(and)i Fy(\015)3262 4793 y Fu(2)3331 4781 y Fs(2)j Fv([0)p Fy(;)3530 4748 y Fu(1)p 3530 4762 34 4 v 3530 4810 a(2)3573 4781 y Fv(])c Fm(with)-9 4881 y Fy(\015)34 4893 y Fu(2)94 4881 y Fs(\024)22 b Fy(\015)224 4893 y Fu(1)262 4881 y Fm(.)31 b(Let)26 b Fv(^)-48 b Fy(u)491 4893 y Fu(0)549 4881 y Fm(and)26 b Fv(^)-45 b Fy(v)738 4893 y Fu(0)797 4881 y Fm(be)21 b(de\256ned)i(by)e(\(3.19\).)34 b(Ther)m(e)21 b(e)n(xists)f(a)i(constant)g Fy(C)28 b Fm(suc)o(h)21 b(that)h(for)f(all)g Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)3324 4893 y Fu(+)3379 4881 y Fm(,)f(we)f(have)1153 5102 y Fs(j)5 b Fv(^)-47 b Fy(u)1224 5114 y Fu(0)1261 5102 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fs(k)p Fr(v)1768 5114 y Fu(+)1823 5102 y Fs(k)1865 5114 y Ft(\013;\015)1963 5122 y Fo(1)1994 5114 y Ft(;\015)2049 5122 y Fo(2)2328 5046 y Fv(1)p 2117 5083 464 4 v 2117 5160 a(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2384 5100 y(p)p 2453 5100 48 4 v 60 x Fy(x)p Fv(\))2532 5136 y Ft(\013)2613 5102 y Fm(,)1160 5323 y Fs(j)s Fv(^)-45 b Fy(v)1223 5335 y Fu(0)1261 5323 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fs(k)p Fr(v)1768 5335 y Fu(+)1823 5323 y Fs(k)1865 5335 y Ft(\013;\015)1963 5343 y Fo(1)1994 5335 y Ft(;\015)2049 5343 y Fo(2)2154 5267 y Fv(1)p 2117 5304 117 4 v 2117 5321 a Fs(p)p 2186 5321 48 4 v 60 x Fy(x)2464 5267 y Fv(1)p 2253 5304 464 4 v 2253 5381 a(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2520 5321 y(p)p 2590 5321 48 4 v 2590 5381 a Fy(x)p Fv(\))2669 5357 y Ft(\013)2750 5323 y Fm(,)1082 5580 y Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(u)1224 5592 y Fu(0)1261 5580 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fs(k)p Fr(v)1768 5592 y Fu(+)1823 5580 y Fs(k)1865 5592 y Ft(\013;\015)1963 5600 y Fo(1)1994 5592 y Ft(;\015)2049 5600 y Fo(2)2244 5524 y Fy(x)2301 5462 y Fo(1)p Fc(\000)p Fk(\015)2405 5474 y Fo(1)p 2302 5480 137 3 v 2356 5514 a(2)p 2117 5561 464 4 v 2117 5637 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2384 5577 y(p)p 2453 5577 48 4 v 60 x Fy(x)p Fv(\))2532 5613 y Ft(\013)2604 5463 y Fn(\022)2675 5524 y Fv(1)g(+)g Fs(j)p Fy(k)s Fs(j)p 2675 5561 235 4 v 2746 5637 a(j)p Fy(k)s Fs(j)2920 5463 y Fn(\023)2981 5480 y Ft(\015)3016 5488 y Fo(1)3089 5580 y Fm(,)1838 5909 y Fz(26)p eop %%Page: 27 27 bop 1089 184 a Fs(j)p Fy(D)5 b Fv(^)-45 b Fy(v)1223 196 y Fu(0)1261 184 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fs(k)p Fr(v)1768 196 y Fu(+)1823 184 y Fs(k)1865 196 y Ft(\013;\015)1963 204 y Fo(1)1994 196 y Ft(;\015)2049 204 y Fo(2)2370 128 y Fv(1)p 2117 165 549 4 v 2117 241 a(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2384 182 y(p)p 2453 182 48 4 v 59 x Fy(x)p Fv(\))2532 217 y Ft(\013)p Fq(\000)p Fu(1)2853 68 y Fs(p)p 2922 68 V 60 x Fy(x)p 2685 165 452 4 v 2685 241 a Fs(j)p Fy(k)s(x)p Fs(j)2824 217 y Ft(\015)2859 225 y Fo(2)2914 241 y Fv(+)g Fs(j)p Fy(k)s(x)p Fs(j)3170 184 y Fm(,)570 345 y Fn(\014)570 395 y(\014)570 445 y(\014)603 441 y Fv(^)-48 b Fy(u)645 453 y Fu(0)682 441 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b Fs(\000)f Fy(e)1017 406 y Fu(\003)1062 414 y Fc(\000)1112 406 y Fu(\()p Ft(k)q Fu(\))p Ft(x)1242 441 y Fy(u)1290 453 y Fu(+)1345 441 y Fv(\(0\))1451 345 y Fn(\014)1451 395 y(\014)1451 445 y(\014)1501 441 y Fs(\024)23 b Fy(C)28 b Fs(k)p Fr(v)1768 453 y Fu(+)1823 441 y Fs(k)1865 453 y Ft(\013;\015)1963 461 y Fo(1)1994 453 y Ft(;\015)2049 461 y Fo(2)2244 384 y Fy(x)2301 323 y Fk(\015)2332 335 y Fo(1)2365 323 y Fc(\000)p Fo(1)p 2302 341 137 3 v 2356 374 a(2)p 2117 421 464 4 v 2117 498 a Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2384 438 y(p)p 2453 438 48 4 v 60 x Fy(x)p Fv(\))2532 474 y Ft(\013)2613 441 y Fy(:)-9 644 y Fm(Pr)l(oof)o(.)31 b Fz(By)21 b(hypothesis,)i(there)e(e)o(xists)f(a)h (constant)h Fy(C)27 b Fz(such)22 b(that)f(for)h(all)f Fy(k)k Fs(2)f Fr(R)p Fz(,)d(we)g(ha)n(v)o(e)1038 839 y Fs(j)p Fy(u)1109 851 y Fu(+)1164 839 y Fv(\()p Fy(k)s Fv(\))p Fs(j)i(\024)g Fy(C)28 b Fs(k)p Fr(v)1587 851 y Fu(+)1642 839 y Fs(k)1684 851 y Ft(\013;\015)1782 859 y Fo(1)1813 851 y Ft(;\015)1868 859 y Fo(2)2067 783 y Fv(1)p 1914 820 347 4 v 1914 896 a(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))2214 872 y Ft(\013)2294 839 y Fz(,)1045 1060 y Fs(j)p Fy(v)1108 1072 y Fu(+)1164 1060 y Fv(\()p Fy(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fs(k)p Fr(v)1587 1072 y Fu(+)1642 1060 y Fs(k)1684 1072 y Ft(\013;\015)1782 1080 y Fo(1)1813 1072 y Ft(;\015)1868 1080 y Fo(2)2067 1003 y Fv(1)p 1914 1041 V 1914 1117 a(\(1)19 b(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))2214 1093 y Ft(\013)2294 1060 y Fz(,)967 1295 y Fs(j)p Fy(D)r(u)1109 1307 y Fu(+)1164 1295 y Fv(\()p Fy(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fs(k)p Fr(v)1587 1307 y Fu(+)1642 1295 y Fs(k)1684 1307 y Ft(\013;\015)1782 1315 y Fo(1)1813 1307 y Ft(;\015)1868 1315 y Fo(2)1918 1178 y Fn(\022)1989 1239 y Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p 1989 1276 235 4 v 2061 1352 a(j)p Fy(k)s Fs(j)2234 1178 y Fn(\023)2295 1196 y Ft(\015)2330 1204 y Fo(1)2543 1239 y Fv(1)p 2391 1276 347 4 v 2391 1352 a(\(1)g(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2690 1328 y Ft(\013)2770 1295 y Fz(,)974 1532 y Fs(j)p Fy(D)r(v)1108 1544 y Fu(+)1164 1532 y Fv(\()p Fy(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fs(k)p Fr(v)1587 1544 y Fu(+)1642 1532 y Fs(k)1684 1544 y Ft(\013;\015)1782 1552 y Fo(1)1813 1544 y Ft(;\015)1868 1552 y Fo(2)1918 1415 y Fn(\022)1989 1476 y Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p 1989 1513 235 4 v 2061 1589 a(j)p Fy(k)s Fs(j)2234 1415 y Fn(\023)2295 1433 y Ft(\015)2330 1441 y Fo(2)2543 1476 y Fv(1)p 2391 1513 347 4 v 2391 1589 a(\(1)g(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2690 1565 y Ft(\013)1320 1769 y Fs(\024)23 b Fy(C)28 b Fs(k)p Fr(v)1587 1781 y Fu(+)1642 1769 y Fs(k)1684 1781 y Ft(\013;\015)1782 1789 y Fo(1)1813 1781 y Ft(;\015)1868 1789 y Fo(2)1918 1652 y Fn(\022)1989 1713 y Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p 1989 1750 235 4 v 2061 1826 a(j)p Fy(k)s Fs(j)2234 1652 y Fn(\023)2295 1669 y Ft(\015)2330 1677 y Fo(1)2543 1713 y Fv(1)p 2391 1750 347 4 v 2391 1826 a(\(1)g(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2690 1802 y Ft(\013)2770 1769 y Fy(:)-9 1978 y Fz(Furthermore,)26 b(we)e(ha)n(v)o(e)g(that)f(lim)1029 1990 y Ft(k)q Fq(!)p Fu(0)1183 1978 y Fy(v)1223 1990 y Fu(+)1278 1978 y Fv(\()p Fy(k)s Fv(\))k(=)f(0)p Fz(.)38 b(This)23 b(holds)i(because)f Fy(v)2322 1990 y Fu(+)2401 1978 y Fz(is)f(odd,)j(and)e Fy(D)r(v)2911 1990 y Fu(+)2967 1978 y Fv(\()p Fy(k)s Fv(\))g Fz(di)n(v)o(er)n(ges)g(only)h (lik)o(e)e(a)-9 2077 y(po)n(wer)i(la)o(w)f(as)g Fy(k)30 b Fs(!)d Fv(0)p Fz(,)e(so)f(that)g Fy(v)1028 2089 y Fu(+)1083 2077 y Fv(\()p Fy(k)s Fv(\))k Fs(\030)f Fz(s)-5 b(i)g(g)g(n)q Fv(\()p Fy(k)s Fv(\))25 b Fz(for)g(small)e Fy(k)28 b Fz(is)23 b(ruled)i(out.)41 b(By)24 b(Lemma)g(C.2,)i(and)f(since)e Fy(\015)3388 2089 y Fu(2)3453 2077 y Fs(\024)k Fy(\015)3588 2089 y Fu(1)3625 2077 y Fz(,)e(we)-9 2177 y(ha)n(v)o(e)1246 2336 y Fs(j)p Fy(v)1309 2348 y Fu(+)1365 2336 y Fv(\()p Fy(k)s Fv(\))p Fs(j)e Fv(=)g Fs(j)p Fy(v)1672 2348 y Fu(+)1727 2336 y Fv(\()p Fy(k)s Fv(\))c Fs(\000)f Fy(e)1978 2301 y Fu(\003)2023 2309 y Fc(\000)2072 2301 y Fu(\()p Ft(k)q Fu(\))2165 2336 y Fy(v)2205 2348 y Fu(+)2260 2336 y Fv(\(0\))p Fs(j)1521 2527 y(\024)23 b Fy(C)1688 2410 y Fn(\022)1830 2471 y Fs(j)p Fy(k)s Fs(j)p 1759 2508 235 4 v 1759 2584 a Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2004 2410 y Fn(\023)2065 2427 y Fu(1)p Fq(\000)p Ft(\015)2185 2435 y Fo(2)2398 2471 y Fv(1)p 2245 2508 347 4 v 2245 2584 a(\(1)g(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2544 2560 y Ft(\013)2602 2527 y Fs(k)p Fr(v)2694 2539 y Fu(+)2749 2527 y Fs(k)2791 2539 y Ft(\013;\015)2889 2547 y Fo(1)2921 2539 y Ft(;\015)2976 2547 y Fo(2)1521 2776 y Fs(\024)23 b Fy(C)1688 2659 y Fn(\022)1830 2720 y Fs(j)p Fy(k)s Fs(j)p 1759 2757 235 4 v 1759 2833 a Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2004 2659 y Fn(\023)2065 2676 y Fu(1)p Fq(\000)p Ft(\015)2185 2684 y Fo(1)2398 2720 y Fv(1)p 2245 2757 347 4 v 2245 2833 a(\(1)g(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2544 2809 y Ft(\013)2602 2776 y Fs(k)p Fr(v)2694 2788 y Fu(+)2749 2776 y Fs(k)2791 2788 y Ft(\013;\015)2889 2796 y Fo(1)2921 2788 y Ft(;\015)2976 2796 y Fo(2)3035 2776 y Fz(,)703 3026 y Fs(j)p Fy(u)774 3038 y Fu(+)828 3026 y Fv(\()p Fy(k)s Fv(\))h Fs(\000)f Fy(e)1079 2991 y Fu(\003)1124 2999 y Fc(\000)1174 2991 y Fu(\()p Ft(k)q Fu(\))1266 3026 y Fy(u)1314 3038 y Fu(+)1369 3026 y Fv(\(0\))p Fs(j)23 b(\024)g Fy(C)1688 2909 y Fn(\022)1830 2970 y Fs(j)p Fy(k)s Fs(j)p 1759 3007 235 4 v 1759 3083 a Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2004 2909 y Fn(\023)2065 2926 y Fu(1)p Fq(\000)p Ft(\015)2185 2934 y Fo(1)2398 2970 y Fv(1)p 2245 3007 347 4 v 2245 3083 a(\(1)g(+)g Fs(j)p Fy(k)s Fs(j)p Fv(\))2544 3059 y Ft(\013)2602 3026 y Fs(k)p Fr(v)2694 3038 y Fu(+)2749 3026 y Fs(k)2791 3038 y Ft(\013;\015)2889 3046 y Fo(1)2921 3038 y Ft(;\015)2976 3046 y Fo(2)3035 3026 y Fy(:)-9 3229 y Fz(Finally)-6 b(,)21 b(we)h(note)f(that)1220 3388 y Fy(P)1273 3400 y Fu(1)p Ft(;u)1369 3388 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b Fs(\000)f Fy(e)1704 3353 y Fu(\003)1749 3361 y Fc(\000)1799 3353 y Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2000 3388 y Fv(=)23 b Fs(\000)p Fy(P)2206 3400 y Fu(2)p Ft(;u)2302 3388 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fz(,)-9 3546 y(so)d(that)236 3705 y Fy(P)289 3717 y Fu(1)p Ft(;u)386 3705 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fy(u)628 3717 y Fu(+)683 3705 y Fv(\()p Fy(k)s Fv(\))19 b Fs(\000)f Fy(e)934 3671 y Fu(\003)979 3679 y Fc(\000)1028 3671 y Fu(\()p Ft(k)q Fu(\))p Ft(x)1159 3705 y Fy(u)1207 3717 y Fu(+)1261 3705 y Fv(\(0\))24 b(=)e Fs(\000)p Fy(P)1596 3717 y Fu(2)p Ft(;u)1692 3705 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fy(u)1934 3717 y Fu(+)1990 3705 y Fv(\()p Fy(k)s Fv(\))19 b(+)f Fy(e)2241 3671 y Fu(\003)2286 3679 y Fc(\000)2335 3671 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2616 3613 y Fn(\020)2665 3705 y Fy(u)2713 3717 y Fu(+)2768 3705 y Fv(\()p Fy(k)s Fv(\))h Fs(\000)f Fy(e)3019 3671 y Fu(\003)3064 3679 y Fc(\000)3113 3671 y Fu(\()p Ft(k)q Fu(\))3206 3705 y Fy(u)3254 3717 y Fu(+)3309 3705 y Fv(\(0\))3415 3613 y Fn(\021)3501 3705 y Fy(:)-9 3883 y Fz(The)30 b(inequalities)f(\(3.28\)\261\(3.32\))35 b(follo)n(w)30 b(easily)g(by)h(using)g(these)e(inequalities,)j(Lemmas)d(C.1)i(and)g(C.3.)57 b(W)-7 b(e)30 b(omit)g(the)-9 3983 y(details.)p 286 3983 43 43 v -9 4219 a Fd(C.6)102 b(Pr)n(oof)26 b(of)f(Pr)n(oposition)h(3.3)-9 4376 y Fz(In)16 b(this)f(section,)i(we)e(sho)n(w)i(that)e(the)h(e)o (xpressions)g(gi)n(v)o(en)g(in)g(Proposition)h(3.3)g(solv)o(e)f(the)f(F)o (ourier)i(transform)f(of)h(\(3.12\)\261\(3.14\),)-9 4476 y(namely)623 4682 y Fy(@)667 4694 y Ft(x)709 4565 y Fn(\022)775 4626 y Fv(^)-47 b Fy(u)775 4739 y Fv(^)i Fy(v)818 4565 y Fn(\023)879 4682 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1184 4565 y Fn(\022)1262 4631 y Fv(0)99 b Fs(\000)p Fy(ik)1245 4731 y(ik)134 b Fv(1)1542 4565 y Fn(\023)14 b(\022)1683 4626 y Fv(^)-47 b Fy(u)1683 4739 y Fv(^)i Fy(v)1725 4565 y Fn(\023)1786 4682 y Fv(\()p Fy(x;)14 b(k)s Fv(\))20 b(+)2083 4565 y Fn(\022)2567 4626 y Fv(0)2150 4739 y(^)-48 b Fy(\021)2185 4751 y Fu(+)2240 4739 y Fv(\()p Fy(k)s Fv(\))p Fy(e)2389 4715 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2714 4739 y Fv(+)25 b(^)-49 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))3032 4565 y Fn(\023)3116 4682 y Fz(,)414 b(\(C.10\))687 4914 y(lim)675 4965 y Ft(x)p Fq(!)p Fu(1)831 4913 y Fv(^)826 4914 y Fr(u)p Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(\021)34 b Fz(lim)1184 4965 y Ft(x)p Fq(!)p Fu(1)1335 4797 y Fn(\022)1401 4858 y Fv(^)-47 b Fy(u)p Fv(\()p Fy(x;)14 b(k)s Fv(\))1401 4971 y(^)-45 b Fy(v)s Fv(\()p Fy(x;)14 b(k)s Fv(\))1638 4797 y Fn(\023)1722 4914 y Fv(=)1810 4797 y Fn(\022)1877 4858 y Fv(^)-48 b Fy(u)1919 4870 y Fu(+)1974 4858 y Fv(\()p Fy(k)s Fv(\))1878 4971 y(^)j Fy(v)1915 4983 y Fu(+)1970 4971 y Fv(\()p Fy(k)s Fv(\))2084 4797 y Fn(\023)2168 4914 y Fs(\021)2261 4913 y Fv(^)2256 4914 y Fr(v)2306 4926 y Fu(+)2362 4914 y Fv(\()p Fy(k)s Fv(\))23 b Fy(:)1033 b Fz(\(C.11\))-9 5118 y(Once)23 b(we)g(ha)n(v)o(e)g(sho)n(wn)i(this,)e(the)g(proof)i(is)d(completed)h(by)h (remarking)g(that)f(Lemmas)f(3.1)j(and)e(3.2)h(ensure)g(that)e(the)h (right\261)-9 5217 y(hand)i(sides)f(of)g(\(2.12\))j(and)d(\(2.13\))i(are)e (continuous)i(in)e Fy(x;)14 b(y)s Fz(,)25 b(so)g(that)e(these)h(equations)g (are)g(strongly)h(satis\256ed.)39 b(The)24 b(proof)-9 5317 y(relies)18 b(mainly)i(in)f(Duhamel')-5 b(s)21 b(formula.)32 b(Although)21 b(we)f(could)g(check)g(directly)f(that)g(the)g(inte)o(gral)g(e) o(xpressions)i(we)f(ga)n(v)o(e)f(are)-9 5416 y(solution)j(of)g(\(C.10\),)h (we)e(prefer)h(to)f(gi)n(v)o(e)g(a)g(constructi)n(v)o(e)h(proof.)33 b(As)21 b(we)g(noted)i(in)e(section)g(3.1,)h(the)g(linear)e(operator)1503 5623 y Fy(L)p Fv(\()p Fy(k)s Fv(\))j(=)1781 5506 y Fn(\022)1858 5572 y Fv(0)99 b Fs(\000)p Fy(ik)1842 5672 y(ik)134 b Fv(1)2138 5506 y Fn(\023)2236 5623 y Fz(,)1838 5909 y(27)p eop %%Page: 28 28 bop -9 145 a Fz(has)34 b(both)i(unstable)f(and)g(center)n(\261stable)g(eigen) m(v)n(alues)f Fv(\003)1769 157 y Fq(\006)1824 145 y Fz(.)72 b(W)-7 b(e)34 b(will)g(sho)n(w)i(belo)n(w)f(that)f(the)h(e)o(xpression)h(we)e (ga)n(v)o(e)h(in)-9 244 y(Proposition)17 b(3.3)g(for)f Fy(\021)675 256 y Fu(+)746 244 y Fz(is)g(the)f(good)j(choice)d(to)h(satisfy)f(the)g (initial)f(conditions)j(for)g Fy(u)e Fz(and)h Fy(v)s Fz(.)29 b(W)-7 b(e)16 b(be)o(gin)g(with)g(diagonalyzing)-9 344 y(the)21 b(matrix)g Fy(L)p Fv(\()p Fy(k)s Fv(\))p Fz(.)31 b(W)-7 b(e)21 b(ha)n(v)o(e)h(that)1421 443 y Fy(L)p Fv(\()p Fy(k)s Fv(\))h(=)f Fy(S)5 b Fv(\()p Fy(k)s Fv(\))p Fy(O)r Fv(\()p Fy(k)s Fv(\))p Fy(S)g Fv(\()p Fy(k)s Fv(\))2205 409 y Fq(\000)p Fu(1)2319 443 y Fz(,)-9 593 y(where)1320 831 y Fy(S)g Fv(\()p Fy(k)s Fv(\))23 b(=)1597 689 y Fn( )1742 766 y Fv(1)1991 733 y Fq(\000)p Ft(ik)p 1956 747 181 4 v 1956 795 a Fu(\003)2001 803 y Fo(+)2049 795 y Fu(\()p Ft(k)q Fu(\))1707 850 y Fq(\000)p Ft(ik)p 1673 864 V 1673 911 a Fu(\003)1718 919 y Fo(+)1765 911 y Fu(\()p Ft(k)q Fu(\))2026 883 y Fv(1)2147 689 y Fn(!)2250 831 y Fz(,)1310 1088 y Fy(O)r Fv(\()p Fy(k)s Fv(\))h(=)1597 971 y Fn(\022)1658 1037 y Fv(\003)1716 1049 y Fq(\000)1772 1037 y Fv(\()p Fy(k)s Fv(\))174 b(0)1749 1137 y(0)g(\003)2023 1149 y Fu(+)2078 1137 y Fv(\()p Fy(k)s Fv(\))2188 971 y Fn(\023)2286 1088 y Fz(,)1231 1345 y Fy(S)5 b Fv(\()p Fy(k)s Fv(\))1397 1311 y Fq(\000)p Fu(1)1509 1345 y Fv(=)1607 1289 y(\003)1665 1301 y Fu(+)1720 1289 y Fv(\()p Fy(k)s Fv(\))p 1607 1326 224 4 v 1629 1402 a(\001\()p Fy(k)s Fv(\))1854 1203 y Fn( )1999 1281 y Fv(1)2274 1248 y Ft(ik)p 2213 1262 181 4 v 2213 1309 a Fu(\003)2258 1317 y Fo(+)2305 1309 y Fu(\()p Ft(k)q Fu(\))1990 1365 y Ft(ik)p 1929 1379 V 1929 1426 a Fu(\003)1974 1434 y Fo(+)2022 1426 y Fu(\()p Ft(k)q Fu(\))2283 1398 y Fv(1)2404 1203 y Fn(!)2507 1345 y Fy(:)-9 1598 y Fz(The)19 b(unstable)h(modes)g(corresponding)j(to)c Fv(\003)1335 1610 y Fu(+)1409 1598 y Fz(can)h(be)f(con)m(v)o(erted)h(to)g (stable)e(modes)j(by)f(running)i(their)d(e)n(v)n(olution)i(backw)o(ard)-9 1697 y(in)g Fy(x)p Fz(.)32 b(See)22 b(for)g(e)o(xample)f([Gallay].)32 b(T)-7 b(o)23 b(do)f(this,)g(de\256ne)g(ne)n(w)g(v)n(ariables)f Fv(\()s(^)-45 b Fy(v)2257 1709 y Ft(s)2293 1697 y Fy(;)17 b Fv(^)-45 b Fy(v)2370 1709 y Ft(u)2414 1697 y Fv(\))21 b Fz(\(the)h(subscript) g Fy(u)f Fz(is)h(for)g(unstable)g(and)g(the)-9 1797 y(subscript)g Fy(s)f Fz(for)h(center)n(\261stable\))f(by)1292 1910 y Fn(\022)1360 1971 y Fv(^)-45 b Fy(v)1397 1983 y Ft(s)1356 2084 y Fv(^)g Fy(v)1393 2096 y Ft(u)1436 1910 y Fn(\023)1498 2027 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b(=)g Fy(S)5 b Fv(\()p Fy(k)s Fv(\))1969 1993 y Fq(\000)p Fu(1)2058 1910 y Fn(\022)2125 1971 y Fv(^)-48 b Fy(u)2125 2084 y Fv(^)j Fy(v)2167 1910 y Fn(\023)2228 2027 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)1082 b Fz(\(C.12\))-9 2259 y(F)o(or)22 b Fv(\()s(^)-45 b Fy(v)201 2271 y Ft(s)237 2259 y Fy(;)17 b Fv(^)-45 b Fy(v)314 2271 y Ft(u)357 2259 y Fv(\))p Fz(,)22 b(we)f(get)g(the)g(system)1015 2492 y Fy(@)1059 2504 y Ft(x)1101 2375 y Fn(\022)1169 2436 y Fv(^)-45 b Fy(v)1206 2448 y Ft(s)1166 2549 y Fv(^)f Fy(v)1202 2561 y Ft(u)1246 2375 y Fn(\023)1307 2492 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fy(O)r Fv(\()p Fy(k)s Fv(\))1788 2375 y Fn(\022)1857 2436 y Fv(^)-45 b Fy(v)1894 2448 y Ft(s)1853 2549 y Fv(^)g Fy(v)1890 2561 y Ft(u)1933 2375 y Fn(\023)1994 2492 y Fv(\()p Fy(x;)14 b(k)s Fv(\))20 b(+)2291 2375 y Fn(\022)2360 2436 y Fv(^)-46 b Fy(n)2406 2448 y Ft(s)2356 2549 y Fv(^)g Fy(n)2402 2561 y Ft(u)2445 2375 y Fn(\023)2506 2492 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fz(,)806 b(\(C.13\))-9 2719 y(with)1287 2732 y Fn(\022)1356 2793 y Fv(^)-46 b Fy(n)1402 2805 y Ft(s)1352 2906 y Fv(^)g Fy(n)1398 2918 y Ft(u)1441 2732 y Fn(\023)1502 2849 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fy(S)5 b Fv(\()p Fy(k)s Fv(\))1974 2815 y Fq(\000)p Fu(1)2063 2732 y Fn(\022)2225 2793 y Fv(0)2128 2906 y(^)-46 b Fy(n)p Fv(\()p Fy(x;)14 b(k)s Fv(\))2369 2732 y Fn(\023)2453 2849 y Fz(,)-9 3060 y(where)28 b Fv(^)-46 b Fy(n)o Fv(\()p Fy(x;)14 b(k)s Fv(\))27 b(=)32 b(^)-48 b Fy(\021)624 3072 y Fu(+)679 3060 y Fv(\()p Fy(k)s Fv(\))p Fy(e)828 3030 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1154 3060 y Fv(+)26 b(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fz(.)38 b(F)o(ollo)n(wing)24 b([Gallay],)h(we)e(impose)h(for)g(the)g(center)n (-stable)f(modes)h(initial)-9 3159 y(conditions)i(at)f Fy(x)30 b Fv(=)f(1)c Fz(and)i(for)f(the)g(unstable)g(modes)g(at)g Fy(x)j Fv(=)g Fs(1)p Fz(.)44 b(W)-7 b(e)26 b(set)j Fv(^)-46 b Fy(v)2357 3171 y Ft(s)2393 3159 y Fv(\(1)p Fy(;)14 b(k)s Fv(\))29 b(=)34 b(^)-47 b Fy(\027)2746 3171 y Ft(s)2781 3159 y Fv(\()p Fy(k)s Fv(\))27 b Fz(for)f(some)h(\(e)n(v)o(en\))f(function)-5 3259 y Fv(^)-46 b Fy(\027)32 3271 y Ft(s)91 3259 y Fz(to)24 b(be)g(chosen)h(later) e(on,)j(and)e(we)g(set)g(lim)1352 3271 y Ft(x)p Fq(!1)1543 3259 y Fv(^)-45 b Fy(v)1580 3271 y Ft(u)1624 3259 y Fv(\()p Fy(x;)14 b(k)s Fv(\))27 b(=)g(0)c Fz(\(in)h(order)i(to)e(satisfy)f(the)h (boundary)j(conditions)e(for)g Fr(v)h Fz(at)-9 3359 y(in\256nity\).)31 b(From)23 b(\(C.12\))f(and)g(\(C.13\),)h(we)f(get)f(the)g(inte)o(gral)f (equation)738 3586 y Fv(^)-45 b Fy(v)775 3598 y Ft(s)810 3586 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fy(e)1155 3552 y Fu(\003)1200 3560 y Fc(\000)1249 3552 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))1521 3586 y Fv(^)-47 b Fy(\027)1557 3598 y Ft(s)1593 3586 y Fv(\()p Fy(k)s Fv(\))19 b(+)1867 3530 y Fy(ik)p 1815 3567 180 4 v 1815 3643 a Fv(\001\()p Fy(k)s Fv(\))2018 3473 y Fn(Z)2101 3494 y Ft(x)2064 3662 y Fu(1)2121 3586 y Fy(dx)2211 3598 y Fu(0)2270 3586 y Fy(e)2309 3552 y Fu(\003)2354 3560 y Fc(\000)2404 3552 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)2646 3560 y Fo(0)2678 3552 y Fu(\))2712 3586 y Fv(^)-46 b Fy(n)p Fv(\()p Fy(x)2837 3598 y Fu(0)2875 3586 y Fy(;)14 b(k)s Fv(\))23 b Fz(,)730 3815 y Fv(^)-45 b Fy(v)767 3827 y Ft(u)810 3815 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fs(\000)1191 3759 y Fv(\003)1249 3771 y Fu(+)1303 3759 y Fv(\()p Fy(k)s Fv(\))p 1191 3796 224 4 v 1212 3872 a(\001\()p Fy(k)s Fv(\))1437 3702 y Fn(Z)1520 3723 y Fq(1)1483 3891 y Ft(x)1569 3815 y Fy(dx)1659 3827 y Fu(0)1718 3815 y Fy(e)1757 3781 y Fu(\003)1802 3789 y Fo(+)1849 3781 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)2091 3789 y Fo(0)2123 3781 y Fu(\))2158 3815 y Fv(^)-47 b Fy(n)p Fv(\()p Fy(x)2282 3827 y Fu(0)2320 3815 y Fy(;)14 b(k)s Fv(\))23 b Fy(:)-9 4058 y Fz(Since)i Fv(^)-46 b Fy(n)o Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)29 b(^)-48 b Fy(\021)597 4070 y Fu(+)652 4058 y Fv(\()p Fy(k)s Fv(\))p Fy(e)801 4028 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1126 4058 y Fv(+)25 b(^)-49 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fz(,)23 b(we)e(\256nd)h(that)352 4284 y Fv(^)-45 b Fy(v)389 4296 y Ft(s)425 4284 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e Fy(e)769 4250 y Fu(\003)814 4258 y Fc(\000)864 4250 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))1136 4284 y Fv(^)-47 b Fy(\027)1172 4296 y Ft(s)1207 4284 y Fv(\()p Fy(k)s Fv(\))19 b(+)1482 4228 y Fy(ik)p 1429 4265 180 4 v 1429 4341 a Fv(\001\()p Fy(k)s Fv(\))1817 4228 y(1)p 1629 4265 418 4 v 1629 4341 a(\003)1687 4353 y Fq(\000)1742 4341 y Fv(\()p Fy(k)s Fv(\))g(+)f Fs(j)p Fy(k)s Fs(j)2070 4192 y Fn(\020)2120 4284 y Fy(e)2159 4250 y Fu(\003)2204 4258 y Fc(\000)2253 4250 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2539 4284 y Fs(\000)g Fy(e)2661 4250 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2967 4192 y Fn(\021)3036 4284 y Fv(^)-48 b Fy(\021)3071 4296 y Fu(+)3127 4284 y Fv(\()p Fy(k)s Fv(\))703 4512 y(+)848 4456 y Fy(ik)p 796 4493 180 4 v 796 4569 a Fv(\001\()p Fy(k)s Fv(\))999 4399 y Fn(Z)1082 4420 y Ft(x)1045 4588 y Fu(1)1102 4512 y Fy(dx)1192 4524 y Fu(0)1251 4512 y Fy(e)1290 4478 y Fu(\003)1335 4486 y Fc(\000)1385 4478 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)1627 4486 y Fo(0)1659 4478 y Fu(\))1695 4512 y Fv(^)g Fy(q)s Fv(\()p Fy(x)1808 4524 y Fu(0)1846 4512 y Fy(;)14 b(k)s Fv(\))23 b Fz(,)344 4742 y Fv(^)-45 b Fy(v)381 4754 y Ft(u)425 4742 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e Fs(\000)805 4686 y Fv(\003)863 4698 y Fu(+)918 4686 y Fv(\()p Fy(k)s Fv(\))p 805 4723 224 4 v 827 4799 a(\001\()p Fy(k)s Fv(\))1236 4686 y(1)p 1048 4723 417 4 v 1048 4799 a(\003)1106 4811 y Fu(+)1161 4799 y Fv(\()p Fy(k)s Fv(\))d(+)f Fs(j)p Fy(k)s Fs(j)1475 4742 y Fy(e)1514 4707 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1826 4742 y Fv(^)-48 b Fy(\021)1861 4754 y Fu(+)1916 4742 y Fv(\()p Fy(k)s Fv(\))19 b Fs(\000)2138 4686 y Fv(\003)2196 4698 y Fu(+)2251 4686 y Fv(\()p Fy(k)s Fv(\))p 2138 4723 224 4 v 2160 4799 a(\001\()p Fy(k)s Fv(\))2385 4629 y Fn(Z)2468 4649 y Fq(1)2431 4817 y Ft(x)2517 4742 y Fy(dx)2607 4754 y Fu(0)2666 4742 y Fy(e)2705 4707 y Fu(\003)2750 4715 y Fo(+)2797 4707 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)3039 4715 y Fo(0)3071 4707 y Fu(\))3107 4742 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)3220 4754 y Fu(0)3258 4742 y Fy(;)14 b(k)s Fv(\))23 b Fy(:)-9 4968 y Fz(In)f(matrix)f(notation,)h(we)f(ha)n(v)o(e) 1144 4991 y Fn(\022)1212 5052 y Fv(^)-45 b Fy(v)1249 5064 y Ft(s)1208 5165 y Fv(^)g Fy(v)1245 5177 y Ft(u)1288 4991 y Fn(\023)1349 5109 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fy(A)p Fv(\()p Fy(x;)14 b(k)s Fv(\))1911 4991 y Fn(\022)1988 5052 y Fv(^)-47 b Fy(\027)2024 5064 y Ft(s)1979 5165 y Fv(^)f Fy(\021)2014 5177 y Fu(+)2069 4991 y Fn(\023)2130 5109 y Fv(\()p Fy(k)s Fv(\))19 b(+)f Fy(b)p Fv(\()p Fy(x;)c(k)s Fv(\))24 b Fz(,)934 b(\(C.14\))-9 5303 y(where)645 5457 y Fy(A)p Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1012 5315 y Fn( )1078 5387 y Fy(e)1117 5357 y Fu(\003)1162 5365 y Fc(\000)1211 5357 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))1613 5355 y Ft(ik)p 1572 5369 144 4 v 1572 5416 a Fu(\001\()p Ft(k)q Fu(\))1873 5355 y(1)p 1735 5369 311 4 v 1735 5416 a(\003)1780 5424 y Fc(\000)1829 5416 y Fu(\()p Ft(k)q Fu(\)+)p Fq(j)p Ft(k)q Fq(j)2069 5320 y Fn(\000)2107 5387 y Fy(e)2146 5357 y Fu(\003)2191 5365 y Fc(\000)2240 5357 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2526 5387 y Fs(\000)18 b Fy(e)2648 5357 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2954 5320 y Fn(\001)1258 5516 y Fv(0)508 b Fs(\000)1883 5475 y Fu(\003)1928 5483 y Fo(+)1974 5475 y Fu(\()p Ft(k)q Fu(\))p 1882 5497 181 4 v 1901 5544 a(\001\()p Ft(k)q Fu(\))2220 5483 y(1)p 2083 5497 308 4 v 2083 5544 a(\003)2128 5552 y Fo(+)2175 5544 y Fu(\()p Ft(k)q Fu(\)+)p Fq(j)p Ft(k)q Fq(j)2401 5516 y Fy(e)2440 5486 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2992 5315 y Fn(!)3095 5457 y Fz(,)1838 5909 y(28)p eop %%Page: 29 29 bop -9 145 a Fz(and)1006 302 y Fy(b)p Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b(=)1347 185 y Fn(\022)1524 195 y Ft(ik)p 1482 209 144 4 v 1482 256 a Fu(\001\()p Ft(k)q Fu(\))1649 160 y Fn(R)1705 181 y Ft(x)1688 257 y Fu(1)1725 227 y Fy(dx)1815 239 y Fu(0)1874 227 y Fy(e)1913 197 y Fu(\003)1958 205 y Fc(\000)2007 197 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)2249 205 y Fo(0)2281 197 y Fu(\))2317 227 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2430 239 y Fu(0)2468 227 y Fy(;)14 b(k)s Fv(\))1408 378 y Fs(\000)1483 337 y Fu(\003)1528 345 y Fo(+)1575 337 y Fu(\()p Ft(k)q Fu(\))p 1483 359 181 4 v 1501 407 a(\001\()p Ft(k)q Fu(\))1687 311 y Fn(R)1743 332 y Fq(1)1726 408 y Ft(x)1791 378 y Fy(dx)1881 390 y Fu(0)1940 378 y Fy(e)1979 354 y Fu(\003)2024 362 y Fo(+)2071 354 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Ft(x)2313 362 y Fo(0)2345 354 y Fu(\))2382 378 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2495 390 y Fu(0)2533 378 y Fy(;)14 b(k)s Fv(\))2648 185 y Fn(\023)2732 302 y Fy(:)-9 537 y Fz(In)i(order)h(to)f(complete)f(the)h (proof,)j(we)c(no)n(w)i(ha)n(v)o(e)f(to)g(eliminate)e(from)j(\(C.12\))23 b Fv(^)-48 b Fy(\021)2341 549 y Fu(+)2412 537 y Fz(and)21 b Fv(^)-46 b Fy(\027)2592 549 y Ft(s)2643 537 y Fz(in)16 b(f)o(a)n(v)n(or)h(of) 22 b Fv(^)-47 b Fy(u)3050 549 y Fu(+)3120 537 y Fz(and)20 b Fv(^)-45 b Fy(v)3299 549 y Fu(+)3354 537 y Fz(.)29 b(Ev)n(aluating)-9 637 y(\(C.14\))22 b(at)f Fy(x)i Fv(=)g(1)e Fz(gi)n(v)o(es)1145 667 y Fn(\022)1211 728 y Fv(^)-47 b Fy(\027)1247 740 y Fq(\000)1211 841 y Fv(^)g Fy(\027)1247 853 y Fu(+)1303 667 y Fn(\023)1364 784 y Fv(\(1)p Fy(;)14 b(k)s Fv(\))23 b(=)g Fy(A)p Fv(\(1)p Fy(;)14 b(k)s Fv(\))1915 667 y Fn(\022)1981 728 y Fv(^)-47 b Fy(\027)2017 740 y Fu(+)1982 841 y Fv(^)f Fy(\021)2017 853 y Fu(+)2072 667 y Fn(\023)2133 784 y Fv(\()p Fy(k)s Fv(\))19 b(+)f Fy(b)p Fv(\(1)p Fy(;)c(k)s Fv(\))23 b Fy(:)-9 978 y Fz(From)f(this,)f (we)g(get)952 1091 y Fn(\022)1018 1152 y Fv(^)-46 b Fy(\027)1055 1164 y Fu(+)1019 1265 y Fv(^)e Fy(\021)1054 1277 y Fu(+)1110 1091 y Fn(\023)1171 1208 y Fv(\()p Fy(k)s Fv(\))23 b(=)g Fy(A)p Fv(\(1)p Fy(;)14 b(k)s Fv(\))1643 1174 y Fq(\000)p Fu(1)1746 1091 y Fn(\024\022)1856 1152 y Fv(^)-47 b Fy(\027)1892 1164 y Fq(\000)1856 1265 y Fv(^)g Fy(\027)1892 1277 y Fu(+)1948 1091 y Fn(\023)2009 1208 y Fv(\(1)p Fy(;)14 b(k)s Fv(\))k Fs(\000)g Fy(b)p Fv(\(1)p Fy(;)c(k)s Fv(\))2524 1091 y Fn(\025)1304 1441 y Fv(=)23 b Fy(A)p Fv(\(1)p Fy(;)14 b(k)s Fv(\))1643 1407 y Fq(\000)p Fu(1)1746 1324 y Fn(\024)1790 1441 y Fy(S)5 b Fv(\()p Fy(k)s Fv(\))1956 1407 y Fq(\000)p Fu(1)2045 1324 y Fn(\022)2112 1385 y Fv(^)-48 b Fy(u)2154 1397 y Fu(+)2113 1498 y Fv(^)j Fy(v)2150 1510 y Fu(+)2209 1324 y Fn(\023)2270 1441 y Fv(\()p Fy(k)s Fv(\))19 b Fs(\000)f Fy(b)p Fv(\(1)p Fy(;)c(k)s Fv(\))2707 1324 y Fn(\025)2787 1441 y Fz(,)-9 1668 y(and)21 b(therefore)488 1764 y Fn(\022)555 1824 y Fv(^)-48 b Fy(u)555 1938 y Fv(^)i Fy(v)597 1764 y Fn(\023)658 1881 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e Fy(S)5 b Fv(\()p Fy(k)s Fv(\))1129 1764 y Fn(\022)1195 1824 y Fv(^)-46 b Fy(\027)1232 1836 y Fq(\000)1196 1938 y Fv(^)f Fy(\027)1232 1950 y Fu(+)1288 1764 y Fn(\023)1349 1881 y Fv(\()p Fy(x;)14 b(k)s Fv(\))876 2113 y(=)22 b Fy(S)5 b Fv(\()p Fy(k)s Fv(\))1143 1996 y Fn(\022)1204 2113 y Fy(A)p Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fy(A)p Fv(\(1)p Fy(;)g(k)s Fv(\))1711 2079 y Fq(\000)p Fu(1)1815 1996 y Fn(\024)1859 2113 y Fy(S)5 b Fv(\()p Fy(k)s Fv(\))2025 2079 y Fq(\000)p Fu(1)2115 1996 y Fn(\022)2181 2057 y Fv(^)-47 b Fy(u)2224 2069 y Fu(+)2182 2170 y Fv(^)i Fy(v)2219 2182 y Fu(+)2278 1996 y Fn(\023)2339 2113 y Fv(\()p Fy(k)s Fv(\))19 b Fs(\000)f Fy(b)p Fv(\(1)p Fy(;)c(k)s Fv(\))2776 1996 y Fn(\025)2838 2113 y Fv(+)k Fy(b)p Fv(\()p Fy(x;)c(k)s Fv(\))3151 1996 y Fn(\023)3249 2113 y Fy(:)-9 2340 y Fz(The)24 b(e)n(v)n(aluation)g(of)g(the)g(matrix)g(products)i(is)e(not)h(dif)n (\256cult)f(b)n(ut)h(lengthy)-6 b(.)40 b(The)24 b(results)g(are)g(the)g (equations)h(as)f(stated)f(in)h(the)-9 2440 y(proposition.)-9 2681 y Fd(C.7)102 b(Pr)n(oof)26 b(of)f(Pr)n(oposition)h(3.6)-9 2838 y Fz(In)16 b(this)f(section,)i(we)e(sho)n(w)i(that)e(the)h(e)o (xpressions)g(gi)n(v)o(en)g(in)g(Proposition)h(3.6)g(solv)o(e)f(the)f(F)o (ourier)i(transform)f(of)h(\(3.15\)\261\(3.17\),)-9 2937 y(namely)-6 b(,)888 3184 y Fy(@)932 3196 y Ft(x)974 3067 y Fn(\022)1043 3128 y Fv(^)-49 b Fy(p)1041 3241 y Fv(^)h Fy(\021)1079 3067 y Fn(\023)1140 3184 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1446 3067 y Fn(\022)1523 3133 y Fv(0)99 b Fs(\000)p Fy(ik)1507 3233 y(ik)134 b Fv(0)1803 3067 y Fn(\023)14 b(\022)1947 3128 y Fv(^)-49 b Fy(p)1945 3241 y Fv(^)h Fy(\021)1983 3067 y Fn(\023)2044 3184 y Fv(\()p Fy(x;)14 b(k)s Fv(\))20 b(+)2341 3067 y Fn(\022)2402 3127 y Fs(\000)2468 3105 y Fv(^)2467 3127 y Fy(h)2515 3139 y Fu(1)2435 3220 y Fv(^)2434 3242 y Fy(h)2482 3254 y Fu(2)2551 3067 y Fn(\023)2613 3184 y Fv(\()p Fy(x;)14 b(k)s Fv(\))45 b Fz(,)678 b(\(C.15\))957 3358 y(lim)945 3409 y Ft(x)p Fq(!)p Fu(1)1102 3358 y Fv(^)-48 b Fy(\021)s Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f(0)f Fy(:)-9 3568 y Fz(Once)32 b(we)f(ha)n(v)o(e)h(sho)n(wn)i (this,)g(we)e(can)g(use)g(Lemma)f(3.5)i(to)f(get)f(that)g(the)h (right\261hand)i(sides)d(of)i(\(2.14\))h(and)e(\(2.15\))i(are)-9 3668 y(well)26 b(de\256ned)i(and)g(continuous)h(in)e Fy(x;)14 b(y)s Fz(,)28 b(so)g(that)e(these)h(equations)h(are)e(strongly)j (satis\256ed.)47 b(The)28 b(proof)h(that)d(the)h(inte)o(gral)-9 3768 y(e)o(xpression)17 b(solv)o(es)g(\(C.15\))g(is)g(v)o(ery)g(similar)e(to) i(the)f(one)h(of)g(Proposition)h(3.3,)h(b)n(ut)f(with)e Fv(\003)2655 3780 y Fq(\006)2727 3768 y Fz(replaced)g(by)i Fs(\006j)p Fy(k)s Fs(j)p Fz(.)28 b(In)18 b(particular)m(,)-9 3867 y(there)27 b(are)g(also)h(stable)f(and)h(unstable)g(modes.)51 b(The)28 b(proof)h(relies)e(on)i(Duhamel')-5 b(s)28 b(formula,)j(and)d(will)f(be)h(a)f (constructi)n(v)o(e)-9 3967 y(proof.)32 b(The)22 b(idea)f(is)g(to)g (diagonalize)g(in)g(\(C.15\))i(the)e(matrix)1486 4197 y Fy(M)9 b Fv(\()p Fy(k)s Fv(\))23 b(=)1796 4080 y Fn(\022)1874 4147 y Fv(0)99 b Fs(\000)p Fy(ik)1857 4246 y(ik)135 b Fv(0)2154 4080 y Fn(\023)2252 4197 y Fy(:)-9 4435 y Fz(W)-7 b(e)21 b(ha)n(v)o(e)h(that) e Fy(M)32 b Fv(=)23 b Fy(R)q(D)r(R)850 4405 y Fq(\000)p Fu(1)959 4435 y Fz(where)484 4649 y Fy(R)q Fv(\()p Fy(k)s Fv(\))p Fy(D)r Fv(\()p Fy(k)s Fv(\))p Fy(R)903 4615 y Fq(\000)p Fu(1)992 4649 y Fv(\()p Fy(k)s Fv(\))h Fs(\021)1213 4532 y Fn(\022)1405 4599 y Fv(1)374 b(1)1275 4698 y Fy(i)34 b Fz(s)-5 b(i)g(g)g(n)r Fv(\()p Fy(k)s Fv(\))83 b Fs(\000)p Fy(i)34 b Fz(s)-5 b(i)g(g)g(n)r Fv(\()p Fy(k)s Fv(\))2025 4532 y Fn(\023)14 b(\022)2161 4599 y Fs(\000j)p Fy(k)s Fs(j)108 b Fv(0)2219 4698 y(0)140 b Fs(j)p Fy(k)s Fs(j)2493 4532 y Fn(\023)14 b(\022)2639 4566 y Fu(1)p 2639 4580 34 4 v 2639 4627 a(2)2765 4599 y Fs(\000)2844 4566 y Ft(i)p 2840 4580 V 2840 4627 a Fu(2)2918 4599 y Fz(s)-5 b(i)g(g)g(n)q Fv(\()p Fy(k)s Fv(\))2639 4667 y Fu(1)p 2639 4681 V 2639 4728 a(2)2823 4667 y Ft(i)p 2818 4681 V 2818 4728 a Fu(2)2896 4700 y Fz(s)g(i)g(g)g(n)q Fv(\()p Fy(k)s Fv(\))3156 4532 y Fn(\023)3254 4649 y Fy(:)-9 4883 y Fz(W)e(e)21 b(de\256ne)h(the)f(ne)n(w)h(v)n(ariables)e Fv(\()7 b(^)-49 b Fy(p)1037 4895 y Ft(s)1073 4883 y Fy(;)21 b Fv(^)-49 b Fy(p)1152 4895 y Ft(u)1195 4883 y Fv(\))22 b Fz(by)1289 4998 y Fn(\022)1361 5059 y Fv(^)-49 b Fy(p)1396 5071 y Ft(s)1357 5172 y Fv(^)g Fy(p)1392 5184 y Ft(u)1435 4998 y Fn(\023)1496 5115 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e Fy(R)1865 5081 y Fq(\000)p Fu(1)1954 5115 y Fv(\()p Fy(k)s Fv(\))2064 4998 y Fn(\022)2134 5059 y Fv(^)-49 b Fy(p)2132 5172 y Fv(^)h Fy(\021)2170 4998 y Fn(\023)2231 5115 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)-9 5348 y Fz(F)o(or)e Fv(\()7 b(^)-49 b Fy(p)203 5360 y Ft(s)238 5348 y Fy(;)21 b Fv(^)-49 b Fy(p)317 5360 y Ft(u)360 5348 y Fv(\))p Fz(,)22 b(we)f(get)g(the)g(equations)1012 5597 y Fy(@)1056 5609 y Ft(x)1098 5480 y Fn(\022)1170 5541 y Fv(^)-49 b Fy(p)1205 5553 y Ft(s)1166 5654 y Fv(^)g Fy(p)1201 5666 y Ft(u)1244 5480 y Fn(\023)1305 5597 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fy(D)r Fv(\()p Fy(k)s Fv(\))1792 5480 y Fn(\022)1864 5541 y Fv(^)-49 b Fy(p)1899 5553 y Ft(s)1860 5654 y Fv(^)g Fy(p)1895 5666 y Ft(u)1939 5480 y Fn(\023)2000 5597 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)2296 5480 y Fn(\022)2362 5518 y Fv(^)2361 5540 y Fy(h)2409 5552 y Ft(s)2358 5633 y Fv(^)2357 5655 y Fy(h)2405 5667 y Ft(u)2448 5480 y Fn(\023)2509 5597 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fz(,)803 b(\(C.16\))1838 5909 y(29)p eop %%Page: 30 30 bop -9 145 a Fz(with)1233 174 y Fn(\022)1299 212 y Fv(^)1298 233 y Fy(h)1346 245 y Ft(s)1295 327 y Fv(^)1294 349 y Fy(h)1342 361 y Ft(u)1385 174 y Fn(\023)1446 291 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fy(R)1816 256 y Fq(\000)p Fu(1)1905 291 y Fv(\()p Fy(k)s Fv(\))2015 174 y Fn(\022)2076 233 y Fs(\000)2142 212 y Fv(^)2141 233 y Fy(h)2189 245 y Fu(1)2109 327 y Fv(^)2108 349 y Fy(h)2156 361 y Fu(2)2226 174 y Fn(\023)2287 291 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)1023 b Fz(\(C.17\))-9 481 y(From)22 b(\(C.16\))h(and)f(\(C.17\),)g(we)g(get)f(the)g(inte)o(gral)f(equations)885 703 y Fv(^)-49 b Fy(p)920 715 y Ft(s)955 703 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fy(e)1300 668 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1614 703 y Fv(^)-50 b Fy(\045)1649 715 y Ft(s)1684 703 y Fv(\()p Fy(k)s Fv(\))19 b(+)1896 590 y Fn(Z)1979 610 y Ft(x)1942 778 y Fu(1)2000 703 y Fy(dx)2090 715 y Fu(0)2149 703 y Fy(e)2188 668 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Ft(x)2470 676 y Fo(0)2501 668 y Fu(\))2532 681 y Fv(^)2531 703 y Fy(h)2579 715 y Ft(s)2614 703 y Fv(\()p Fy(x)2693 715 y Fu(0)2731 703 y Fy(;)14 b(k)s Fv(\))23 b Fz(,)877 929 y Fv(^)-49 b Fy(p)912 941 y Ft(u)955 929 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fs(\000)1340 816 y Fn(Z)1422 836 y Fq(1)1385 1005 y Ft(x)1471 929 y Fy(dx)1561 941 y Fu(0)1620 929 y Fy(e)1659 895 y Fq(j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Ft(x)1889 903 y Fo(0)1920 895 y Fu(\))1951 907 y Fv(^)1950 929 y Fy(h)1998 941 y Ft(u)2042 929 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fz(,)-9 1152 y(where)29 b Fv(^)-50 b Fy(\045)262 1164 y Ft(s)298 1152 y Fv(\()p Fy(k)s Fv(\))21 b Fz(is)g(a)g(function)i(to)e(be)h(determined.)31 b(From)e Fv(^)-49 b Fy(p)1741 1164 y Ft(s)1798 1152 y Fz(and)29 b Fv(^)-50 b Fy(p)1983 1164 y Ft(u)2048 1152 y Fz(we)21 b(compute)h Fv(\()7 b(^)-49 b Fy(p;)20 b Fv(^)-48 b Fy(\021)s Fv(\))p Fz(.)31 b(W)-7 b(e)22 b(get)155 1376 y Fv(^)-49 b Fy(p)p Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fy(e)535 1341 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))848 1376 y Fv(^)-49 b Fy(\045)884 1388 y Ft(s)919 1376 y Fv(\()p Fy(k)s Fv(\))19 b(+)1131 1263 y Fn(Z)1214 1283 y Ft(x)1177 1451 y Fu(1)1234 1376 y Fy(dx)1324 1388 y Fu(0)1384 1376 y Fy(e)1423 1341 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Ft(x)1705 1349 y Fo(0)1736 1341 y Fu(\))1767 1354 y Fv(^)1766 1376 y Fy(h)1814 1388 y Ft(s)1849 1376 y Fv(\()p Fy(x)1928 1388 y Fu(0)1966 1376 y Fy(;)14 b(k)s Fv(\))19 b Fs(\000)2183 1263 y Fn(Z)2266 1283 y Fq(1)2229 1451 y Ft(x)2314 1376 y Fy(dx)2404 1388 y Fu(0)2464 1376 y Fy(e)2503 1341 y Fq(j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Ft(x)2733 1349 y Fo(0)2764 1341 y Fu(\))2795 1354 y Fv(^)2794 1376 y Fy(h)2842 1388 y Ft(u)2885 1376 y Fv(\()p Fy(x)2964 1388 y Fu(0)3002 1376 y Fy(;)14 b(k)s Fv(\))23 b Fz(,)152 1605 y Fv(^)-48 b Fy(\021)s Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fy(i)34 b Fz(s)-5 b(i)g(g)g(n)r Fv(\()p Fy(k)s Fv(\))811 1488 y Fn(\022)872 1605 y Fy(e)911 1571 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1225 1605 y Fv(^)-50 b Fy(\045)1260 1617 y Ft(s)1296 1605 y Fv(\()p Fy(k)s Fv(\))19 b(+)1508 1492 y Fn(Z)1591 1512 y Ft(x)1554 1681 y Fu(1)1611 1605 y Fy(dx)1701 1617 y Fu(0)1760 1605 y Fy(e)1799 1571 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Ft(x)2081 1579 y 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y Fu(\))585 1918 y Fv(^)584 1940 y Fy(h)632 1952 y Fu(+)687 1940 y Fv(\()p Fy(x)766 1952 y Fu(0)804 1940 y Fy(;)14 b(k)s Fv(\))p Fz(.)75 b(Inserting)37 b(back)f(this)g(e)o(xpression)h(in)f(the)g(inte)o(gral)f(equations)h(for)44 b Fv(^)-49 b Fy(p)36 b Fz(and)42 b Fv(^)-48 b Fy(\021)39 b Fz(gi)n(v)o(es)d(the)-9 2040 y(desired)21 b(result,)g(we)h(omit)f(the)g (details.)-9 2322 y FB(A)m(ppendix)31 b(D:)g(Pr)n(oof)f(of)h(Lemma)f(3.8)-9 2509 y Fz(In)24 b(order)h(to)f(pro)o(v)o(e)h(Lemma)e(3.8,)j(we)e(bound)i (each)d(of)i(the)e(quadratic)h(terms)g(separately)-6 b(.)38 b(F)o(or)25 b(concision,)g(we)f(reformulate)-9 2609 y(Lemma)c(3.8)j(with)e (the)g(help)h(of)g(the)f(auxiliary)g(functions)i Fy(w)1801 2621 y Ft(\013)1870 2609 y Fz(and)39 b Fv(\026)-59 b Fy(w)2073 2621 y Ft(\013)2120 2609 y Fz(:)-9 2770 y FA(De\256nition)22 b(D)n(.1)42 b Fm(F)-9 b(or)22 b(all)f Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)1142 2782 y Fu(+)1198 2770 y Fm(,)e(we)g(de\256ne) 1410 2986 y Fy(w)1469 2998 y Ft(\013)1516 2986 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)2043 2930 y(1)p 1832 2967 464 4 v 1832 3043 a(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2099 2983 y(p)p 2168 2983 48 4 v 60 x Fy(x)q Fv(\))2248 3019 y Ft(\013)2328 2986 y Fm(,)p 1410 3201 62 4 v 1410 3247 a Fy(w)1472 3259 y Ft(\014)1516 3247 y Fv(\()p Fy(x;)c(k)s Fv(\))24 b(=)1822 3130 y Fn(\022)1958 3191 y Fy(x)2005 3160 y Fu(1)p Ft(=)p Fu(2)p 1893 3228 283 4 v 1893 3304 a Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fy(x)2185 3130 y Fn(\023)2246 3137 y Ft(\014)2328 3247 y Fy(:)-9 3473 y FA(Pr)n(oposition)k(D)n(.2)43 b Fm(Let)23 b Fy(h)775 3485 y Fu(1)834 3473 y Fm(and)i Fy(h)1033 3485 y Fu(2)1092 3473 y Fm(satisfy)d(Condition)i(3.7.)36 b(Then)24 b(for)f(all)f Fy(\027)31 b Fs(2)25 b Fv(\(1)p Fy(=)p Fv(2)p Fy(;)14 b Fv(1\))p Fm(,)22 b(ther)m(e)g(e)n(xists)g(a)h(constant)h Fy(C)29 b Fm(suc)o(h)-9 3573 y(that)21 b(for)g(all)g Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)g Fv(\012)732 3585 y Fu(+)787 3573 y Fm(,)d(we)g(have)382 3749 y Fs(j)6 b Fv(^)-48 b Fy(q)442 3761 y Fu(1)p Ft(;p)533 3749 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fs(j)7 b Fv(^)-49 b Fy(q)912 3761 y Fu(2)p Ft(;p)1004 3749 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b(H)g(x)1563 3715 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1719 3749 y Fy(w)1778 3761 y Ft(\013)1826 3749 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)490 3884 y Fs(j)6 b Fv(^)-48 b Fy(q)550 3896 y Fu(1)587 3884 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fs(j)7 b Fv(^)-49 b Fy(q)966 3896 y Fu(2)1004 3884 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b(H)g(x)1563 3850 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1719 3884 y Fs(j)p Fy(k)s Fs(j)22 b Fy(w)1892 3896 y Ft(\013)p Fu(+1)2023 3884 y Fv(\()p Fy(x;)14 b(k)s Fv(\))20 b(+)e Fy(C)27 b(H)h(x)2550 3850 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2707 3884 y Fs(j)p Fy(k)s Fs(j)p Fy(w)2858 3896 y Ft(\013)p Fq(\000)p Fu(1)2990 3884 y Fv(\()p Fy(x;)14 b(k)s Fv(\))k(\026)-60 b Fy(w)3243 3896 y Fu(2)3282 3884 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)348 4027 y Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)479 4039 y Fu(1)516 4027 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)966 4039 y Fu(2)1004 4027 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b(H)g(x)1573 3966 y Fk(\015)s Fc(\000)p Fo(1)p 1573 3980 108 3 v 1613 4013 a(2)1694 4027 y Fy(w)1753 4039 y Ft(\013)1801 4027 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(C)28 b(H)g(x)2328 3993 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2484 4027 y Fy(w)2543 4039 y Ft(\013)p Fq(\000)p Fu(1)2676 4027 y Fv(\()p Fy(x;)14 b(k)s Fv(\))k(\026)-60 b Fy(w)2929 4039 y Fu(2)2968 4027 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fm(,)240 4170 y Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)371 4182 y Fu(1)p Ft(;p)462 4170 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)912 4182 y Fu(2)p Ft(;p)1004 4170 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b(H)g(x)1573 4109 y Fk(\015)s Fc(\000)p Fo(1)p 1573 4123 V 1613 4155 a(2)1694 4170 y Fy(w)1753 4182 y Ft(\013)1801 4170 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(C)28 b(H)g(x)2328 4135 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2484 4170 y Fy(w)2543 4182 y Ft(\013)p Fq(\000)p Fu(1)2676 4170 y Fv(\()p Fy(x;)14 b(k)s Fv(\))k(\026)-60 b Fy(w)2929 4182 y Fu(2)2968 4170 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1304 4305 y(+)k Fy(C)28 b(H)g(w)1630 4317 y Ft(\013)p Fq(\000)p Fu(1)1784 4305 y Fy(x)1831 4270 y Fu(1)p Ft(=)p Fu(2)p Fq(\000)p Ft(\027)2046 4305 y Fs(j)p Fy(k)s Fs(j)2138 4270 y Fq(\000)p Ft(\027)2254 4305 y Fy(:)-9 4481 y Fm(Pr)l(oof)o(.)j Fz(See)21 b(the)g(follo)n(wing)h (subsections.)p 1314 4481 43 43 v -9 4721 a Fd(D)n(.1)102 b(Pr)n(eliminary)27 b(Bounds)-9 4878 y Fz(W)-7 b(e)21 b(be)o(gin)h(with)f(a)g(preparatory)i (lemma.)-9 5039 y FA(Lemma)d(D)n(.3)43 b Fm(Let)21 b Fy(\025)i(>)g Fv(0)p Fm(.)30 b(Ther)m(e)22 b(e)n(xists)e(a)i(constant)g Fy(C)28 b Fm(suc)o(h)21 b(that)h(for)f(all)g Fy(k)26 b Fs(2)d Fr(R)e Fm(and)i Fy(x)p Fm(,)f Fy(x)2808 5051 y Fu(0)2869 5039 y Fs(\025)g Fv(1)f Fm(we)g(have)1234 5215 y Fy(e)1273 5181 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5502 48 4 v 60 x Fy(x)p Fs(j)2708 5532 y Ft(\013)p 2377 5599 501 4 v 2377 5675 a Fv(\(1)19 b(+)f Fs(j)p Fy(k)2622 5622 y Fs(p)p 2691 5622 85 4 v 53 x Fy(x)2738 5687 y Fu(0)2775 5675 y Fs(j)p Fv(\))2830 5651 y Ft(\013)2911 5618 y Fy(:)1838 5909 y Fz(30)p eop %%Page: 31 31 bop -9 145 a Fz(F)o(or)22 b Fy(x)176 157 y Fu(0)236 145 y Fs(\025)h Fy(x=)p Fv(2)e Fz(we)g(ha)n(v)o(e)h(that)981 374 y Fy(e)1020 340 y Fq(\000)p Ft(\025)p Fq(j)p Ft(k)q Fq(jj)p Ft(x)p Fq(\000)p Ft(x)1335 348 y Fo(0)1366 340 y Fq(j)1490 318 y Fs(j)p Fy(k)1559 258 y Fs(p)p 1628 258 48 4 v 60 x Fy(x)q Fs(j)1699 288 y Ft(\013)p 1400 355 437 4 v 1400 431 a Fv(1)c(+)g Fs(j)p Fy(k)1612 378 y Fs(p)p 1681 378 85 4 v 53 x Fy(x)1728 443 y Fu(0)1766 431 y Fs(j)1789 407 y Ft(\013)1869 374 y Fs(\024)23 b Fy(C)2173 318 y Fs(j)p Fy(k)2242 258 y Fs(p)p 2312 258 48 4 v 2312 318 a Fy(x)p Fs(j)2382 288 y Ft(\013)p 2054 355 496 4 v 2054 443 a Fv(1)18 b(+)g Fs(j)p Fy(k)2266 372 y Fn(p)p 2348 372 131 4 v 2348 443 a Fy(x=)p Fv(2)p Fs(j)2502 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(the)g(de\256nitions)h(of)g Fy(q)1804 1185 y Ft(i)1853 1173 y Fz(and)g Fy(q)2034 1185 y Ft(i;p)2116 1173 y Fz(:)1106 1362 y Fy(g)1146 1374 y Fq(\006)1201 1362 y Fv(\()p Fy(k)s(;)14 b(x)1363 1374 y Fu(0)1401 1362 y Fy(;)g(x)p Fv(\))24 b(=)f Fy(e)1668 1327 y Fq(\000j)p Ft(k)q Fq(j)d(j)p Ft(x)p Fq(\000)p Ft(x)1964 1335 y Fo(0)1996 1327 y Fq(j)2038 1362 y Fs(\006)e Fy(e)2160 1327 y Fq(\000j)p Ft(k)q Fq(j)j Fu(\()p Ft(x)p Fu(+)p Ft(x)2462 1335 y Fo(0)2494 1327 y Fq(\000)p Fu(2\))2632 1362 y Fy(:)-9 1540 y Fz(The)g(follo)n(wing)h(lemma)f(summarizes)g(the)g (properties)h(of)g(these)f(functions)i(using)f(Lemma)f(D.3.)-9 1702 y FA(Lemma)f(D)n(.4)43 b Fm(Ther)m(e)21 b(e)n(xists)g(a)g(constant)i Fy(C)k Fm(suc)o(h)22 b(that)f(for)h(all)f Fy(k)26 b Fs(2)d Fr(R)p Fm(,)e Fy(x;)14 b(x)2292 1714 y Fu(0)2353 1702 y Fs(\025)23 b Fv(1)p Fm(,)e(we)f(have)303 1880 y Fv(0)j Fs(\024)g Fy(g)496 1892 y Fq(\006)551 1880 y Fv(\()p Fy(k)s(;)14 b(x)713 1892 y Fu(0)751 1880 y Fy(;)g(x)p Fv(\))22 b Fy(w)948 1892 y Ft(\013)996 1880 y Fv(\()p Fy(x)1075 1892 y Fu(0)1113 1880 y Fy(;)14 b(k)s Fv(\))23 b Fs(\024)f Fy(C)28 b(e)1464 1845 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)1807 1853 y Fo(0)1838 1845 y Fq(j)1883 1880 y Fy(w)1942 1892 y Ft(\013)1990 1880 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)1359 b Fz(\(D.2\))303 2004 y Fv(0)23 b Fs(\024)g Fy(g)496 2016 y Fq(\006)551 2004 y Fv(\()p Fy(k)s(;)14 b(x)713 2016 y Fu(0)751 2004 y Fy(;)g(x)p Fv(\))22 b Fy(w)948 2016 y Ft(\013)996 2004 y Fv(\()p Fy(x)1075 2016 y Fu(0)1113 2004 y Fy(;)14 b(k)s Fv(\))23 b Fs(\024)f Fy(C)28 b(g)1465 2016 y Fq(\006)1520 2004 y Fv(\()p Fy(k)s(=)p Fv(2)p Fy(;)14 b(x)1766 2016 y Fu(0)1803 2004 y Fy(;)g(x)p Fv(\))22 b Fy(w)2000 2016 y Ft(\013)2048 2004 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)1301 b Fz(\(D.3\))332 2171 y Fs(j)p Fy(@)399 2183 y Ft(k)440 2171 y Fy(g)480 2183 y Fq(\006)536 2171 y Fv(\()p Fy(k)s(;)14 b(x)698 2183 y Fu(0)735 2171 y Fy(;)g(x)p Fv(\))p Fs(j)h Fy(w)948 2183 y Ft(\013)996 2171 y Fv(\()p Fy(x)1075 2183 y Fu(0)1113 2171 y Fy(;)f(k)s Fv(\))23 b Fs(\024)1351 2115 y Fy(C)p 1348 2152 92 4 v 1348 2228 a Fs(j)p Fy(k)s Fs(j)1450 2171 y Fy(e)1489 2137 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)1832 2145 y Fo(0)1863 2137 y Fq(j)1908 2171 y Fy(w)1967 2183 y Ft(\013)2015 2171 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)1334 b Fz(\(D.4\))332 2395 y Fs(j)p Fy(@)399 2407 y Ft(k)440 2395 y Fy(g)480 2407 y Fq(\000)536 2395 y Fv(\()p Fy(k)s(;)14 b(x)698 2407 y Fu(0)735 2395 y Fy(;)g(x)p Fv(\))p Fs(j)h Fy(w)948 2407 y Ft(\013)996 2395 y Fv(\()p Fy(x)1075 2407 y Fu(0)1113 2395 y Fy(;)f(k)s Fv(\))23 b Fs(\024)1351 2339 y Fy(C)p 1348 2376 V 1348 2452 a Fs(j)p Fy(k)s Fs(j)1464 2303 y Fn(\020)1514 2395 y Fy(e)1553 2361 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)1896 2369 y Fo(0)1927 2361 y Fq(j)1969 2395 y Fs(\000)18 b Fy(e)2091 2361 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(j)p Fu(\()p Ft(x)p Fu(+)p Ft(x)2439 2369 y Fo(0)2470 2361 y Fq(\000)p Fu(2\))2586 2303 y Fn(\021)2670 2395 y Fy(w)2729 2407 y Ft(\013)2777 2395 y Fv(\()p Fy(x;)c(k)s Fv(\))1311 2584 y(+)k Fs(jj)p Fy(x)h Fs(\000)f Fy(x)1636 2596 y Fu(0)1673 2584 y Fs(j)h(\000)f Fv(\()p Fy(x)h Fv(+)f Fy(x)2026 2596 y Fu(0)2082 2584 y Fs(\000)g Fv(2\))p Fs(j)c Fy(e)2315 2549 y Fq(\000j)p Ft(k)q(=)p Fu(4)p Fq(j)p Fu(\()p Fq(j)p Ft(x)p Fq(\000)p Ft(x)2684 2557 y Fo(0)2715 2549 y Fq(j)p Fu(+\()p Ft(x)p Fu(+)p Ft(x)2939 2557 y Fo(0)2970 2549 y Fq(\000)p Fu(2\)\))3111 2584 y Fy(w)3170 2596 y Ft(\013)3218 2584 y Fv(\()p Fy(x;)g(k)s Fv(\))24 b Fm(,)131 b Fz(\(D.5\))333 2708 y Fs(j)p Fy(@)400 2720 y Ft(k)441 2708 y Fy(g)481 2720 y Fu(+)536 2708 y Fv(\()p Fy(k)s(;)14 b(x)698 2720 y Fu(0)735 2708 y Fy(;)g(x)p Fv(\))p Fs(j)h Fy(w)948 2720 y Ft(\013)996 2708 y Fv(\()p Fy(x)1075 2720 y Fu(0)1113 2708 y Fy(;)f(k)s Fv(\))23 b Fs(\024)f Fy(C)42 b Fz(max)p Fs(f)p Fv(\()p Fy(x)1706 2720 y Fu(0)1762 2708 y Fs(\000)18 b Fv(1\))p Fy(;)c Fv(\()p Fy(x)19 b Fs(\000)f Fv(1\))p Fs(g)i Fy(e)2312 2674 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)2655 2682 y Fo(0)2687 2674 y Fq(j)2732 2708 y Fy(w)2791 2720 y Ft(\013)2838 2708 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)509 b Fz(\(D.6\))-9 2886 y Fm(Pr)l(oof)o(.)31 b Fz(There)21 b(e)o(xists)g(a)g(constant)g Fy(C)28 b Fz(such)22 b(that)e(for)j(all)d Fy(\033)26 b Fs(2)e(f)p Fv(1)p Fy(;)14 b Fs(\000)p Fv(1)p Fs(g)19 b Fz(and)i Fv(0)i Fs(\024)g Fy(a)g Fs(\024)f Fy(b)p Fz(,)f(we)h(ha)n(v)o(e)335 3064 y Fv(0)h Fs(\024)f Fy(e)526 3030 y Fq(\000)p Ft(a)636 3064 y Fv(+)c Fy(\033)s(e)808 3030 y Fq(\000)p Ft(b)917 3064 y Fs(\024)23 b Fv(2)p Fy(e)1086 3030 y Fq(\000)p Ft(a)1200 3064 y Fs(\024)f Fv(2)p Fy(e)1368 3030 y Fq(\000)p Ft(a=)p Fu(2)1527 3064 y Fy(e)1566 3030 y Fq(\000)p Ft(a=)p Fu(2)1748 3064 y Fz(,)1819 b(\(D.7\))335 3189 y Fv(0)23 b Fs(\024)f Fy(e)526 3154 y Fq(\000)p Ft(a)636 3189 y Fv(+)c Fy(\033)s(e)808 3154 y Fq(\000)p Ft(b)917 3189 y Fs(\024)23 b Fv(\()p Fy(e)1076 3154 y Fq(\000)p Ft(a=)p Fu(2)1253 3189 y Fv(+)18 b Fy(e)1375 3154 y Fq(\000)p Ft(b=)p Fu(2)1527 3189 y Fv(\)\()p Fy(e)1630 3154 y Fq(\000)p Ft(a=)p Fu(2)1808 3189 y Fv(+)g Fy(\033)s(e)1980 3154 y Fq(\000)p Ft(b=)p Fu(2)2132 3189 y Fv(\))24 b Fs(\024)e Fv(2\()p Fy(e)2388 3154 y Fq(\000)p Ft(a=)p Fu(2)2565 3189 y Fv(+)c Fy(\033)s(e)2737 3154 y Fq(\000)p Ft(b=)p Fu(2)2890 3189 y Fv(\))p Fy(e)2961 3154 y Fq(\000)p Ft(a=)p Fu(2)3143 3189 y Fz(,)424 b(\(D.8\))458 3313 y Fy(ae)541 3279 y Fq(\000)p Ft(a)651 3313 y Fv(+)18 b Fy(be)809 3279 y Fq(\000)p Ft(b)917 3313 y Fs(\024)23 b Fv(\()p Fy(e)1076 3279 y Fq(\000)p Ft(a=)p Fu(2)1253 3313 y Fv(+)18 b Fy(e)1375 3279 y Fq(\000)p Ft(b=)p Fu(2)1527 3313 y Fv(\)\()p Fy(ae)1674 3279 y Fq(\000)p Ft(a=)p Fu(2)1852 3313 y Fv(+)g Fy(be)2010 3279 y Fq(\000)p Ft(b=)p Fu(2)2161 3313 y Fv(\))917 3438 y Fs(\024)23 b Fv(\()p Fy(e)1076 3403 y Fq(\000)p Ft(a=)p Fu(2)1253 3438 y Fv(+)18 b Fy(e)1375 3403 y Fq(\000)p Ft(b=)p Fu(2)1527 3438 y Fv(\)\()p Fy(e)1630 3403 y Fq(\000)p Ft(a=)p Fu(4)1808 3438 y Fv(+)g Fy(e)1930 3403 y Fq(\000)p Ft(b=)p Fu(4)2082 3438 y Fv(\)\()p Fy(ae)2229 3403 y Fq(\000)p Ft(a=)p Fu(4)2407 3438 y Fv(+)g Fy(be)2565 3403 y Fq(\000)p Ft(b=)p Fu(4)2716 3438 y Fv(\))917 3562 y Fs(\024)23 b Fy(C)k Fv(\()p Fy(e)1162 3528 y Fq(\000)p Ft(a=)p Fu(2)1340 3562 y Fv(+)18 b Fy(e)1462 3528 y Fq(\000)p Ft(b=)p Fu(2)1614 3562 y Fv(\))p Fy(e)1685 3528 y Fq(\000)p Ft(a=)p Fu(4)1867 3562 y Fz(,)1700 b(\(D.9\))458 3687 y Fy(ae)541 3652 y Fq(\000)p Ft(a)651 3687 y Fs(\000)18 b Fy(be)809 3652 y Fq(\000)p Ft(b)917 3687 y Fv(=)23 b(\()p Fy(e)1076 3652 y Fq(\000)p Ft(a=)p Fu(2)1253 3687 y Fs(\000)18 b Fy(e)1375 3652 y Fq(\000)p Ft(b=)p Fu(2)1527 3687 y Fv(\)\()p Fy(ae)1674 3652 y Fq(\000)p Ft(a=)p Fu(2)1852 3687 y Fv(+)g Fy(be)2010 3652 y Fq(\000)p Ft(b=)p Fu(2)2161 3687 y Fv(\))h(+)f(\()p Fy(a)h Fs(\000)f Fy(b)p Fv(\))p Fy(e)2580 3652 y Fq(\000)p Fu(\()p Ft(a)p Fu(+)p Ft(b)p Fu(\))p Ft(=)p Fu(2)917 3811 y Fs(\024)23 b Fv(\()p Fy(e)1076 3777 y Fq(\000)p Ft(a=)p Fu(2)1253 3811 y Fs(\000)18 b Fy(e)1375 3777 y Fq(\000)p Ft(b=)p Fu(2)1527 3811 y Fv(\)\()p Fy(e)1630 3777 y Fq(\000)p Ft(a=)p Fu(4)1808 3811 y Fv(+)g Fy(e)1930 3777 y Fq(\000)p Ft(b=)p Fu(4)2082 3811 y Fv(\)\()p Fy(ae)2229 3777 y Fq(\000)p Ft(a=)p Fu(4)2407 3811 y Fv(+)g Fy(be)2565 3777 y Fq(\000)p Ft(b=)p Fu(4)2716 3811 y Fv(\))h(+)f(\()p Fy(a)h Fs(\000)f Fy(b)p Fv(\))p Fy(e)3135 3777 y Fq(\000)p Fu(\()p Ft(a)p Fu(+)p Ft(b)p Fu(\))p Ft(=)p Fu(2)917 3936 y Fs(\024)23 b Fy(C)k Fv(\()p Fy(e)1162 3901 y Fq(\000)p Ft(a=)p Fu(2)1340 3936 y Fs(\000)18 b Fy(e)1462 3901 y Fq(\000)p Ft(b=)p Fu(2)1614 3936 y Fv(\))p Fy(e)1685 3901 y Fq(\000)p Ft(a=)p Fu(4)1862 3936 y Fv(+)g Fs(j)p Fy(a)h Fs(\000)f Fy(b)p Fs(j)p Fy(e)2212 3901 y Fq(\000)p Fu(\()p Ft(a)p Fu(+)p Ft(b)p Fu(\))p Ft(=)p Fu(2)917 4085 y Fs(\024)23 b Fy(C)1105 3993 y Fn(\020)1154 4085 y Fy(e)1193 4051 y Fq(\000)p Ft(a=)p Fu(2)1371 4085 y Fs(\000)18 b Fy(e)1493 4051 y Fq(\000)p Ft(b=)p Fu(2)1663 4085 y Fv(+)g Fs(j)p Fy(a)h Fs(\000)f Fy(b)p Fs(j)p Fy(e)2013 4051 y Fq(\000)p Fu(\()p Ft(a)p Fu(+)p Ft(b)p Fu(\))p Ft(=)p Fu(4)2303 3993 y Fn(\021)2367 4085 y Fy(e)2406 4051 y Fq(\000)p Ft(a=)p Fu(4)2587 4085 y Fy(:)936 b Fz(\(D.10\))-9 4288 y(Since)31 b Fy(x)25 b Fv(+)f Fy(x)419 4300 y Fu(0)480 4288 y Fs(\000)g Fv(2)37 b Fs(\025)g(j)p Fy(x)19 b Fs(\000)f Fy(x)969 4300 y Fu(0)1007 4288 y Fs(j)p Fz(,)35 b(for)d Fy(x)p Fz(,)k Fy(x)1367 4300 y Fu(0)1442 4288 y Fs(\025)h Fv(1)p Fz(,)d(we)e(can)g (use)g(these)f(inequalities)g(with)h Fy(a)37 b Fv(=)g Fs(j)p Fy(k)s Fs(jj)p Fy(x)25 b Fs(\000)e Fy(x)3385 4300 y Fu(0)3423 4288 y Fs(j)32 b Fz(and)g Fy(b)38 b Fv(=)-9 4388 y Fs(j)p Fy(k)s Fs(j)p Fv(\()p Fy(x)25 b Fv(+)e Fy(x)322 4400 y Fu(0)384 4388 y Fs(\000)h Fv(2\))p Fz(.)63 b(Inequality)33 b(\(D.2\))g(follo)n(ws)g(from)g (\(D.7\))h(using)f(\(D.1\))h(with)e Fy(\025)38 b Fv(=)f(1)p Fy(=)p Fv(2)p Fz(.)62 b(Inequality)33 b(\(D.3\))g(follo)n(ws)-9 4487 y(from)25 b(\(D.8\))i(using)f(\(D.1\))g(with)f Fy(\025)j Fv(=)f(1)p Fy(=)p Fv(2)p Fz(.)40 b(Inequality)26 b(\(D.4\))g(follo)n(ws)f (from)h(\(D.9\))h(using)f(\(D.1\))g(with)f Fy(\025)j Fv(=)f(1)p Fy(=)p Fv(4)p Fz(.)40 b(Finally)-9 4587 y(inequality)21 b(\(D.5\))i(follo)n (ws)f(from)g(\(D.10\),)i(using)e(\(D.1\))h(with)e Fy(\025)j Fv(=)e(1)p Fy(=)p Fv(4)p Fz(.)p 2219 4587 43 43 v -9 4686 a(Looking)28 b(back)e(at)f(the)h(bounds)j(on)e Fy(h)1139 4698 y Fu(1)1176 4686 y Fy(;)14 b(h)1261 4698 y Fu(2)1324 4686 y Fz(of)27 b(Condition)g(3.7,)h (and)f(using)g(the)f(bounds)j(of)e(Lemma)e(D.4,)k(we)d(see)g(that)f(we)-9 4786 y(will)20 b(ha)n(v)o(e)i(to)f(bound)j(inte)o(grals)c(of)i(the)f(form) 1602 4891 y Fn(Z)1684 5004 y Fy(dx)1774 5016 y Fu(0)1833 5004 y Fy(x)1880 4961 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1880 5026 y(0)2058 5004 y Fy(e)2097 4970 y Fq(\000j)p Ft(k)q Fq(jj)p Ft(x)p Fq(\000)p Ft(x)2373 4978 y Fo(0)2404 4970 y Fq(j)2451 5004 y Fz(,)1074 b(\(D.11\))1288 5109 y Fn(Z)1371 5222 y Fy(dx)1461 5234 y Fu(0)1520 5222 y Fv(\()p Fy(x)1599 5234 y Fu(0)1655 5222 y Fs(\000)18 b Fv(1\))j Fy(x)1880 5179 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1880 5244 y(0)2058 5222 y Fy(e)2097 5188 y Fq(\000j)p Ft(k)q Fq(jj)p Ft(x)p Fq(\000)p Ft(x)2373 5196 y Fo(0)2404 5188 y Fq(j)2451 5222 y Fz(,)1074 b(\(D.12\))1559 5327 y Fn(Z)1642 5440 y Fy(dx)1732 5452 y Fu(0)1791 5440 y Fy(x)1838 5396 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1838 5462 y(0)2016 5440 y Fy(g)2056 5452 y Fq(\000)2112 5440 y Fv(\()p Fy(k)s(;)14 b(x)2274 5452 y Fu(0)2312 5440 y Fy(;)g(x)p Fv(\))23 b Fz(,)1074 b(\(D.13\))-9 5660 y(either)20 b(on)i Fv([1)p Fy(;)14 b(x)p Fv(])22 b Fz(or)g(on)g Fv([)p Fy(x;)14 b Fs(1)p Fv(\))p Fz(.)32 b(The)21 b(three)g(follo)n(wing)i(lemmas)d (deal)h(with)g(these)g(three)g(types)h(of)g(inte)o(grals.)1838 5909 y(31)p eop %%Page: 32 32 bop -9 145 a FA(Lemma)20 b(D)n(.5)43 b Fm(Ther)m(e)21 b(e)n(xists)g(a)g (constant)i Fy(C)k Fm(suc)o(h)22 b(that)f(for)h(all)f Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)2256 157 y Fu(+)2311 145 y Fm(,)e(we)g(have)1272 243 y Fn(Z)1355 264 y Fq(1)1318 432 y Ft(x)1403 356 y Fy(dx)1493 368 y Fu(0)1553 356 y Fy(x)1600 313 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1600 379 y(0)1778 356 y Fy(e)1817 322 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)2160 330 y Fo(0)2191 322 y Fq(j)2238 356 y Fs(\024)h Fy(C)28 b(x)2459 322 y Fq(\000)p Fu(1)2587 356 y Fv(\026)-59 b Fy(w)2629 368 y Fu(1)2666 356 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)641 b Fz(\(D.14\))1300 470 y Fn(Z)1383 490 y Ft(x)1346 658 y Fu(1)1403 583 y Fy(dx)1493 595 y Fu(0)1553 583 y Fy(x)1600 540 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1600 605 y(0)1778 583 y Fy(e)1817 548 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)2160 556 y Fo(0)2191 548 y Fq(j)2238 583 y Fs(\024)22 b Fy(C)49 b(x)2480 548 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2658 583 y Fv(\()17 b(\026)-59 b Fy(w)2749 595 y Fu(2)2787 583 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(w)3142 595 y Fu(1)3180 583 y Fv(\()p Fy(x;)c(k)s Fv(\)\))24 b Fm(,)95 b Fz(\(D.15\))310 696 y Fn(Z)393 717 y Fq(1)356 885 y Fu(1)441 809 y Fy(dx)531 821 y Fu(0)590 809 y Fy(x)637 766 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)637 831 y(0)829 809 y Fz(min)p Fv(\()p Fy(x)20 b Fs(\000)e Fv(1)p Fy(;)c(x)1269 821 y Fu(0)1324 809 y Fs(\000)k Fv(1\))p Fy(e)1520 775 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(j)p Fz(max)p Fu(\()p Ft(x)p Fq(\000)p Fu(1)p Ft(;x)2068 783 y Fo(0)2099 775 y Fq(\000)p Fu(1\))2238 809 y Fs(\024)k Fy(C)28 b(x)2459 775 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2654 809 y Fv(\026)-59 b Fy(w)2696 821 y Fu(2)2733 809 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)572 b Fz(\(D.16\))-9 1016 y Fm(Pr)l(oof)o(.)31 b Fz(De\256ne,)22 b(for)g Fy(t)h Fs(\025)g Fv(0)e Fz(the)g(function)h Fy(I)1289 1028 y Fu(1)1327 1016 y Fy(;)14 b(I)1400 1028 y Fu(2)1438 1016 y Fy(;)g(I)1511 1028 y Fu(3)1569 1016 y Fz(and)22 b Fy(I)1749 1028 y Fu(4)1808 1016 y Fz(by)h(the)e(equations)1078 1225 y Fy(I)1114 1237 y Fu(1)1152 1225 y Fv(\()p Fy(t)p Fv(\))i(=)g(\(1)18 b(+)g Fy(t)p Fv(\))1608 1112 y Fn(Z)1691 1133 y Fq(1)1654 1301 y Fu(1)1740 1225 y Fy(ds)j(s)1882 1191 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2060 1225 y Fy(e)2099 1191 y Fq(\000)p Ft(t)p Fu(\()p Ft(s)p Fq(\000)p Fu(1\))2371 1225 y Fz(,)1078 1464 y Fy(I)1114 1476 y Fu(2)1152 1464 y Fv(\()p Fy(t)p Fv(\))i(=)g(\(1)18 b(+)g Fy(t)p Fv(\))1608 1351 y Fn(Z)1691 1372 y Fu(1)1654 1540 y(1)p Ft(=)p Fu(2)1737 1464 y Fy(ds)k(s)1880 1430 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2057 1464 y Fy(e)2096 1430 y Ft(t)p Fu(\()p Ft(s)p Fq(\000)p Fu(1\))2316 1464 y Fz(,)1078 1724 y Fy(I)1114 1736 y Fu(3)1152 1724 y Fv(\()p Fy(t)p Fv(\))h(=)1438 1667 y Fy(t)p 1367 1705 173 4 v 1367 1781 a(t)18 b Fs(\000)g Fv(1)1564 1611 y Fn(Z)1647 1631 y Fu(\()p Ft(t)p Fu(+1\))p Ft(=)p Fu(2)1610 1799 y(1)1857 1724 y Fy(dx)1947 1736 y Fu(0)2007 1724 y Fy(x)2054 1681 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2054 1746 y(0)2233 1724 y Fz(,)1078 1961 y Fy(I)1114 1973 y Fu(4)1152 1961 y Fv(\()p Fy(t)p Fv(\))23 b(=)1357 1844 y Fn(\022)1581 1905 y Fy(t)1611 1874 y Fu(2)p 1428 1942 374 4 v 1428 2026 a Fv(\()p Fy(t)c Fs(\000)f Fv(1\))1666 2002 y Fu(2)1703 1958 y Fs(p)p 1772 1958 30 4 v 68 x Fy(t)1826 1848 y Fn(Z)1909 1868 y Ft(t)1872 2036 y Fu(1)1916 1961 y Fy(dx)2006 1973 y Fu(0)2066 1961 y Fv(\()p Fy(x)2145 1973 y Fu(0)2201 1961 y Fs(\000)g Fv(1\))p Fy(x)2405 1918 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2405 1983 y(0)2562 1844 y Fn(\023)2660 1961 y Fy(:)-9 2177 y Fz(The)f(functions)h Fy(I)509 2189 y Fu(1)547 2177 y Fy(;)c(I)620 2189 y Fu(2)658 2177 y Fy(;)g(I)731 2189 y Fu(3)786 2177 y Fz(and)k Fy(I)962 2189 y Fu(4)1017 2177 y Fz(can)f(be)g(e)o(xpressed)h(in)g(terms)f(of)h(elementary) f(functions,)i(in)e(particular)m(,)h(the)o(y)f(are)g(bounded.)-9 2277 y(W)-7 b(e)21 b(therefore)h(ha)n(v)o(e)f(the)h(bounds)387 2368 y Fn(Z)470 2389 y Fq(1)433 2557 y Ft(x)519 2481 y Fy(dx)609 2493 y Fu(0)668 2481 y Fy(x)715 2438 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)715 2504 y(0)893 2481 y Fy(e)932 2447 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)1275 2455 y Fo(0)1306 2447 y Fq(j)1353 2481 y Fv(=)h Fy(x)1488 2447 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1658 2368 y Fn(Z)1741 2389 y Fq(1)1704 2557 y Fu(1)1790 2481 y Fy(dx)1880 2493 y Fu(0)1939 2481 y Fy(x)1986 2438 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1986 2504 y(0)2164 2481 y Fy(e)2203 2447 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(j)p Ft(x)p Fu(\()p Ft(x)2500 2455 y Fo(0)2532 2447 y Fq(\000)p Fu(1\))2670 2481 y Fs(\024)2767 2425 y Fy(C)28 b(x)2901 2395 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)p 2767 2462 291 4 v 2771 2538 a Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fy(x)3105 2481 y Fz(sup)3103 2553 y Ft(x)p Fq(\025)p Fu(0)3239 2481 y Fy(I)3275 2493 y Fu(1)3313 2481 y Fv(\()p Fy(x)p Fv(\))24 b Fz(,)271 2623 y Fn(Z)354 2643 y Ft(x)317 2811 y Fu(\()p Ft(x)p Fu(+1\))p Ft(=)p Fu(2)540 2736 y Fy(dx)630 2748 y Fu(0)689 2736 y Fy(x)736 2693 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)736 2758 y(0)893 2736 y Fy(e)932 2701 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)1275 2709 y Fo(0)1306 2701 y Fq(j)1353 2736 y Fv(=)f Fy(x)1488 2701 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1658 2623 y Fn(Z)1741 2643 y Fu(1)1714 2786 y Fk(x)p Fo(+1)p 1714 2799 104 3 v 1736 2831 a(2)p Fk(x)1811 2736 y Fy(dx)1901 2748 y Fu(0)1960 2736 y Fy(x)2007 2693 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2007 2758 y(0)2185 2736 y Fy(e)2224 2701 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(j)p Ft(x)p Fu(\(1)p Fq(\000)p Ft(x)2606 2709 y Fo(0)2637 2701 y Fu(\))2690 2736 y Fs(\024)2788 2679 y Fy(C)k(x)2921 2649 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)p 2788 2717 291 4 v 2792 2793 a Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fy(x)3126 2736 y Fz(sup)3123 2808 y Ft(x)p Fq(\025)p Fu(0)3260 2736 y Fy(I)3296 2748 y Fu(2)3333 2736 y Fv(\()p Fy(x)p Fv(\))24 b Fz(,)418 2888 y Fn(Z)501 2908 y Fu(\()p Ft(x)p Fu(+1\))p Ft(=)p Fu(2)464 3076 y(1)724 3001 y Fy(dx)814 3013 y Fu(0)883 2944 y Fy(e)922 2914 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)1265 2922 y Fo(0)1296 2914 y Fq(j)p 883 2981 437 4 v 1026 3078 a Fy(x)1073 3035 y Fu(3)p Ft(=)p Fu(2)1073 3100 y(0)1353 3001 y Fs(\024)f Fy(e)1480 2966 y Fq(\000j)p Ft(k)q(=)p Fu(4)p Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1863 2944 y Fy(x)c Fs(\000)f Fv(1)p 1863 2981 191 4 v 1935 3058 a Fy(x)2101 3001 y Fz(sup)2099 3073 y Ft(x)p Fq(\025)p Fu(1)2235 3001 y Fy(I)2271 3013 y Fu(3)2309 3001 y Fv(\()p Fy(x)p Fv(\))1353 3254 y Fs(\024)1563 3198 y Fy(C)p 1451 3235 311 4 v 1451 3311 a Fv(1)g(+)g Fy(k)1640 3287 y Fu(2)1677 3311 y Fy(x)1724 3287 y Fu(2)1785 3137 y Fn(\022)1846 3254 y Fv(\(1)h(+)f Fy(k)2068 3220 y Fu(2)2105 3254 y Fy(x)2152 3220 y Fu(2)2189 3254 y Fv(\))2231 3198 y Fy(x)i Fs(\000)e Fv(1)p 2231 3235 191 4 v 2303 3311 a Fy(x)2432 3254 y(e)2471 3220 y Fq(\000j)p Ft(k)q(=)p Fu(4)p Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2845 3137 y Fn(\023)1353 3479 y Fs(\024)1563 3423 y Fy(C)p 1451 3460 311 4 v 1451 3536 a Fv(1)g(+)g Fy(k)1640 3512 y Fu(2)1677 3536 y Fy(x)1724 3512 y Fu(2)1785 3387 y Fn(\020)1835 3479 y Fv(1)g(+)g Fy(k)2024 3445 y Fu(2)2061 3479 y Fy(x)p Fv(\()p Fy(x)i Fs(\000)e Fv(1\))p Fy(e)2403 3445 y Fq(\000j)p Ft(k)q(=)p Fu(4)p Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2776 3387 y Fn(\021)1353 3721 y Fs(\024)1563 3665 y Fy(C)p 1451 3702 V 1451 3778 a Fv(1)g(+)g Fy(k)1640 3754 y Fu(2)1677 3778 y Fy(x)1724 3754 y Fu(2)1785 3579 y Fn( )1851 3721 y Fv(1)g(+)g Fz(sup)1994 3794 y Ft(\030)r Fq(\025)p Fu(0)2112 3721 y Fv(\()p Fs(j)p Fy(k)s Fs(j)p Fy(\030)23 b Fv(+)18 b Fy(\030)2418 3687 y Fu(2)2455 3721 y Fv(\))p Fy(e)2526 3687 y Fq(\000)p Ft(\030)r(=)p Fu(4)2682 3579 y Fn(!)1353 3977 y Fs(\024)23 b Fy(C)1575 3920 y Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p 1537 3958 V 1537 4034 a Fv(1)g(+)g Fy(k)1726 4010 y Fu(2)1763 4034 y Fy(x)1810 4010 y Fu(2)1353 4194 y Fs(\024)23 b Fy(C)1541 4077 y Fn(\022)1602 4194 y Fy(x)1649 4160 y Fq(\000)p Fu(1)1777 4194 y Fv(\026)-59 b Fy(w)1819 4206 y Fu(2)1857 4194 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(x)2200 4160 y Fq(\000)p Fu(1)2441 4138 y Fv(1)p 2321 4175 283 4 v 2321 4251 a(1)g(+)g Fs(j)p Fy(k)s Fs(j)p Fy(x)2613 4077 y Fn(\023)2711 4194 y Fy(:)-9 4414 y Fz(Inequality)33 b(\(D.14\))j(follo)n(ws)e(at)f(once)g(by)i (de\256nition)f(of)51 b Fv(\026)-59 b Fy(w)1841 4426 y Fu(1)1879 4414 y Fz(,)36 b(and)e(inequality)f(\(D.15\))j(follo)n(ws)e(since)f Fv(\(1)25 b(+)f Fs(j)p Fy(k)s Fs(j)p Fy(x)p Fv(\))3576 4384 y Fq(\000)p Fu(1)3706 4414 y Fs(\024)-9 4514 y Fv(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)258 4454 y(p)p 327 4454 48 4 v 60 x Fy(x)q Fv(\))407 4484 y Fq(\000)p Fu(1)496 4514 y Fz(.)33 b(Finally)-6 b(,)22 b(to)g(sho)n(w)g(\(D.16\),)j(we)c(consider)i(separately)e(the)h(inte)o (grals)f(o)o(v)o(er)h Fv([1)p Fy(;)14 b(x)p Fv(])21 b Fz(and)i Fv([)p Fy(x;)14 b Fs(1)p Fv(\))p Fz(.)33 b(On)22 b Fv([)p Fy(x;)14 b Fs(1)p Fv(\))p Fz(,)-9 4613 y(we)21 b(use)g(\(D.14\))j(to)d(sho)n(w)i(that) 410 4820 y Fv(\()p Fy(x)c Fs(\000)f Fv(1\))679 4707 y Fn(Z)762 4727 y Fq(1)725 4895 y Ft(x)810 4820 y Fy(dx)900 4832 y Fu(0)960 4820 y Fy(x)1007 4777 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1007 4842 y(0)1184 4820 y Fy(e)1223 4785 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(j)p Fu(\()p Ft(x)1482 4793 y Fo(0)1514 4785 y Fq(\000)p Fu(1\))1652 4820 y Fs(\024)23 b Fv(\()p Fy(x)c Fs(\000)f Fv(1\))p Fy(e)2034 4785 y Fq(\000j)p Ft(k)q(=)p Fu(4)p Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2421 4707 y Fn(Z)2504 4727 y Fq(1)2467 4895 y Fu(1)2553 4820 y Fy(dx)2643 4832 y Fu(0)2702 4820 y Fy(x)2749 4777 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2749 4842 y(0)2906 4820 y Fy(e)2945 4785 y Fq(\000j)p Ft(k)q(=)p Fu(4)p Fq(j)p Fu(\()p Ft(x)3204 4793 y Fo(0)3236 4785 y Fq(\000)p Fu(1\))1652 5058 y Fs(\024)23 b Fy(C)1916 5002 y(x)1963 4972 y Fu(1)p Ft(=)p Fu(2)p 1837 5039 311 4 v 1837 5115 a Fv(1)17 b(+)i Fy(k)2026 5091 y Fu(2)2063 5115 y Fy(x)2110 5091 y Fu(2)2157 5058 y Fv(\(1)f(+)h Fs(j)p Fy(k)s Fs(j)p Fy(x)p Fv(\))2514 5002 y Fy(x)g Fs(\000)f Fv(1)p 2514 5039 191 4 v 2586 5115 a Fy(x)2715 5058 y(e)2754 5024 y Fq(\000j)p Ft(k)q(=)p Fu(4)p Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1652 5233 y Fs(\024)23 b Fy(C)28 b(x)1874 5199 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2069 5233 y Fv(\026)-59 b Fy(w)2111 5245 y Fu(2)2148 5233 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fz(,)-9 5398 y(and)d(on)i Fv([1)p Fy(;)14 b(x)p Fv(])p Fz(,)21 b(we)g(ha)n(v)o(e)344 5623 y Fy(e)383 5588 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))770 5510 y Fn(Z)853 5530 y Ft(x)816 5698 y Fu(1)873 5623 y Fy(dx)963 5635 y Fu(0)1023 5623 y Fv(\()p Fy(x)1102 5635 y Fu(0)1158 5623 y Fs(\000)d Fv(1\))j Fy(x)1383 5579 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1383 5645 y(0)1563 5623 y Fv(=)1740 5566 y Fy(x)1787 5536 y Fu(1)p Ft(=)p Fu(2)p 1660 5603 311 4 v 1660 5680 a Fv(1)d(+)g Fy(k)1849 5656 y Fu(2)1886 5680 y Fy(x)1933 5656 y Fu(2)1995 5506 y Fn(\022)2056 5623 y Fv(\(1)g(+)g Fy(k)2277 5588 y Fu(2)2314 5623 y Fy(x)2361 5588 y Fu(2)2399 5623 y Fv(\))2441 5566 y(\()p Fy(x)i Fs(\000)e Fv(1\))2697 5536 y Fu(2)p 2441 5603 293 4 v 2545 5680 a Fy(x)2592 5656 y Fu(2)2744 5623 y Fy(e)2783 5588 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))3156 5506 y Fn(\023)3231 5623 y Fy(I)3267 5635 y Fu(4)3305 5623 y Fv(\()p Fy(x)p Fv(\))1838 5909 y Fz(32)p eop %%Page: 33 33 bop 1563 207 a Fs(\024)22 b Fy(C)1826 151 y(x)1873 121 y Fu(1)p Ft(=)p Fu(2)p 1747 188 311 4 v 1747 264 a Fv(1)c(+)g Fy(k)1936 240 y Fu(2)1973 264 y Fy(x)2020 240 y Fu(2)2081 65 y Fn( )2147 207 y Fv(1)g(+)g Fz(sup)2291 279 y Ft(\030)r(>)p Fu(0)2422 207 y Fy(\030)2462 173 y Fu(2)2499 207 y Fy(e)2538 173 y Fq(\000)p Ft(\030)2626 65 y Fn(!)2708 207 y Fz(sup)2706 279 y Ft(x)p Fq(\025)p Fu(1)2842 207 y Fy(I)2878 219 y Fu(4)2916 207 y Fv(\()p Fy(x)p Fv(\))1563 474 y Fs(\024)k Fy(C)1826 418 y(x)1873 387 y Fu(1)p Ft(=)p Fu(2)p 1747 455 V 1747 531 a Fv(1)c(+)g Fy(k)1936 507 y Fu(2)1973 531 y Fy(x)2020 507 y Fu(2)2091 474 y Fs(\024)k Fy(C)28 b(x)2312 439 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2507 474 y Fv(\026)-59 b Fy(w)2549 486 y Fu(2)2586 474 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)-9 683 y Fz(This)d(achie)n(v)o(es)f(the)h(proof.)p 855 683 43 43 v -9 862 a FA(Lemma)f(D)n(.6)43 b Fm(F)-9 b(or)22 b(all)f Fy(\027)28 b Fs(2)c Fv(\(1)p Fy(=)p Fv(2)p Fy(;)14 b Fv(1\))p Fm(,)20 b(ther)m(e)g(e)n(xists)h(a)g(constant)i Fy(C)k Fm(suc)o(h)22 b(that)f(for)h(all)f Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(2)h Fv(\012)2941 874 y Fu(+)2996 862 y Fm(,)d(we)g(have)907 975 y Fn(Z)990 996 y Fq(1)953 1164 y Ft(x)1039 1088 y Fy(dx)1129 1100 y Fu(0)1188 1088 y Fy(e)1227 1054 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)1570 1062 y Fo(0)1601 1054 y Fq(j)1646 1088 y Fv(\()p Fy(x)1725 1100 y Fu(0)1781 1088 y Fs(\000)d Fv(1\))j Fy(x)2006 1045 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2006 1111 y(0)2186 1088 y Fs(\024)i Fy(C)k(x)2407 1054 y Fu(1)p Ft(=)p Fu(2)p Fq(\000)p Ft(\027)2622 1088 y Fs(j)p Fy(k)s Fs(j)2714 1054 y Fq(\000)p Ft(\027)2831 1088 y Fy(:)692 b Fz(\(D.17\))-9 1310 y Fm(Pr)l(oof)o(.)31 b Fz(W)-7 b(e)21 b(ha)n(v)o(e)469 1420 y Fn(Z)552 1440 y Fq(1)516 1608 y Ft(x)601 1533 y Fy(dx)691 1545 y Fu(0)750 1533 y Fy(e)789 1498 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)1132 1506 y Fo(0)1163 1498 y Fq(j)1209 1533 y Fv(\()p Fy(x)1288 1545 y Fu(0)1344 1533 y Fs(\000)d Fv(1\))j Fy(x)1569 1489 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1569 1555 y(0)1749 1533 y Fs(\024)h Fy(C)1916 1420 y Fn(Z)1999 1440 y Fq(1)1962 1608 y Ft(x)2047 1533 y Fy(dx)2137 1545 y Fu(0)2196 1533 y Fy(e)2235 1498 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)2578 1506 y Fo(0)2610 1498 y Fq(j)2655 1533 y Fy(x)2702 1489 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2702 1555 y(0)1749 1771 y Fs(\024)g Fy(C)28 b Fs(j)p Fy(k)s Fs(j)2015 1737 y Fq(\000)p Ft(\027)2122 1658 y Fn(Z)2205 1679 y Fq(1)2168 1847 y Ft(x)2254 1771 y Fy(dx)2344 1783 y Fu(0)2413 1715 y Fs(j)p Fy(k)s Fv(\()p Fy(x)19 b Fs(\000)f Fy(x)2710 1727 y Fu(0)2748 1715 y Fv(\))p Fs(j)2803 1685 y Ft(\027)2844 1715 y Fy(e)2883 1685 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)3226 1693 y Fo(0)3257 1685 y Fq(j)p 2413 1752 869 4 v 2591 1849 a Fy(x)2638 1806 y Fu(1)p Ft(=)p Fu(2)2638 1871 y(0)2764 1849 y Fv(\()p Fy(x)2843 1861 y Fu(0)2899 1849 y Fs(\000)g Fy(x)p Fv(\))3061 1825 y Ft(\027)1749 2022 y Fs(\024)k Fy(C)28 b Fs(j)p Fy(k)s Fs(j)2015 1987 y Fq(\000)p Ft(\027)2129 2022 y Fy(x)2176 1987 y Fu(1)p Ft(=)p Fu(2)p Fq(\000)p Ft(\027)2384 1909 y Fn(Z)2467 1929 y Fq(1)2430 2097 y Fu(1)2516 2022 y Fy(ds)21 b(s)2658 1987 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2814 2022 y Fv(\()p Fy(s)e Fs(\000)f Fv(1\))3061 1987 y Fq(\000)p Ft(\027)3177 2022 y Fy(:)-9 2248 y Fz(Since)j Fy(\027)28 b Fs(2)23 b Fv(\(1)p Fy(=)p Fv(2)p Fy(;)14 b Fv(1\))p Fz(,)20 b(the)h(inte)o(gral)g(is)g(con)m(v)o(er)n (gent,)i(and)f(\(D.17\))h(follo)n(ws.)p 2270 2248 43 43 v -9 2427 a FA(Lemma)d(D)n(.7)43 b Fm(Ther)m(e)21 b(e)n(xists)g(a)g(constant)i Fy(C)k Fm(suc)o(h)22 b(that)f(for)h(all)f Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)2256 2439 y Fu(+)2311 2427 y Fm(,)e(we)g(have)985 2540 y Fn(Z)1068 2561 y Fq(1)1031 2729 y Fu(1)1116 2653 y Fy(dx)1206 2665 y Fu(0)1266 2653 y Fy(x)1313 2610 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1313 2676 y(0)1469 2653 y Fy(g)1509 2665 y Fq(\000)1565 2653 y Fv(\()p Fy(k)s(=)p Fv(2)p Fy(;)14 b(x)1811 2665 y Fu(0)1848 2653 y Fy(;)g(x)p Fv(\))23 b Fs(\024)g Fy(C)49 b Fs(j)p Fy(k)s Fs(j)21 b Fy(x)2343 2619 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2538 2653 y Fv(\026)-59 b Fy(w)2580 2665 y Fu(2)2618 2653 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fm(,)690 b Fz(\(D.18\))904 2880 y Fs(j)p Fy(k)s Fs(j)1010 2767 y Fn(Z)1093 2787 y Fq(1)1056 2955 y Fu(1)1142 2880 y Fy(dx)1232 2892 y Fu(0)1291 2880 y Fy(x)1348 2804 y Fk(\015)s Fc(\000)p Fo(1)p 1348 2818 108 3 v 1388 2851 a(2)1338 2902 y Fu(0)1469 2880 y Fy(g)1509 2892 y Fq(\000)1565 2880 y Fv(\()p Fy(k)s(=)p Fv(2)p Fy(;)14 b(x)1811 2892 y Fu(0)1848 2880 y Fy(;)g(x)p Fv(\))23 b Fs(\024)g Fy(C)28 b(x)2219 2819 y Fk(\015)s Fc(\000)p Fo(1)p 2219 2833 V 2258 2865 a(2)2363 2880 y Fy(:)1160 b Fz(\(D.19\))-9 3101 y Fm(Pr)l(oof)o(.)31 b Fz(De\256ne)22 b(for)g Fy(t)h Fs(\025)g Fv(0)d Fz(the)h(function)i Fy(I)1268 3113 y Fu(5)1306 3101 y Fy(;)14 b(I)1379 3113 y Fu(6)1438 3101 y Fz(and)21 b Fy(I)1617 3113 y Fu(7)1676 3101 y Fz(by)i(the)e(equations)1119 3337 y Fy(I)1155 3349 y Fu(5)1193 3337 y Fv(\()p Fy(t)p Fv(\))i(=)1408 3281 y(1)18 b(+)g Fy(t)1581 3251 y Fu(2)p 1408 3318 211 4 v 1498 3394 a Fy(t)1628 3337 y(e)1667 3303 y Fq(\000)p Ft(t)1762 3224 y Fn(Z)1845 3245 y Fu(1)1808 3413 y(0)1861 3337 y Fy(ds)j(s)2003 3303 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2173 3270 y Fn(\000)2211 3337 y Fy(e)2250 3303 y Ft(ts)2329 3337 y Fs(\000)d Fy(e)2451 3303 y Fq(\000)p Ft(ts)2563 3270 y Fn(\001)2638 3337 y Fz(,)1119 3570 y Fy(I)1155 3582 y Fu(6)1193 3570 y Fv(\()p Fy(t)p Fv(\))23 b(=)1408 3514 y(1)18 b(+)g Fy(t)1581 3484 y Fu(2)p 1408 3551 V 1498 3627 a Fy(t)1642 3570 y Fz(sinh)q Fv(\()p Fy(t)p Fv(\))1892 3457 y Fn(Z)1975 3477 y Fq(1)1938 3646 y Fu(1)2024 3570 y Fy(ds)j(s)2166 3536 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2344 3570 y Fy(e)2383 3536 y Fq(\000)p Ft(ts)2518 3570 y Fz(,)1102 3820 y Fy(I)1138 3832 y Fu(7)1175 3820 y Fv(\()p Fy(x)p Fv(\))j(=)1414 3763 y Fy(x)1471 3706 y Fo(1)p Fc(\000)p Fk(\015)p 1471 3720 108 3 v 1511 3753 a Fo(2)p 1408 3801 191 4 v 1408 3877 a Fy(x)19 b Fs(\000)f Fv(1)1622 3707 y Fn(Z)1705 3727 y Fu(\()p Ft(x)p Fu(+1\))p Ft(=)p Fu(2)1668 3895 y(1)1929 3820 y Fy(dx)2019 3832 y Fu(0)2078 3820 y Fy(x)2135 3744 y Fk(\015)s Fc(\000)p Fo(1)p 2135 3758 108 3 v 2175 3791 a(2)2125 3842 y Fu(0)2279 3820 y Fy(:)-9 4040 y Fz(These)25 b(functions)j(can)e(be)g(e)o(xpressed)h(in)f(terms)f(of)i (elementary)f(functions,)i(in)e(particular)m(,)h(the)o(y)f(are)g(bounded.)48 b(T)-7 b(o)26 b(pro)o(v)o(e)-9 4140 y(\(D.18\),)d(we)e(split)g(the)g(inte)o (gration)g(in)h(tw)o(o)f(parts.)32 b(W)-7 b(e)21 b(\256rst)g(deal)g(with)h Fv([)p Fy(x;)14 b Fs(1)p Fv(\))p Fz(.)31 b(Using)22 b(\(D.14\),)i(we)d(ha)n (v)o(e)247 4256 y Fn(Z)330 4277 y Fq(1)293 4445 y Ft(x)379 4369 y Fy(dx)469 4381 y Fu(0)528 4369 y Fy(x)575 4326 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)575 4391 y(0)753 4369 y Fy(g)793 4381 y Fq(\000)848 4369 y Fv(\()p Fy(k)s(=)p Fv(2)p Fy(;)14 b(x)1094 4381 y Fu(0)1131 4369 y Fy(;)g(x)p Fv(\))24 b(=)e Fy(e)1397 4335 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1785 4369 y Fz(sinh)p Fv(\()p Fs(j)p Fy(k)s(=)p Fv(2)p Fs(j)p Fv(\()p Fy(x)d 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Fv(\026)-59 b Fy(w)2548 4744 y Fu(2)2586 4732 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)-9 4960 y Fz(W)-7 b(e)21 b(no)n(w)h(turn)g(to)g Fv([1)p Fy(;)14 b(x)p Fv(])p Fz(.)31 b(W)-7 b(e)21 b(ha)n(v)o(e)945 5073 y Fn(Z)1028 5093 y Ft(x)991 5262 y Fu(1)1048 5186 y Fy(dx)1138 5198 y Fu(0)1219 5186 y Fy(x)1266 5143 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1266 5208 y(0)1443 5186 y Fy(g)1483 5198 y Fq(\000)1539 5186 y Fv(\()p Fy(k)s(=)p Fv(2)p Fy(;)14 b(x)1785 5198 y Fu(0)1822 5186 y Fy(;)g(x)p Fv(\))24 b Fs(\024)e Fy(W)2127 5198 y Fu(1)2165 5186 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(W)2539 5198 y Fu(2)2577 5186 y Fv(\()p Fy(x;)c(k)s Fv(\))23 b Fz(,)-9 5407 y(where)273 5626 y Fy(W)351 5638 y Fu(1)388 5626 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fy(e)733 5592 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(j)p Ft(x)983 5513 y Fn(Z)1066 5533 y Ft(x)1029 5701 y Fu(1)1087 5626 y Fy(dx)1177 5638 y Fu(0)1236 5626 y Fy(x)1283 5583 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1283 5648 y(0)1453 5534 y Fn(\020)1503 5626 y Fy(e)1542 5592 y Fq(j)p Ft(k)q(=)p Fu(2)p Fq(j)p Ft(x)1723 5600 y Fo(0)1777 5626 y Fs(\000)18 b Fy(e)1899 5592 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(j)p Ft(x)2132 5600 y Fo(0)2168 5534 y Fn(\021)2241 5626 y Fs(\024)23 b Fy(C)2459 5570 y Fs(j)p Fy(k)s Fs(j)p Fy(x)2598 5540 y Fu(1)p Ft(=)p Fu(2)p 2425 5607 311 4 v 2425 5683 a Fv(1)18 b(+)g Fy(k)2614 5659 y Fu(2)2651 5683 y Fy(x)2698 5659 y Fu(2)2746 5626 y Fy(I)2782 5638 y Fu(5)2820 5626 y Fv(\()p Fy(k)s(x=)p Fv(2\))k Fz(,)1838 5909 y(33)p eop %%Page: 34 34 bop 273 180 a Fy(W)351 192 y Fu(2)388 180 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f(2)p Fy(e)775 146 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))1162 180 y Fz(sinh)p Fv(\()p Fs(j)p Fy(k)s(=)p Fv(2)p Fs(j)p Fv(\))1557 67 y Fn(Z)1640 87 y Ft(x)1603 256 y Fu(1)1660 180 y Fy(dx)1750 192 y Fu(0)1809 180 y Fy(x)1856 137 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1856 202 y(0)2013 180 y Fy(e)2052 146 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(j)p Ft(x)2285 154 y Fo(0)2344 180 y Fs(\024)f Fy(C)2595 124 y Fs(j)p Fy(k)s Fs(j)p 2528 161 227 4 v 2528 237 a Fv(1)c(+)g Fy(k)2717 213 y Fu(2)2764 180 y Fy(e)2803 146 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))3176 180 y Fy(I)3212 192 y Fu(6)3250 180 y Fv(\()p Fy(k)s(=)p Fv(2\))23 b Fz(,)-9 404 y(so)e(that)g(we)g(ha)n(v)o(e)958 501 y Fn(Z)1041 522 y Ft(x)1004 690 y Fu(1)1061 614 y Fy(dx)1151 626 y Fu(0)1210 614 y Fy(x)1257 571 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1257 636 y(0)1435 614 y Fy(g)1475 626 y Fq(\000)1531 614 y Fv(\()p Fy(k)s(=)p Fv(2)p Fy(;)14 b(x)1777 626 y Fu(0)1813 614 y Fy(;)g(x)p Fv(\))24 b Fs(\024)f Fy(C)k Fs(j)p Fy(k)s Fs(j)21 b Fy(x)2287 580 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2482 614 y Fv(\026)-59 b Fy(w)2524 626 y Fu(2)2562 614 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)-9 838 y Fz(This)d(completes)g(the)g(proof)i(of)f(\(D.18\).)33 b(W)-7 b(e)21 b(no)n(w)i(turn)f(to)f(\(D.19\).)33 b(W)-7 b(e)22 b(split)e(again)h(the)g(inte)o(gral)g(in)g(tw)o(o)h(parts.)31 b(W)-7 b(e)21 b(ha)n(v)o(e)396 1065 y Fs(j)p Fy(k)s Fs(j)502 952 y Fn(Z)585 973 y Fq(1)548 1141 y Fu(\()p Ft(x)p Fu(+1\))p Ft(=)p Fu(2)772 1065 y Fy(dx)862 1077 y Fu(0)921 1065 y Fy(x)978 990 y Fk(\015)s Fc(\000)p Fo(1)p 978 1004 108 3 v 1018 1037 a(2)968 1088 y Fu(0)1121 1065 y Fy(g)1161 1077 y Fq(\000)1216 1065 y Fv(\()p Fy(k)s(=)p Fv(2)p Fy(;)14 b(x)1462 1077 y Fu(0)1499 1065 y Fy(;)g(x)p Fv(\))24 b Fs(\024)e Fy(C)28 b(x)1870 1004 y Fk(\015)s Fc(\000)p Fo(1)p 1870 1018 V 1910 1051 a(2)2005 952 y Fn(Z)2088 973 y Fq(1)2051 1141 y(\0001)2152 1065 y Fy(dx)2242 1077 y Fu(0)2301 1065 y Fs(j)p Fy(k)s Fs(j)p Fy(e)2432 1031 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)2775 1039 y Fo(0)2806 1031 y Fq(j)2853 1065 y Fs(\024)23 b Fy(C)k(x)3084 1004 y Fk(\015)s Fc(\000)p Fo(1)p 3085 1018 V 3124 1051 a(2)3229 1065 y Fz(,)360 1325 y Fs(j)p Fy(k)s Fs(j)466 1212 y Fn(Z)548 1232 y Fu(\()p Ft(x)p Fu(+1\))p Ft(=)p Fu(2)511 1401 y(1)772 1325 y Fy(dx)862 1337 y Fu(0)921 1325 y Fy(x)978 1249 y Fk(\015)s Fc(\000)p Fo(1)p 978 1263 V 1018 1296 a(2)968 1347 y Fu(0)1121 1325 y Fy(g)1161 1337 y Fq(\000)1216 1325 y Fv(\()p Fy(k)s(=)p Fv(2)p Fy(;)14 b(x)1462 1337 y Fu(0)1499 1325 y Fy(;)g(x)p Fv(\))24 b Fs(\024)e Fy(C)28 b Fs(j)p Fy(k)s Fs(j)p Fv(\()p Fy(x)19 b Fs(\000)f Fv(1\))p Fy(e)2199 1291 y Fq(\000j)p Ft(k)q(=)p Fu(4)p Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2594 1325 y Fy(x)2651 1264 y Fk(\015)s Fc(\000)p Fo(1)p 2651 1278 V 2690 1311 a(2)2793 1325 y Fy(I)2829 1337 y Fu(7)2867 1325 y Fv(\()p Fy(x)p Fv(\))24 b Fs(\024)f Fy(C)k(x)3233 1264 y Fk(\015)s Fc(\000)p Fo(1)p 3234 1278 V 3273 1311 a(2)3378 1325 y Fy(:)-9 1549 y Fz(This)21 b(completes)g(the)g(proof)i(of)f(the)f(lemma.)p 1380 1549 43 43 v -9 1790 a Fd(D)n(.2)102 b(Bound)24 b(on)32 b Fa(^)-56 b Fb(q)731 1805 y Ff(1)796 1790 y Fd(and)25 b Fb(D)10 b Fa(^)-56 b Fb(q)1113 1805 y Ff(1)-9 1947 y Fz(Using)18 b(\(D.3\))i(and)e(\(D.18\),)j (it)c(is)h(no)n(w)g(an)g(easy)g(task)g(to)g(sho)n(w)h(that)e(there)h(e)o (xists)f(a)g(constant)h Fy(C)24 b Fz(such)19 b(that)e(for)i(all)e Fv(\()p Fy(x;)d(k)s Fv(\))23 b Fs(2)h Fv(\012)3694 1959 y Fu(+)3749 1947 y Fz(,)818 2179 y Fs(j)6 b Fv(^)-48 b Fy(q)878 2191 y Fu(1)915 2179 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b(H)1441 2062 y Fn(\022)1502 2066 y(Z)1585 2087 y Fq(1)1548 2255 y Fu(1)1634 2179 y Fy(dx)1724 2191 y Fu(0)1783 2179 y Fy(x)1830 2136 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1830 2201 y(0)1987 2179 y Fy(g)2027 2191 y Fq(\000)2082 2179 y Fv(\()p Fy(k)s(=)p Fv(2)p Fy(;)14 b(x)2328 2191 y Fu(0)2365 2179 y Fy(;)g(x)p Fv(\))2481 2062 y Fn(\023)2557 2179 y Fy(w)2616 2191 y Ft(\013)p Fq(\000)p Fu(1)2748 2179 y Fv(\()p Fy(x;)g(k)s Fv(\))1156 2369 y Fs(\024)22 b Fy(C)28 b(H)g Fs(j)p Fy(k)s Fs(j)21 b Fy(x)1587 2335 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1744 2369 y Fy(w)1803 2381 y Ft(\013)p Fq(\000)p Fu(1)1936 2369 y Fv(\()p Fy(x;)14 b(k)s Fv(\))j(\026)-59 b Fy(w)2189 2381 y Fu(2)2227 2369 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)-9 2552 y Fz(Similarly)-6 b(,)21 b(using)h(\(D.3\),)h(\(D.5\),)g(\(D.16\),)g(\(D.18\))h(and)d(\(D.19\),)j(we)d (sho)n(w)h(that)f(there)g(e)o(xists)f(a)h(constant)g Fy(C)28 b Fz(such)21 b(that)g(for)h(all)-9 2651 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(2)h Fv(\012)347 2663 y Fu(+)402 2651 y Fz(,)367 2882 y Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)498 2894 y Fu(1)535 2882 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)k(H)1057 2826 y(w)1116 2838 y Ft(\013)p Fq(\000)p Fu(1)1249 2826 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p 1057 2863 387 4 v 1204 2939 a Fs(j)p Fy(k)s Fs(j)1467 2769 y Fn(Z)1550 2789 y Fq(1)1513 2957 y Fu(1)1599 2882 y Fy(dx)1689 2894 y Fu(0)1748 2882 y Fy(x)1795 2839 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1795 2904 y(0)1973 2882 y Fy(g)2013 2894 y Fq(\000)2069 2882 y Fv(\()p Fy(k)s(=)p Fv(2)p Fy(;)g(x)2315 2894 y Fu(0)2352 2882 y Fy(;)g(x)p Fv(\))836 3110 y(+)k Fy(C)27 b(H)h(w)1161 3122 y Ft(\013)1209 3110 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1417 2997 y Fn(Z)1501 3018 y Fq(1)1464 3186 y Fu(1)1550 3110 y Fy(dx)1640 3122 y Fu(0)1699 3110 y Fy(x)1756 3035 y Fk(\015)s Fc(\000)p Fo(1)p 1756 3049 108 3 v 1796 3082 a(2)1746 3132 y Fu(0)1899 3110 y Fs(j)p Fy(k)s Fs(j)p Fy(g)2031 3122 y Fq(\000)2086 3110 y Fv(\()p Fy(k)s(=)p Fv(2)p Fy(;)g(x)2332 3122 y Fu(0)2369 3110 y Fy(;)g(x)p Fv(\))836 3337 y(+)k Fy(C)27 b(H)h(w)1161 3349 y Ft(\013)p Fq(\000)p Fu(1)1294 3337 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1502 3223 y Fn(Z)1586 3244 y Fq(1)1549 3412 y Fu(1)1635 3337 y Fy(dx)1725 3349 y Fu(0)1784 3337 y Fy(x)1831 3293 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1831 3359 y(0)2009 3337 y Fz(min)p Fv(\()p Fy(x)19 b Fs(\000)f Fv(1)p Fy(;)c(x)2448 3349 y Fu(0)2504 3337 y Fs(\000)k Fv(1\))p Fy(e)2700 3302 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(j)p Fz(max)p Fu(\()p Ft(x)p Fq(\000)p Fu(1)p Ft(;x)3248 3310 y Fo(0)3279 3302 y Fq(\000)p Fu(1\))776 3534 y Fs(\024)23 b Fy(C)k(H)h(x)1104 3473 y Fk(\015)s Fc(\000)p Fo(1)p 1105 3487 V 1144 3520 a(2)1247 3534 y Fy(w)1306 3546 y Ft(\013)1354 3534 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(C)27 b(H)h(x)1880 3500 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2058 3534 y Fy(w)2117 3546 y Ft(\013)p Fq(\000)p Fu(1)2250 3534 y Fv(\()p Fy(x;)14 b(k)s Fv(\))k(\026)-60 b Fy(w)2503 3546 y Fu(2)2542 3534 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fy(:)-9 3775 y Fd(D)n(.3)102 b(Bound)24 b(on)32 b Fa(^)-56 b Fb(q)731 3790 y Ff(2)796 3775 y Fd(and)25 b Fb(D)10 b Fa(^)-56 b Fb(q)1113 3790 y Ff(2)-9 3932 y Fz(Using)22 b(\(D.2\),)h(\(D.14\))g(and)f (\(D.15\),)i(we)d(sho)n(w)i(that)d(there)h(e)o(xists)g(a)g(constant)g Fy(C)28 b Fz(such)22 b(that)f(for)h(all)e Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)3325 3944 y Fu(+)3380 3932 y Fz(,)833 4164 y Fs(j)6 b Fv(^)-48 b Fy(q)893 4176 y Fu(2)930 4164 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b(H)1456 4047 y Fn(\022)1517 4051 y(Z)1600 4072 y Fq(1)1563 4240 y Fu(1)1649 4164 y Fy(dx)1739 4176 y Fu(0)1798 4164 y Fy(x)1845 4121 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1845 4186 y(0)2023 4164 y Fy(e)2062 4130 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)2405 4138 y Fo(0)2436 4130 y Fq(j)2460 4047 y Fn(\023)2535 4164 y Fs(j)p Fy(k)s Fs(j)p Fy(w)2686 4176 y Ft(\013)2733 4164 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1171 4354 y Fs(\024)22 b Fy(C)28 b(H)g(x)1489 4320 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1667 4354 y Fv(\()17 b(\026)-59 b Fy(w)1758 4366 y Fu(2)1796 4354 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(w)2151 4366 y Fu(1)2189 4354 y Fv(\()p Fy(x;)c(k)s Fv(\)\))p Fs(j)p Fy(k)s Fs(j)22 b Fy(w)2588 4366 y Ft(\013)2636 4354 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1171 4479 y Fs(\024)22 b Fy(C)28 b(H)g(x)1489 4444 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1646 4479 y Fs(j)p Fy(k)s Fs(j)p Fv(\()p Fy(w)1829 4491 y Ft(\013)p Fu(+1)1979 4479 y Fv(+)18 b Fy(w)2121 4491 y Ft(\013)2186 4479 y Fv(\026)-59 b Fy(w)2228 4491 y Fu(2)2266 4479 y Fv(\)\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)-9 4661 y Fz(Similarly)-6 b(,)38 b(using)e(\(D.3\),)41 b(\(D.4\),)f(\(D.14\))e(and)d(\(D.15\),)41 b(we)36 b(sho)n(w)g(that)f(there)g (e)o(xists)f(a)h(constant)g Fy(C)41 b Fz(such)36 b(that)f(for)h(all)-9 4761 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(2)h Fv(\012)347 4773 y Fu(+)402 4761 y Fz(,)800 4993 y Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)931 5005 y Fu(2)969 4993 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b(H)1495 4876 y Fn(\022)1556 4880 y(Z)1639 4901 y Fq(1)1602 5069 y Fu(1)1688 4993 y Fy(dx)1778 5005 y Fu(0)1837 4993 y Fy(x)1884 4950 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1884 5015 y(0)2062 4993 y Fy(e)2101 4959 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)2444 4967 y Fo(0)2475 4959 y Fq(j)2499 4876 y Fn(\023)2574 4993 y Fy(w)2633 5005 y Ft(\013)p Fq(\000)p Fu(1)2765 4993 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1210 5183 y Fs(\024)22 b Fy(C)28 b(H)g(x)1528 5149 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1706 5183 y Fv(\()p Fy(w)1797 5195 y Ft(\013)1863 5183 y Fv(+)18 b Fy(w)2005 5195 y Ft(\013)p Fq(\000)p Fu(1)2155 5183 y Fv(\026)-59 b Fy(w)2197 5195 y Fu(2)2235 5183 y Fv(\)\()p Fy(x;)14 b(k)s Fv(\))24 b Fz(,)-9 5366 y(as)d(claimed.)1838 5909 y(34)p eop %%Page: 35 35 bop -9 145 a Fd(D)n(.4)102 b(Bound)24 b(on)32 b Fa(^)-56 b Fb(q)731 160 y Ff(1)p Fh(;p)852 145 y Fd(and)24 b Fb(D)10 b Fa(^)-56 b Fb(q)1168 160 y Ff(1)p Fh(;p)-9 302 y Fz(Using)22 b(\(D.3\),)h(\(D.14\))g(and)f(\(D.18\),)i(we)d(sho)n(w)i(that)d(there)h(e)o (xists)g(a)g(constant)g Fy(C)28 b Fz(such)22 b(that)f(for)h(all)e Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)3325 314 y Fu(+)3380 302 y Fz(,)331 525 y Fs(j)6 b Fv(^)-48 b Fy(q)391 537 y Fu(1)p Ft(;p)483 525 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b(w)957 537 y Ft(\013)p Fq(\000)p Fu(1)1089 525 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1297 408 y Fn(\022)1359 412 y(Z)1442 432 y Ft(x)1405 601 y Fu(1)1462 525 y Fy(dx)1552 537 y Fu(0)1611 525 y Fy(x)1658 482 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1658 547 y(0)1836 525 y Fy(g)1876 537 y Fq(\000)1932 525 y Fv(\()p Fy(k)s(=)p Fv(2)p Fy(;)g(x)2178 537 y Fu(0)2215 525 y Fy(;)g(x)p Fv(\))19 b(+)2433 412 y Fn(Z)2516 432 y Fq(1)2479 601 y Ft(x)2565 525 y Fy(dx)2655 537 y Fu(0)2714 525 y Fy(x)2761 482 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2761 547 y(0)2917 525 y Fy(e)2956 491 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)3299 499 y Fo(0)3331 491 y Fq(j)3354 408 y Fn(\023)723 715 y Fs(\024)k Fy(C)28 b(w)957 727 y Ft(\013)p Fq(\000)p Fu(1)1089 715 y Fv(\()p Fy(x;)14 b(k)s Fv(\))22 b Fy(x)1352 681 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1523 715 y Fv(\()p Fs(j)p Fy(k)s Fs(j)17 b Fv(\026)-59 b Fy(w)1706 727 y Fu(2)1744 715 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(w)2099 727 y Fu(1)2137 715 y Fv(\()p Fy(x;)c(k)s Fv(\)\))24 b Fs(\024)e Fy(C)28 b(x)2608 681 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2786 715 y Fy(w)2845 727 y Ft(\013)2893 715 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fy(:)-9 888 y Fz(T)-7 b(o)31 b(bound)i Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)483 900 y Fu(1)p Ft(;p)574 888 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)p Fz(,)34 b(we)d(split)f(the)h(inte)o(gration)f(in)h(the)g(de\256nition)g(of)38 b Fv(^)-48 b Fy(q)2412 900 y Fu(1)p Ft(;p)2534 888 y Fz(in)30 b(tw)o(o)i(parts,)h Fv([1)p Fy(;)14 b(x)p Fv(])31 b Fz(and)g Fv([)p Fy(x;)14 b Fs(1)p Fv(\))p Fz(.)60 b(On)-9 988 y Fv([1)p Fy(;)14 b(x)p Fv(])p Fz(,)21 b(the)h(inte)o(grand)g(is)f(the)g(same)g(as)h (the)f(one)h(of)28 b Fv(^)-48 b Fy(q)1562 1000 y Fu(1)1600 988 y Fz(,)21 b(so)h(that)f(we)h(can)f(repeat)g(the)h(ar)n(guments)h(of)f (Subsection)g(D.2.)33 b(All)21 b(we)-9 1088 y(ha)n(v)o(e)j(to)g(do)h(is)f(to) g(bound)i(the)e(inte)o(gral)g(o)o(v)o(er)g Fv([)p Fy(x;)14 b Fs(1)p Fv(\))p Fz(.)40 b(F)o(or)25 b(all)e Fy(\027)32 b Fs(2)27 b Fv(\(1)p Fy(=)p Fv(2)p Fy(;)14 b Fv(1\))p Fz(,)23 b(there)h(e)o(xists)g(a)f (constant)i Fy(C)30 b Fz(such)25 b(that)e(for)i(all)-9 1187 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(2)h Fv(\012)347 1199 y Fu(+)402 1187 y Fz(,)668 1379 y Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)799 1391 y Fu(1)p Ft(;p)890 1379 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)k(H)h(x)1459 1318 y Fk(\015)s Fc(\000)p Fo(1)p 1460 1332 108 3 v 1499 1365 a(2)1602 1379 y Fy(w)1661 1391 y Ft(\013)1709 1379 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(C)28 b(H)g(x)2236 1345 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2413 1379 y Fy(w)2472 1391 y Ft(\013)p Fq(\000)p Fu(1)2605 1379 y Fv(\()p Fy(x;)14 b(k)s Fv(\))k(\026)-60 b Fy(w)2858 1391 y Fu(2)2897 1379 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1191 1554 y(+)k Fy(C)27 b(H)1450 1441 y Fn(Z)1533 1461 y Fq(1)1496 1629 y Ft(x)1582 1554 y Fy(dx)1672 1566 y Fu(0)1731 1554 y Fs(j)p Fy(@)1798 1566 y Ft(k)1839 1554 y Fy(g)1879 1566 y Fu(+)1934 1554 y Fv(\()p Fy(k)s(;)14 b(x)2096 1566 y Fu(0)2133 1554 y Fy(;)g(x)p Fv(\))p Fs(j)22 b Fy(w)2353 1566 y Ft(\013)p Fq(\000)p Fu(1)2486 1554 y Fv(\()p Fy(x)2565 1566 y Fu(0)2603 1554 y Fy(;)14 b(k)s Fv(\))21 b Fy(x)2786 1510 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2786 1576 y(0)1191 1780 y Fv(+)d Fy(C)27 b(H)1450 1667 y Fn(Z)1533 1687 y Fq(1)1496 1856 y Ft(x)1582 1780 y Fy(dx)1672 1792 y Fu(0)1731 1780 y Fy(x)1788 1704 y Fk(\015)s Fc(\000)p Fo(1)p 1788 1718 V 1828 1751 a(2)1778 1802 y Fu(0)1931 1780 y Fs(j)p Fy(k)s Fs(j)21 b Fy(e)2083 1746 y Fq(\000j)p Ft(k)q Fq(jj)p Ft(x)p Fq(\000)p Ft(x)2359 1754 y Fo(0)2390 1746 y Fq(j)2435 1780 y Fy(w)2494 1792 y Ft(\013)2542 1780 y Fv(\()p Fy(x)2621 1792 y Fu(0)2659 1780 y Fy(;)14 b(k)s Fv(\))1131 1978 y Fs(\024)23 b Fy(C)k(H)h(x)1459 1916 y Fk(\015)s Fc(\000)p Fo(1)p 1460 1930 V 1499 1963 a(2)1602 1978 y Fy(w)1661 1990 y Ft(\013)1709 1978 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(C)28 b(H)g(x)2236 1943 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2413 1978 y Fy(w)2472 1990 y Ft(\013)p Fq(\000)p Fu(1)2605 1978 y Fv(\()p Fy(x;)14 b(k)s Fv(\))k(\026)-60 b Fy(w)2858 1990 y Fu(2)2897 1978 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1191 2149 y(+)k Fy(C)27 b(H)h(w)1516 2161 y Ft(\013)p Fq(\000)p Fu(1)1649 2149 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1857 2036 y Fn(Z)1941 2057 y Fq(1)1904 2225 y Ft(x)1990 2149 y Fy(dx)2080 2161 y Fu(0)2139 2149 y Fy(e)2178 2115 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)2521 2123 y Fo(0)2552 2115 y Fq(j)2576 2149 y Fv(\()p Fy(x)2655 2161 y Fu(0)2711 2149 y Fs(\000)k Fv(1\))j Fy(x)2936 2106 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2936 2171 y(0)1131 2347 y Fs(\024)i Fy(C)k(H)h(x)1459 2286 y Fk(\015)s Fc(\000)p Fo(1)p 1460 2300 V 1499 2332 a(2)1602 2347 y Fy(w)1661 2359 y Ft(\013)1709 2347 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(C)28 b(H)g(x)2236 2312 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2413 2347 y Fy(w)2472 2359 y Ft(\013)p Fq(\000)p Fu(1)2605 2347 y Fv(\()p Fy(x;)14 b(k)s Fv(\))k(\026)-60 b Fy(w)2858 2359 y Fu(2)2897 2347 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1191 2471 y(+)k Fy(C)27 b(H)h(x)1504 2437 y Fu(1)p Ft(=)p Fu(2)p Fq(\000)p Ft(\027)1720 2471 y Fs(j)p Fy(k)s Fs(j)1812 2437 y Fq(\000)p Ft(\027)1926 2471 y Fy(w)1985 2483 y Ft(\013)p Fq(\000)p Fu(1)2118 2471 y Fv(\()p Fy(x;)14 b(k)s Fv(\))-9 2645 y Fz(as)21 b(claimed.)-9 2884 y Fd(D)n(.5)102 b(Bound)24 b(on)32 b Fa(^)-56 b Fb(q)731 2899 y Ff(2)p Fh(;p)852 2884 y Fd(and)24 b Fb(D)10 b Fa(^)-56 b Fb(q)1168 2899 y Ff(2)p Fh(;p)-9 3041 y Fz(Using)22 b(\(D.2\),)h(\(D.14\))g (and)f(\(D.15\),)i(we)d(sho)n(w)i(that)d(there)h(e)o(xists)g(a)g(constant)g Fy(C)28 b Fz(such)22 b(that)f(for)h(all)e Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)3325 3053 y Fu(+)3380 3041 y Fz(,)887 3261 y Fs(j)6 b Fv(^)-48 b Fy(q)947 3273 y Fu(2)p Ft(;p)1038 3261 y Fv(\()p Fy(k)s(;)14 b(t)p Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b(H)g Fs(j)p Fy(k)s Fs(j)1639 3148 y Fn(Z)1721 3169 y Fq(1)1684 3337 y Fu(1)1770 3261 y Fy(dx)1860 3273 y Fu(0)1919 3261 y Fy(x)1966 3218 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1966 3283 y(0)2165 3261 y Fy(e)2204 3227 y Fq(\000j)p Ft(k)q Fq(jj)p Ft(x)p Fq(\000)p Ft(x)2480 3235 y Fo(0)2511 3227 y Fq(j)2535 3261 y Fy(w)2594 3273 y Ft(\013)2642 3261 y Fv(\()p Fy(x)2721 3273 y Fu(0)2759 3261 y Fy(;)14 b(k)s Fv(\))1261 3448 y Fs(\024)23 b Fy(w)1408 3460 y Ft(\013)1456 3448 y Fv(\()p Fy(x;)14 b(k)s Fv(\))22 b Fs(j)p Fy(k)s Fs(j)f Fy(x)1832 3413 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2002 3448 y Fv(\()q Fy(w)2094 3460 y Fu(1)2131 3448 y Fv(\()p Fy(x;)14 b(k)s Fv(\))20 b(+)35 b(\026)-59 b Fy(w)2487 3460 y Fu(1)2524 3448 y Fv(\()p Fy(x;)14 b(k)s Fv(\)\))1261 3572 y Fs(\024)23 b Fy(C)28 b(x)1483 3538 y Fq(\000)p Fu(1)1593 3572 y Fy(w)1652 3584 y Ft(\013)1700 3572 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)-9 3746 y Fz(Similarly)-6 b(,)38 b(using)e(\(D.3\),)41 b(\(D.4\),)f(\(D.14\))e(and)d(\(D.15\),)41 b(we)36 b(sho)n(w)g(that)f(there)g (e)o(xists)f(a)h(constant)g Fy(C)41 b Fz(such)36 b(that)f(for)h(all)-9 3845 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(2)h Fv(\012)347 3857 y Fu(+)402 3845 y Fz(,)816 4069 y Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)947 4081 y Fu(2)p Ft(;p)1038 4069 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(C)k(H)1564 3952 y Fn(\022)1625 3956 y(Z)1708 3976 y Fq(1)1671 4144 y Fu(1)1757 4069 y Fy(dx)1847 4081 y Fu(0)1906 4069 y Fy(x)1953 4026 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)1953 4091 y(0)2131 4069 y Fy(e)2170 4035 y Fq(\000j)p Ft(k)q(=)p Fu(2)p Fq(jj)p Ft(x)p Fq(\000)p Ft(x)2513 4043 y Fo(0)2544 4035 y Fq(j)2568 3952 y Fn(\023)2643 4069 y Fy(w)2702 4081 y Ft(\013)2750 4069 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1279 4259 y Fs(\024)23 b Fy(C)k(H)h(x)1597 4225 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1775 4259 y Fv(\()p Fy(w)1866 4271 y Ft(\013)1933 4259 y Fv(+)18 b Fy(w)2075 4271 y Ft(\013)p Fq(\000)p Fu(1)2225 4259 y Fv(\026)-59 b Fy(w)2267 4271 y Fu(2)2304 4259 y Fv(\)\()p Fy(x;)14 b(k)s Fv(\))-9 4432 y Fz(as)21 b(claimed.)-9 4714 y FB(A)m(ppendix)31 b(E:)f(Pr)n(oof)h(of)f(Lemma)h(3.10)-9 4902 y Fz(The)22 b(proof)j(of)f(Lemma)e(3.10)i(relies)e(on)i(a)e(certain)g (number)i(of)g(preliminary)f(estimates.)34 b(The)o(y)23 b(are)f(gi)n(v)o(en)h (in)g(the)g(follo)n(wing)-9 5001 y(section.)30 b(Throughout)25 b(this)c(section,)g(we)g(will)g(omit)g(the)g(ar)n(gument)h(`)p Fy(k)s Fz(')g(in)f Fv(\003)2318 5013 y Fq(\006)2395 5001 y Fz(to)h(simplify)f(notation.)-9 5240 y Fd(E.1)102 b(Pr)n(eliminary)27 b(Bounds)-9 5397 y Fz(W)-7 b(e)21 b(be)o(gin)h(by)g(de\256ning)h(the)e(follo) n(wing)h(function)1489 5623 y Fy(g)1529 5635 y Ft(\015)1571 5623 y Fv(\()p Fy(k)s Fv(\))i(=)1792 5506 y Fn(\022)1935 5566 y Fs(j)p Fy(k)s Fs(j)p 1863 5603 235 4 v 1863 5680 a Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2108 5506 y Fn(\023)2169 5523 y Ft(\015)2249 5623 y Fy(:)1838 5909 y Fz(35)p eop %%Page: 36 36 bop -9 145 a Fz(The)21 b(follo)n(wing)h(lemma)f(summarizes)g(some)h(of)g(the) f(properties)h(of)g(the)f(functions)h Fy(g)2546 157 y Ft(\015)2589 145 y Fz(,)f Fv(\003)2689 157 y Fq(\006)2766 145 y Fz(and)1409 356 y Fy(w)1468 368 y Ft(\013)1516 356 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)2043 300 y(1)p 1832 337 464 4 v 1832 414 a(\(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)2099 354 y(p)p 2168 354 48 4 v 60 x Fy(x)p Fv(\))2247 390 y Ft(\013)2328 356 y Fy(:)-9 571 y FA(Lemma)i(E.1)42 b Fm(Let)23 b Fy(p)i Fs(\025)g Fv(1)p Fm(,)e Fy(\013)j Fs(\025)f Fv(1)p Fm(,)e Fv(0)i Fs(\024)h Fy(\015)k Fs(\024)25 b Fv(1)d Fm(and)j Fy(\014)30 b Fs(\025)25 b Fv(0)p Fm(.)35 b(Ther)m(e)23 b(e)n(xists)g(a)g(constant)h Fy(C)29 b Fm(suc)o(h)24 b(that)f(for)g(all)g Fy(x)j Fs(\025)f Fy(x)3554 583 y Fu(0)3617 571 y Fs(\025)g Fv(1)p Fm(,)-9 670 y Fy(k)g Fs(2)f Fr(R)p Fm(,)d(we)g(have)1736 840 y Fs(j)p Fy(k)s Fs(j)1828 806 y Ft(\015)1894 840 y Fs(\024)h(j)p Fv(\003)2062 852 y Fq(\000)2118 840 y Fs(j)2141 806 y Ft(\015)t(=)p Fu(2)2251 840 y Fv(\(1)c(+)g Fs(j)p Fv(\003)2507 852 y Fq(\000)2563 840 y Fs(j)p Fv(\))2618 806 y Ft(\015)t(=)p Fu(2)2751 840 y Fm(,)826 b Fz(\(E.1\))1736 965 y Fs(j)p Fy(k)s Fs(j)1828 930 y Ft(\015)1894 965 y Fs(\024)22 b Fy(g)2021 980 y Ft(\015)t(=)p Fu(2)2131 965 y Fv(\(\003)2221 977 y Fq(\000)2277 965 y Fv(\)\(1)c(+)g Fs(j)p Fv(\003)2565 977 y Fq(\000)2621 965 y Fs(j)p Fv(\))2676 930 y Ft(\015)2742 965 y Fm(,)835 b Fz(\(E.2\))988 1137 y Fy(x)1035 1103 y Ft(\015)t(=)p Fu(2)1146 1137 y Fs(j)p Fy(k)s Fs(j)1238 1103 y Ft(\015)1290 1081 y Fy(x)19 b Fs(\000)f Fv(1)p 1290 1118 191 4 v 1362 1194 a Fy(x)1490 1137 y(e)1529 1103 y Fu(\003)1574 1111 y Fc(\000)1624 1103 y Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=p)1894 1137 y Fs(\024)k Fy(C)30 b Fm(,)1507 b Fz(\(E.3\))1136 1305 y Fy(x)1183 1271 y Ft(\014)s(=)p Fu(2)1296 1305 y Fy(g)1336 1317 y Ft(\014)1380 1305 y Fv(\()p Fy(k)s Fv(\))p Fy(e)1529 1271 y Fu(\003)1574 1279 y Fc(\000)1624 1271 y Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=p)1894 1305 y Fs(\024)22 b Fy(C)30 b Fm(,)1507 b Fz(\(E.4\))1105 1424 y Fy(w)1164 1436 y Ft(\013)1212 1424 y Fv(\()p Fy(x)1291 1436 y Fu(0)1329 1424 y Fy(;)14 b(k)s Fv(\))p 1105 1461 339 4 v 1123 1537 a Fy(w)1182 1549 y Ft(\013)1230 1537 y Fv(\()p Fy(x;)g(k)s Fv(\))1454 1480 y Fy(e)1493 1446 y Fu(\003)1538 1454 y Fc(\000)1587 1446 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)1741 1454 y Fo(0)1772 1446 y Fu(\))p Ft(=p)1894 1480 y Fs(\024)22 b Fy(C)30 b(:)1505 b Fz(\(E.5\))-9 1712 y Fm(Pr)l(oof)o(.)31 b Fz(Inequalities)21 b(\(E.1\))h(and)g(\(E.2\))g(are)f(elementary)-6 b(.)31 b(F)o(or)22 b(inequality)f(\(E.3\),)i(we)e(use)h(that)2856 1679 y Ft(x)p Fq(\000)p Fu(1)p 2856 1693 123 4 v 2898 1740 a Ft(x)3011 1712 y Fs(\024)3099 1644 y Fn(\000)3147 1679 y Ft(x)p Fq(\000)p Fu(1)p 3147 1693 V 3189 1740 a Ft(x)3279 1644 y Fn(\001)3318 1662 y Ft(\015)3381 1712 y Fz(to)g(get)742 1929 y Fs(j)p Fy(k)s Fs(j)834 1894 y Ft(\015)887 1873 y Fy(x)c Fs(\000)g Fv(1)p 887 1910 191 4 v 958 1986 a Fy(x)1087 1929 y(e)1126 1894 y Fu(\003)1171 1902 y Fc(\000)1220 1894 y Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=p)1490 1929 y Fs(\024)1612 1873 y Fv(1)p 1588 1910 91 4 v 1588 1986 a Fy(x)1635 1962 y Ft(\015)1702 1929 y Fz(sup)1702 2001 y Ft(\030)r Fq(\025)p Fu(0)1833 1929 y Fy(\030)1873 1894 y Ft(\015)t(=)p Fu(2)1983 1929 y Fv(\()p Fy(x)h Fs(\000)f Fv(1)g(+)g Fy(\030)t Fv(\))2379 1894 y Ft(\015)t(=)p Fu(2)2490 1929 y Fy(e)2529 1894 y Fq(\000)p Ft(\030)r(=p)2708 1929 y Fs(\024)2863 1873 y Fv(1)p 2805 1910 158 4 v 2805 1987 a Fy(x)2852 1963 y Ft(\015)t(=)p Fu(2)2996 1929 y Fy(:)-9 2162 y Fz(F)o(or)k(inequality)f(\(E.4\),)h(we)g(use)f(that)g Fy(g)1148 2174 y Ft(\014)1192 2162 y Fv(\()p Fy(k)s Fv(\))j Fs(\024)e Fy(g)1453 2177 y Ft(\014)s(=)p Fu(2)1565 2162 y Fv(\(\003)1655 2174 y Fq(\000)1711 2162 y Fv(\))h Fs(\024)g Fv(2)e Fz(to)g(get)741 2347 y Fy(x)788 2313 y Ft(\014)s(=)p Fu(2)901 2347 y Fy(g)941 2359 y Ft(\014)985 2347 y Fv(\()p Fy(k)s Fv(\))p Fy(e)1134 2313 y Fu(\003)1179 2321 y Fc(\000)1229 2313 y Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=p)1499 2347 y Fs(\024)h Fy(x)1633 2313 y Ft(\014)s(=)p Fu(2)1746 2347 y Fy(g)1786 2362 y Ft(\014)s(=)p Fu(2)1897 2347 y Fv(\(\003)1987 2359 y Fq(\000)2043 2347 y Fv(\))p Fy(e)2114 2313 y Fq(\000j)p Fu(\003)2231 2321 y Fc(\000)2280 2313 y Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=p)1499 2496 y Fs(\024)g Fv(\()q Fy(g)1659 2508 y Fu(1)1695 2496 y Fv(\(\003)1785 2508 y Fq(\000)1841 2496 y Fv(\))d(+)f Fs(j)p Fv(\003)2056 2508 y Fq(\000)2112 2496 y Fs(j)p Fv(\()p Fy(x)h Fs(\000)f Fv(1\)\))2422 2454 y Ft(\014)s(=)p Fu(2)2548 2496 y Fy(e)2587 2461 y Fq(\000j)p Fu(\003)2704 2469 y Fc(\000)2753 2461 y Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=p)1499 2635 y Fs(\024)k Fz(sup)1587 2707 y Ft(\030)r Fq(\025)p Fu(0)1718 2635 y Fv(\(1)c(+)g Fy(\030)t Fv(\))1966 2593 y Ft(\014)s(=)p Fu(2)2091 2635 y Fy(e)2130 2601 y Fq(\000)p Ft(\030)r(=p)2310 2635 y Fs(\024)k Fy(C)30 b(:)-9 2889 y Fz(Finally)-6 b(,)30 b(for)g(inequality)e(\(E.5\),)k(we)c(consider)i(\256rst)e Fy(x)1648 2901 y Fu(0)1719 2889 y Fs(\025)1826 2857 y Ft(x)p Fu(+1)p 1826 2871 122 4 v 1871 2918 a(2)1958 2889 y Fz(,)i(then)2194 2849 y Ft(w)2242 2857 y Fk(\013)2284 2849 y Fu(\()p Ft(x)2348 2857 y Fo(0)2380 2849 y Ft(;k)q Fu(\))p 2194 2870 269 4 v 2210 2918 a Ft(w)2258 2926 y Fk(\013)2300 2918 y Fu(\()p Ft(x;k)q Fu(\))2505 2889 y Fs(\024)2613 2849 y Ft(w)2661 2857 y Fk(\013)2703 2849 y Fu(\()p Ft(k)q(;)p Fu(\()p Ft(x)p Fu(+1\))p Ft(=)p Fu(2\))p 2613 2870 440 4 v 2714 2918 a Ft(w)2762 2926 y Fk(\013)2804 2918 y Fu(\()p Ft(x;k)q Fu(\))3095 2889 y Fs(\024)i Fy(C)6 b Fz(.)53 b(On)29 b(the)f(other)-9 3011 y(hand,)22 b(when)g Fy(x)450 3023 y Fu(0)511 3011 y Fs(\024)609 2978 y Ft(x)p Fu(+1)p 609 2992 122 4 v 653 3040 a(2)741 3011 y Fz(,)f(then)404 3180 y Fy(w)463 3192 y Ft(\013)511 3180 y Fv(\()p Fy(x)590 3192 y Fu(0)627 3180 y Fy(;)14 b(k)s Fv(\))p 404 3217 339 4 v 422 3293 a Fy(w)481 3305 y Ft(\013)529 3293 y Fv(\()p Fy(x;)g(k)s Fv(\))752 3236 y Fy(e)791 3202 y Fu(\003)836 3210 y Fc(\000)886 3202 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)1040 3210 y Fo(0)1071 3202 y Fu(\))p Ft(=p)1193 3236 y Fs(\024)1293 3180 y Fy(w)1352 3192 y Ft(\013)1400 3180 y Fv(\(1)p Fy(;)g(k)s Fv(\))p 1290 3217 302 4 v 1290 3293 a Fy(w)1349 3305 y Ft(\013)1397 3293 y Fv(\()p Fy(x;)g(k)s Fv(\))1602 3236 y Fy(e)1641 3202 y Fu(\003)1686 3210 y Fc(\000)1735 3202 y Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(\(2)p Ft(p)p Fu(\))2090 3236 y Fs(\024)22 b Fv(1)c(+)g Fy(x)2367 3202 y Ft(\013=)p Fu(2)2482 3236 y Fy(g)2522 3248 y Ft(\013)2569 3236 y Fv(\()p Fy(k)s Fv(\))p Fy(e)2718 3202 y Fu(\003)2763 3210 y Fc(\000)2813 3202 y Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(\(2)p Ft(p)p Fu(\))3168 3236 y Fs(\024)k Fy(C)30 b(:)-9 3450 y Fz(This)21 b(achie)n(v)o(es)f(the)h(proof.)p 855 3450 43 43 v -9 3549 a(This)g(lemma)f(will)h(help)g(us)h(estimate)e(the)h (deri)n(v)n(ati)n(v)o(es.)-9 3705 y FA(Lemma)f(E.2)42 b Fm(Ther)m(e)21 b(e)n(xists)g(a)g(constant)i Fy(C)k Fm(suc)o(h)22 b(that)f(for)h(all)f Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)2253 3717 y Fu(+)2308 3705 y Fm(,)f(we)e(have)240 3804 y Fn(\014)240 3854 y(\014)240 3904 y(\014)240 3954 y(\014)268 3925 y Fy(D)352 3812 y Fn(Z)435 3832 y Ft(x)399 4000 y Fu(1)456 3925 y Fy(dx)546 3937 y Fu(0)605 3925 y Fy(e)644 3890 y Fu(\003)689 3898 y Fc(\000)738 3890 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)892 3898 y Fo(0)923 3890 y Fu(\))960 3925 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)1073 3937 y Fu(0)1111 3925 y Fy(;)14 b(k)s Fv(\))1226 3804 y Fn(\014)1226 3854 y(\014)1226 3904 y(\014)1226 3954 y(\014)1276 3925 y Fs(\024)23 b Fy(C)1460 3869 y Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p 1460 3906 235 4 v 1532 3982 a(j)p Fy(k)s Fs(j)1719 3812 y Fn(Z)1802 3832 y Ft(x)1765 4000 y Fu(1)1822 3925 y Fy(dx)1912 3937 y Fu(0)1972 3925 y Fy(e)2011 3890 y Fu(\003)2056 3898 y Fc(\000)2105 3890 y Ft(=)p Fu(2\()p Ft(x)p Fq(\000)p Ft(x)2326 3898 y Fo(0)2357 3890 y Fu(\))2388 3833 y Fn(\020)2447 3869 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2589 3881 y Fu(0)2627 3869 y Fy(;)14 b(k)s Fv(\))p Fs(j)p 2447 3906 319 4 v 2489 3982 a Fv(1)k(+)g Fs(j)p Fy(k)s Fs(j)2794 3925 y Fv(+)g Fy(g)2917 3937 y Fu(1)2954 3925 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)3277 3937 y Fu(0)3315 3925 y Fy(;)14 b(k)s Fv(\))p Fs(j)3453 3833 y Fn(\021)3526 3925 y Fm(,)214 4037 y Fn(\014)214 4087 y(\014)214 4136 y(\014)214 4186 y(\014)241 4157 y Fy(D)326 4044 y Fn(Z)409 4065 y Fq(1)372 4233 y Ft(x)458 4157 y Fy(dx)548 4169 y Fu(0)607 4157 y Fy(e)646 4123 y Fu(\003)691 4131 y Fo(+)738 4123 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)892 4131 y Fo(0)923 4123 y Fu(\))960 4157 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)1073 4169 y Fu(0)1111 4157 y Fy(;)14 b(k)s Fv(\))1226 4037 y Fn(\014)1226 4087 y(\014)1226 4136 y(\014)1226 4186 y(\014)1276 4157 y Fs(\024)23 b Fy(C)1464 4044 y Fn(Z)1547 4065 y Fq(1)1510 4233 y Ft(x)1596 4157 y Fy(dx)1686 4169 y Fu(0)1745 4157 y Fy(e)1784 4123 y Fu(\003)1829 4131 y Fo(+)1876 4123 y Ft(=)p Fu(2\()p Ft(x)p Fq(\000)p Ft(x)2097 4131 y Fo(0)2129 4123 y Fu(\))2159 4065 y Fn(\020)2208 4157 y Fs(j)6 b Fv(^)-48 b Fy(q)t Fv(\()p Fy(x)2351 4169 y Fu(0)2389 4157 y Fy(;)14 b(k)s Fv(\))p Fs(j)k Fv(+)g Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2841 4169 y Fu(0)2879 4157 y Fy(;)14 b(k)s Fv(\))p Fs(j)3017 4065 y Fn(\021)3090 4157 y Fy(:)-9 4372 y Fm(Pr)l(oof)o(.)31 b Fz(W)-7 b(e)21 b(ha)n(v)o(e)183 4468 y Fn(\014)183 4518 y(\014)183 4567 y(\014)183 4617 y(\014)211 4588 y Fy(D)296 4475 y Fn(Z)379 4496 y Ft(x)342 4664 y Fu(1)399 4588 y Fy(dx)489 4600 y Fu(0)548 4588 y Fy(e)587 4554 y Fu(\003)632 4562 y Fc(\000)681 4554 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)835 4562 y Fo(0)867 4554 y Fu(\))903 4588 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)1016 4600 y Fu(0)1054 4588 y Fy(;)14 b(k)s Fv(\))1169 4468 y Fn(\014)1169 4518 y(\014)1169 4567 y(\014)1169 4617 y(\014)1220 4588 y Fs(\024)1307 4468 y Fn(\014)1307 4518 y(\014)1307 4567 y(\014)1307 4617 y(\014)1335 4475 y(Z)1418 4496 y Ft(x)1381 4664 y Fu(1)1438 4588 y Fy(dx)1528 4600 y Fu(0)1587 4588 y Fy(e)1626 4554 y Fu(\003)1671 4562 y Fc(\000)1721 4554 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)1875 4562 y Fo(0)1906 4554 y Fu(\))1936 4496 y Fn(\020)1986 4588 y Fv(\003)2044 4600 y Fq(\000)2100 4588 y Fv(\()p Fy(x)19 b Fs(\000)f Fy(x)2328 4600 y Fu(0)2365 4588 y Fv(\))2443 4532 y(2\003)2543 4544 y Fu(+)p 2408 4569 226 4 v 2408 4645 a Fy(k)s Fv(\001\()p Fy(k)s Fv(\))2649 4588 y(^)-48 b Fy(q)s Fv(\()p Fy(x)2762 4600 y Fu(0)2800 4588 y Fy(;)14 b(k)s Fv(\))19 b(+)f Fy(D)8 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)3207 4600 y Fu(0)3245 4588 y Fy(;)14 b(k)s Fv(\))3360 4496 y Fn(\021)3409 4468 y(\014)3409 4518 y(\014)3409 4567 y(\014)3409 4617 y(\014)1220 4846 y Fs(\024)1343 4789 y Fv(1)p 1317 4826 92 4 v 1317 4903 a Fs(j)p Fy(k)s Fs(j)1433 4733 y Fn(Z)1516 4753 y Ft(x)1479 4921 y Fu(1)1536 4846 y Fy(dx)1626 4858 y Fu(0)1685 4846 y Fy(e)1724 4811 y Fu(\003)1769 4819 y Fc(\000)1819 4811 y Ft(=)p Fu(2\()p Ft(x)p Fq(\000)p Ft(x)2040 4819 y Fo(0)2071 4811 y Fu(\))2115 4704 y Fn( )2181 4753 y(\020)2231 4846 y Fz(sup)2231 4918 y Ft(\030)r Fq(\025)p Fu(0)2362 4846 y Fy(\030)t(e)2441 4811 y Fq(\000)p Ft(\030)2529 4753 y Fn(\021)2579 4846 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2721 4858 y Fu(0)2759 4846 y Fy(;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fs(j)p Fy(k)s Fs(jj)p Fy(D)8 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)3304 4858 y Fu(0)3342 4846 y Fy(;)14 b(k)s Fv(\))p Fs(j)3480 4704 y Fn(!)3582 4846 y Fz(,)157 4983 y Fn(\014)157 5032 y(\014)157 5082 y(\014)157 5132 y(\014)185 5103 y Fy(D)270 4990 y Fn(Z)353 5010 y Fq(1)316 5179 y Ft(x)401 5103 y Fy(dx)491 5115 y Fu(0)550 5103 y Fy(e)589 5069 y Fu(\003)634 5077 y Fo(+)681 5069 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)835 5077 y Fo(0)867 5069 y Fu(\))903 5103 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)1016 5115 y Fu(0)1054 5103 y Fy(;)14 b(k)s Fv(\))1169 4983 y Fn(\014)1169 5032 y(\014)1169 5082 y(\014)1169 5132 y(\014)1220 5103 y Fs(\024)1307 4990 y Fn(Z)1390 5010 y Fq(1)1353 5179 y Ft(x)1439 5103 y Fy(dx)1529 5115 y Fu(0)1588 5103 y Fy(e)1627 5069 y Fu(\003)1672 5077 y Fo(+)1719 5069 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)1873 5077 y Fo(0)1905 5069 y Fu(\))1935 5011 y Fn(\020)1984 5103 y Fv(\003)2042 5115 y Fu(+)2097 5103 y Fv(\()p Fy(x)2176 5115 y Fu(0)2233 5103 y Fs(\000)k Fy(x)p Fv(\))p Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2537 5115 y Fu(0)2575 5103 y Fy(;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)3028 5115 y Fu(0)3066 5103 y Fy(;)14 b(k)s Fv(\))p Fs(j)3204 5011 y Fn(\021)1220 5333 y Fs(\024)1307 5220 y Fn(Z)1390 5240 y Fq(1)1353 5408 y Ft(x)1439 5333 y Fy(dx)1529 5345 y Fu(0)1588 5333 y Fy(e)1627 5298 y Fu(\003)1672 5306 y Fo(+)1719 5298 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)1873 5306 y Fo(0)1905 5298 y Fu(\))p Ft(=)p Fu(2)2002 5240 y Fn(\020)o(\020)2101 5333 y Fz(sup)2101 5405 y Ft(\030)r Fq(\025)p Fu(0)2233 5333 y Fy(\030)t(e)2312 5298 y Fq(\000)p Ft(\030)2400 5240 y Fn(\021)2450 5333 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2592 5345 y Fu(0)2630 5333 y Fy(;)14 b(k)s Fv(\))p Fs(j)k Fv(+)g Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)3082 5345 y Fu(0)3120 5333 y Fy(;)14 b(k)s Fv(\))p Fs(j)3258 5240 y Fn(\021)3331 5333 y Fy(:)-9 5560 y Fz(This)21 b(achie)n(v)o(es)f(the)h(proof.)p 855 5560 43 43 v -9 5660 a(The)g(follo)n(wing)h(functions)h(enter)e(the)g (de\256nition)h(of)g Fy(N)1653 5672 y Ft(i)1680 5660 y Fz(.)1838 5909 y(36)p eop %%Page: 37 37 bop -9 145 a FA(De\256nition)22 b(E.3)41 b Fm(F)-9 b(or)22 b(all)f Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)1139 157 y Fu(+)1195 145 y Fm(,)e(we)g(de\256ne)h(the)f(following)g(functions.) 1061 335 y Fv(^)1037 356 y Fy(N)1104 368 y Fu(1)p Ft(;)p Fu(1)1194 356 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1562 300 y Fy(ik)p 1510 337 180 4 v 1510 413 a Fv(\001\()p Fy(k)s Fv(\))1713 243 y Fn(Z)1796 263 y Ft(x)1759 431 y Fu(1)1816 356 y Fy(dx)1906 368 y Fu(0)1965 356 y Fy(e)2004 321 y Fu(\003)2049 329 y Fc(\000)2099 321 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)2253 329 y Fo(0)2284 321 y Fu(\))2320 356 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2433 368 y Fu(0)2471 356 y Fy(;)14 b(k)s Fv(\))23 b Fm(,)1061 563 y Fv(^)1037 584 y Fy(N)1104 596 y Fu(1)p Ft(;)p Fu(2)1194 584 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)1562 528 y Fy(ik)p 1510 565 V 1510 641 a Fv(\001\()p Fy(k)s Fv(\))1713 471 y Fn(Z)1796 492 y Fq(1)1759 660 y Ft(x)1845 584 y Fy(dx)1935 596 y Fu(0)1994 584 y Fy(e)2033 550 y Fu(\003)2078 558 y Fo(+)2125 550 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)2279 558 y Fo(0)2310 550 y Fu(\))2347 584 y Fv(^)-49 b Fy(q)s Fv(\()p Fy(x)2459 596 y Fu(0)2497 584 y Fy(;)14 b(k)s Fv(\))24 b Fm(,)1061 792 y Fv(^)1037 813 y Fy(N)1104 825 y Fu(2)p Ft(;)p Fu(1)1194 813 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fs(\000)1607 756 y Fv(\003)1665 768 y Fq(\000)p 1574 793 V 1574 869 a Fv(\001\()p Fy(k)s Fv(\))1778 700 y Fn(Z)1861 720 y Ft(x)1824 888 y Fu(1)1881 813 y Fy(dx)1971 825 y Fu(0)2030 813 y Fy(e)2069 778 y Fu(\003)2114 786 y Fc(\000)2163 778 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)2317 786 y Fo(0)2349 778 y Fu(\))2385 813 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2498 825 y Fu(0)2536 813 y Fy(;)14 b(k)s Fv(\))23 b Fm(,)1061 1020 y Fv(^)1037 1041 y Fy(N)1104 1053 y Fu(2)p Ft(;)p Fu(2)1194 1041 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fs(\000)1608 985 y Fv(\003)1666 997 y Fu(+)p 1574 1022 V 1574 1098 a Fv(\001\()p Fy(k)s Fv(\))1778 928 y Fn(Z)1861 948 y Fq(1)1824 1117 y Ft(x)1909 1041 y Fy(dx)1999 1053 y Fu(0)2058 1041 y Fy(e)2097 1007 y Fu(\003)2142 1015 y Fo(+)2189 1007 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)2343 1015 y Fo(0)2375 1007 y Fu(\))2411 1041 y Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2524 1053 y Fu(0)2562 1041 y Fy(;)14 b(k)s Fv(\))23 b Fy(:)-9 1270 y Fz(The)e(follo)n(wing)h(lemma)f(summarizes)g (properties)h(of)g(the)f(deri)n(v)n(ati)n(v)o(es)g(of)2254 1249 y Fv(^)2230 1270 y Fy(N)2297 1282 y Ft(i;j)2375 1270 y Fz(.)-9 1421 y FA(Pr)n(oposition)h(E.4)42 b Fm(Ther)m(e)22 b(e)n(xists)e(a)i(constant)g Fy(C)27 b Fm(suc)o(h)22 b(that)g(for)f(all)g Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)2393 1433 y Fu(+)2448 1421 y Fm(,)f(we)e(have)538 1537 y Fn(\014)538 1587 y(\014)538 1637 y(\014)566 1633 y Fy(D)661 1612 y Fv(^)637 1633 y Fy(N)704 1645 y Fu(1)p Ft(;)p Fu(1)794 1633 y Fv(\()p Fy(x;)14 b(k)s Fv(\))988 1537 y Fn(\014)988 1587 y(\014)988 1637 y(\014)1039 1633 y Fs(\024)23 b Fy(C)1227 1520 y Fn(Z)1310 1540 y Ft(x)1273 1708 y Fu(1)1330 1633 y Fy(dx)1420 1645 y Fu(0)1480 1633 y Fy(e)1519 1598 y Fu(\003)1564 1606 y Fc(\000)1613 1598 y Ft(=)p Fu(2\()p Ft(x)p Fq(\000)p Ft(x)1834 1606 y Fo(0)1865 1598 y Fu(\))1896 1540 y Fn(\020)1945 1633 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2087 1645 y Fu(0)2125 1633 y Fy(;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fy(g)2405 1645 y Fu(1)2442 1633 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2765 1645 y Fu(0)2803 1633 y Fy(;)14 b(k)s Fv(\))p Fs(j)2941 1540 y Fn(\021)3014 1633 y Fm(,)538 1764 y Fn(\014)538 1813 y(\014)538 1863 y(\014)566 1859 y Fy(D)661 1838 y Fv(^)637 1859 y Fy(N)704 1871 y Fu(2)p Ft(;)p Fu(1)794 1859 y Fv(\()p Fy(x;)g(k)s Fv(\))988 1764 y Fn(\014)988 1813 y(\014)988 1863 y(\014)1039 1859 y Fs(\024)23 b Fy(C)k(g)1253 1871 y Fu(1)1290 1859 y Fv(\()p Fy(k)s Fv(\))1414 1746 y Fn(Z)1498 1767 y Ft(x)1461 1935 y Fu(1)1518 1859 y Fy(dx)1608 1871 y Fu(0)1667 1859 y Fy(e)1706 1825 y Fu(\003)1751 1833 y Fc(\000)1800 1825 y Ft(=)p Fu(2\()p Ft(x)p Fq(\000)p Ft(x)2021 1833 y Fo(0)2053 1825 y Fu(\))2083 1767 y Fn(\020)2132 1859 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2274 1871 y Fu(0)2312 1859 y Fy(;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fy(g)2592 1871 y Fu(1)2629 1859 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2952 1871 y Fu(0)2990 1859 y Fy(;)14 b(k)s Fv(\))p Fs(j)3128 1767 y Fn(\021)3201 1859 y Fm(,)538 1990 y Fn(\014)538 2040 y(\014)538 2090 y(\014)566 2085 y Fy(D)661 2064 y Fv(^)637 2085 y Fy(N)704 2097 y Fu(1)p Ft(;)p Fu(2)794 2085 y Fv(\()p Fy(x;)g(k)s Fv(\))988 1990 y Fn(\014)988 2040 y(\014)988 2090 y(\014)1039 2085 y Fs(\024)23 b Fy(C)1227 1972 y Fn(Z)1310 1993 y Fq(1)1273 2161 y Ft(x)1359 2085 y Fy(dx)1449 2097 y Fu(0)1508 2085 y Fy(e)1547 2051 y Fu(\003)1592 2059 y Fo(+)1639 2051 y Ft(=)p Fu(2\()p Ft(x)p Fq(\000)p Ft(x)1860 2059 y Fo(0)1892 2051 y Fu(\))1922 1993 y Fn(\020)1971 2085 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2113 2097 y Fu(0)2151 2085 y Fy(;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fy(g)2431 2097 y Fu(1)2468 2085 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2791 2097 y Fu(0)2829 2085 y Fy(;)14 b(k)s Fv(\))p Fs(j)2967 1993 y Fn(\021)3040 2085 y Fm(,)538 2216 y Fn(\014)538 2266 y(\014)538 2316 y(\014)566 2312 y Fy(D)661 2291 y Fv(^)637 2312 y Fy(N)704 2324 y Fu(2)p Ft(;)p Fu(2)794 2312 y Fv(\()p Fy(x;)g(k)s Fv(\))988 2216 y Fn(\014)988 2266 y(\014)988 2316 y(\014)1039 2312 y Fs(\024)23 b Fy(C)1227 2199 y Fn(Z)1310 2219 y Fq(1)1273 2387 y Ft(x)1359 2312 y Fy(dx)1449 2324 y Fu(0)1508 2312 y Fy(e)1547 2277 y Fu(\003)1592 2285 y Fo(+)1639 2277 y Ft(=)p Fu(2\()p Ft(x)p Fq(\000)p Ft(x)1860 2285 y Fo(0)1892 2277 y Fu(\))1922 2220 y Fn(\020)1971 2312 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2113 2324 y Fu(0)2151 2312 y Fy(;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x)2604 2324 y Fu(0)2642 2312 y Fy(;)14 b(k)s Fv(\))p Fs(j)2780 2220 y Fn(\021)2853 2312 y Fy(:)-9 2556 y Fm(pr)l(oof)22 b(of)f(pr)l(op.)33 b(E.4.)62 b Fz(Use)22 b(that)988 2460 y Fn(\014)988 2510 y(\014)988 2560 y(\014)1016 2556 y Fy(D)1150 2523 y Ft(k)p 1097 2537 144 4 v 1097 2584 a Fu(\001\()p Ft(k)q Fu(\))1250 2460 y Fn(\014)1250 2510 y(\014)1250 2560 y(\014)1301 2556 y Fs(\024)1478 2523 y Fu(1)p 1398 2537 193 4 v 1398 2584 a(1+)p Fq(j)p Ft(k)q Fq(j)1558 2568 y Fo(3)1601 2556 y Fz(,)1643 2460 y Fn(\014)1643 2510 y(\014)1643 2560 y(\014)1671 2556 y Fy(D)1776 2515 y Fu(\003)1821 2523 y Fc(\006)p 1752 2537 144 4 v 1752 2584 a Fu(\001\()p Ft(k)q Fu(\))1905 2460 y Fn(\014)1905 2510 y(\014)1905 2560 y(\014)1956 2556 y Fs(\024)2095 2515 y Fu(2)p Fq(j)p Ft(k)q Fq(j)p 2054 2537 193 4 v 2054 2584 a Fu(1+)p Fq(j)p Ft(k)q Fq(j)2214 2568 y Fo(3)2256 2556 y Fz(,)g(together)f(with)h(Lemma)e(E.2.)p 3240 2556 43 43 v -9 2676 a(The)h(follo)n(wing)h(functions)h(will)d(help)i (us)g(estimate)d(the)i(functions)i Fy(N)2083 2688 y Ft(i;j)2182 2676 y Fz(and)f(their)f(deri)n(v)n(ati)n(v)o(es)f(with)h(respect)g(to)h Fy(k)s Fz(.)-9 2827 y FA(De\256nition)g(E.5)41 b Fm(F)-9 b(or)22 b Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)1028 2839 y Fu(+)1084 2827 y Fm(,)e(we)g(de\256ne)h(the)f(following)g(functions)970 3055 y Fv(\026)-59 b Fy(w)1012 3067 y Ft(\013)1082 3055 y Fv(=)1170 2938 y Fn(\022)1306 2999 y Fy(x)1353 2969 y Fu(1)p Ft(=)p Fu(2)p 1241 3036 283 4 v 1241 3112 a Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)p Fy(x)1534 2938 y Fn(\023)1595 2946 y Ft(\013)1679 3055 y Fm(,)535 3285 y Fy(E)596 3297 y Ft(\013;\014)s(;\015)t(;\016)o(;p)865 3285 y Fv(\()p Fy(x;)c(k)s Fv(\))23 b(=)g Fy(g)1210 3297 y Ft(\014)1254 3285 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fy(k)s Fs(j)1456 3250 y Ft(\015)1513 3172 y Fn(Z)1596 3192 y Ft(x)1559 3360 y Fu(1)1616 3285 y Fy(dx)1706 3297 y Fu(0)1766 3285 y Fy(e)1805 3250 y Fu(\003)1850 3258 y Fc(\000)1899 3250 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)2053 3258 y Fo(0)2084 3250 y Fu(\))p Ft(=p)2204 3285 y Fy(x)2251 3249 y Fq(\000)p Ft(\016)2251 3307 y Fu(0)2340 3285 y Fy(w)2399 3297 y Ft(\013)2446 3285 y Fv(\()p Fy(x)2525 3297 y Fu(0)2563 3285 y Fy(;)14 b(k)s Fv(\))24 b Fm(,)650 3511 y Fy(F)703 3523 y Ft(\013;\014)s(;p)865 3511 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b(=)g Fy(g)1210 3523 y Ft(\014)1254 3511 y Fv(\()p Fy(k)s Fv(\))1378 3398 y Fn(Z)1462 3418 y Ft(x)1425 3587 y Fu(1)1482 3511 y Fy(dx)1572 3523 y Fu(0)1631 3511 y Fy(e)1670 3477 y Fu(\003)1715 3485 y Fc(\000)1764 3477 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)1918 3485 y Fo(0)1950 3477 y Fu(\))p Ft(=p)2069 3511 y Fs(j)p Fy(k)s Fs(j)f Fy(x)2230 3468 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2230 3533 y(0)2425 3511 y Fv(\026)-59 b Fy(w)2467 3523 y Fu(2)2504 3511 y Fv(\()p Fy(x)2583 3523 y Fu(0)2621 3511 y Fy(;)14 b(k)s Fv(\))21 b Fy(w)2816 3523 y Ft(\013)p Fq(\000)p Fu(1)2949 3511 y Fv(\()p Fy(x)3028 3523 y Fu(0)3066 3511 y Fy(;)14 b(k)s Fv(\))23 b Fm(,)638 3737 y Fy(G)703 3749 y Ft(\013;\014)s(;p)865 3737 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b(=)1170 3624 y Fn(Z)1253 3645 y Fq(1)1216 3813 y Ft(x)1302 3737 y Fy(dx)1392 3749 y Fu(0)1451 3737 y Fy(e)1490 3703 y Fu(\003)1535 3711 y Fo(+)1582 3703 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)1736 3711 y Fo(0)1767 3703 y Fu(\))p Ft(=p)1887 3737 y Fy(x)1934 3697 y Fq(\000)p Ft(\014)1934 3759 y Fu(0)2052 3737 y Fy(w)2111 3749 y Ft(\013)2159 3737 y Fv(\()p Fy(x)2238 3749 y Fu(0)2276 3737 y Fy(;)14 b(k)s Fv(\))23 b Fm(,)636 3965 y Fy(H)705 3977 y Ft(\013;\015)t(;p)865 3965 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b(=)1180 3908 y(\()p Fs(j)p Fy(k)s Fs(j)p Fy(x)p Fv(\))1383 3878 y Ft(\015)p 1180 3945 247 4 v 1245 3962 a Fs(p)p 1314 3962 48 4 v 60 x Fy(x)1450 3852 y Fn(Z)1533 3872 y Fq(1)1497 4040 y Ft(x)1582 3965 y Fy(dx)1672 3977 y Fu(0)1731 3965 y Fy(e)1770 3930 y Fu(\003)1815 3938 y Fo(+)1862 3930 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)2016 3938 y Fo(0)2048 3930 y Fu(\))p Ft(=p)2167 3965 y Fy(x)2214 3921 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2214 3987 y(0)2409 3965 y Fv(\026)-59 b Fy(w)2451 3977 y Fu(2)2489 3965 y Fv(\()p Fy(x)2568 3977 y Fu(0)2606 3965 y Fy(;)14 b(k)s Fv(\))21 b Fy(w)2801 3977 y Ft(\013)p Fq(\000)p Fu(1)2934 3965 y Fv(\()p Fy(x)3013 3977 y Fu(0)3051 3965 y Fy(;)14 b(k)s Fv(\))23 b Fy(:)-9 4172 y Fz(W)-7 b(e)21 b(summarize)g(the)g (properties)h(of)g(these)f(functions)h(in)g(the)f(follo)n(wing)h (proposition.)-9 4323 y FA(Pr)n(oposition)g(E.6)42 b Fm(Let)21 b Fy(\014)28 b Fs(\025)22 b Fv(0)p Fm(,)f Fv(0)i Fs(\024)f Fy(\015)28 b Fs(\024)23 b Fv(1)p Fm(,)e Fv(0)h Fs(\024)h Fy(\016)j Fs(\024)d Fv(1)p Fm(,)e Fy(\013)i Fs(\025)g Fv(0)p Fm(,)e Fy(\030)27 b Fs(2)c Fr(R)e Fm(and)i Fy(p)g Fs(\025)f Fv(1)f Fm(with)g Fy(\014)h Fv(+)d Fy(\015)27 b Fs(\025)c Fv(0)p Fm(.)30 b(De\256ne)1384 4512 y Fv(~)1372 4534 y Fy(\014)e Fv(=)1534 4417 y Fn(\032)1638 4484 y Fy(\014)23 b Fs(\000)1801 4451 y Fu(1)p 1801 4465 34 4 v 1801 4512 a(2)1927 4484 y Fz(if)83 b Fy(\014)27 b Fs(\025)2234 4451 y Fu(1)p 2234 4465 V 2234 4512 a(2)1638 4584 y Fy(\014)c Fs(\000)18 b Fv(1)94 b Fz(if)83 b Fy(\014)27 b(<)2234 4551 y Fu(1)p 2234 4565 V 2234 4612 a(2)2365 4534 y Fy(:)-9 4749 y Fm(Then)22 b(ther)m(e)e(e)n(xists)h(a)g(constant)i Fy(C)k Fm(suc)o(h)22 b(that)f(for)h(all)f Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)1959 4761 y Fu(+)2014 4749 y Fm(,)e(we)g(have)1095 4913 y Fy(E)1156 4925 y Ft(\013;\014)s(;\015)t(\016)o(;p)1405 4913 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(\024)g Fy(C)28 b(w)1856 4925 y Ft(\013)1903 4913 y Fv(\()p Fy(x;)14 b(k)s Fv(\))22 b Fy(J)2165 4925 y Ft(\014)s(;\015)t(;\016)2320 4913 y Fv(\()p Fy(x)p Fv(\))p Fy(;)1144 b Fz(\(E.6\))1190 5059 y Fy(F)1243 5071 y Ft(\013;\014)s(;p)1405 5059 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(\024)g Fy(C)28 b(w)1856 5071 y Ft(\013)1903 5059 y Fv(\()p Fy(x;)14 b(k)s Fv(\))22 b Fy(x)2166 5024 y Fq(\000)p FD(min)r Fu(\(1)p Ft(=)p Fu(2)p Ft(;)2493 5009 y Fu(~)2484 5024 y Ft(\014)r(=)p Fu(2\))2644 5059 y Fm(,)933 b Fz(\(E.7\))1186 5183 y Fy(G)1251 5195 y Ft(\013;\030)r(;p)1405 5183 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(\024)g Fy(C)6 b(w)1834 5195 y Ft(\013)1882 5183 y Fv(\()p Fy(x;)14 b(k)s Fv(\))22 b Fy(w)2157 5195 y Fu(1)2195 5183 y Fv(\(1)p Fy(;)14 b(k)s Fv(\))21 b Fy(x)2452 5149 y Fq(\000)p Ft(\030)2564 5183 y Fm(,)1013 b Fz(\(E.8\))1176 5308 y Fy(H)1245 5320 y Ft(\013;\015)t(;p)1405 5308 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(\024)g Fy(C)28 b(w)1856 5320 y Ft(\013)p Fq(\000)p Fu(1)1988 5308 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)1371 b Fz(\(E.9\))-9 5472 y Fm(wher)m(e)820 5636 y Fy(J)866 5648 y Ft(\014)s(;\015)t(;\016)1021 5636 y Fv(\()p Fy(x)p Fv(\))25 b(=)d Fy(x)1291 5602 y Fu(1)p Fq(\000)p Ft(\016)r Fq(\000)p FC(min)q Fu(\(1)p Ft(;)p Fu(\()p Ft(\014)s Fu(+)p Ft(\015)t Fu(\))p Ft(=)p Fu(2\))1952 5636 y Fv(+)c Fy(\016)2072 5648 y Fu(1)p Ft(;\016)2196 5636 y Fv(ln\(1)h(+)f Fy(x)p Fv(\))k Fy(x)2589 5602 y Fq(\000)p Fu(\()p Ft(\014)s Fu(+)p Ft(\015)t Fu(\))p Ft(=)p Fu(2)2917 5636 y Fy(:)1838 5909 y Fz(37)p eop %%Page: 38 38 bop -9 145 a Fm(pr)l(oof)22 b(of)f(pr)l(op.)33 b(E.6.)62 b Fz(W)-7 b(e)22 b(be)o(gin)f(with)h(the)f(proof)i(of)f(\(E.6\).)32 b(W)-7 b(e)22 b(ha)n(v)o(e)1123 306 y Fy(E)1184 318 y Ft(\013;\014)s(;\015)t (;\016)o(;p)1452 306 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p 1123 344 525 4 v 1234 420 a Fy(w)1293 432 y Ft(\013)1341 420 y Fv(\()p Fy(x;)g(k)s Fv(\))1680 363 y Fs(\024)23 b Fy(C)k(T)1903 375 y Fu(1)1940 363 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(C)28 b(T)2372 375 y Fu(2)2409 363 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fz(,)-9 579 y(where)943 797 y Fy(T)992 809 y Fu(1)1029 797 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e Fy(g)1374 809 y Ft(\014)1419 797 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fy(k)s Fs(j)1621 763 y Ft(\015)1663 797 y Fy(e)1702 763 y Fu(\003)1747 771 y Fc(\000)1797 763 y Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(\(2)p Ft(p)p Fu(+2\))2226 684 y Fn(Z)2310 704 y Fu(\()p Ft(x)p Fu(+1\))p Ft(=)p Fu(2)2273 873 y(1)2533 797 y Fy(dx)2623 809 y Fu(0)2682 797 y Fy(x)2729 761 y Fq(\000)p Ft(\016)2729 819 y Fu(0)943 1023 y Fy(T)992 1035 y Fu(2)1029 1023 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e Fy(g)1374 1035 y Ft(\014)1419 1023 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fy(k)s Fs(j)1621 989 y Ft(\015)1695 967 y Fv(1)p 1673 1004 84 4 v 1673 1080 a Fy(x)1720 1056 y Ft(\016)1781 910 y Fn(Z)1864 931 y Ft(x)1827 1099 y Fu(\()p Ft(x)p Fu(+1\))p Ft(=)p Fu(2)2051 1023 y Fy(dx)2141 1035 y Fu(0)2200 1023 y Fy(e)2239 989 y Fu(\003)2284 997 y Fc(\000)2333 989 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)2487 997 y Fo(0)2518 989 y Fu(\))p Ft(=)p Fu(\()p Ft(p)p Fu(+1\))2776 1023 y Fy(:)-9 1252 y Fz(F)o(or)g Fy(T)178 1264 y Fu(1)214 1252 y Fz(,)g(we)f(ha)n(v)o(e)383 1491 y Fy(T)432 1503 y Fu(1)468 1491 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(\024)f Fy(C)k(J)906 1503 y Ft(\014)s(;\015)t(;\016)1061 1491 y Fv(\()p Fy(x)p Fv(\))1186 1349 y Fn( )1255 1491 y Fz(sup)1253 1563 y Ft(x)p Fq(\025)p Fu(1)1746 1434 y Fv(1)p 1399 1471 735 4 v 1399 1548 a Fy(x)1446 1524 y Fu(1)p Fq(\000)p Ft(\016)1587 1548 y Fv(+)18 b Fy(\016)1707 1560 y Fu(1)p Ft(;\016)1810 1548 y Fv(ln\(1)g(+)g Fy(x)p Fv(\))2225 1434 y Fy(x)p 2154 1471 191 4 v 2154 1548 a(x)h Fs(\000)f Fv(1)2368 1378 y Fn(Z)2451 1398 y Ft(x)2414 1566 y Fu(1)2471 1491 y Fy(dx)2561 1503 y Fu(0)2620 1491 y Fy(x)2667 1455 y Fq(\000)p Ft(\016)2667 1513 y Fu(0)2756 1349 y Fn(!)2845 1491 y Fs(\024)k Fy(C)28 b(J)3065 1503 y Ft(\014)s(;\015)t(;\016)3220 1491 y Fv(\()p Fy(x)p Fv(\))c Fy(:)-9 1733 y Fz(F)o(or)e Fy(T)178 1745 y Fu(2)214 1733 y Fz(,)g(since)f Fy(\014)h Fv(+)c Fy(\015)28 b Fs(\025)23 b Fv(0)p Fz(,)e(we)g(ha)n(v)o(e)802 1943 y Fy(T)851 1955 y Fu(2)888 1943 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(\024)1212 1887 y Fy(C)p 1203 1924 84 4 v 1203 2000 a(x)1250 1976 y Ft(\016)1297 1943 y Fy(g)1337 1955 y Ft(\014)1381 1943 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fy(k)s Fs(j)1583 1909 y Ft(\015)1695 1887 y Fv(1)p 1636 1924 160 4 v 1636 2000 a Fs(j)p Fv(\003)1717 2012 y Fq(\000)1773 2000 y Fs(j)1820 1851 y Fn(\020)1869 1943 y Fv(1)18 b Fs(\000)g Fy(e)2051 1909 y Fq(\000j)p Fu(\003)2168 1917 y Fc(\000)2217 1909 y Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2416 1851 y Fn(\021)1105 2173 y Fs(\024)1255 2116 y Fy(C)p 1203 2154 169 4 v 1203 2230 a(x)1250 2206 y Ft(\016)r Fq(\000)p Fu(1)1382 2173 y Fy(g)1422 2188 y Ft(\014)s(=)p Fu(2)1533 2173 y Fv(\(\003)1623 2185 y Fq(\000)1679 2173 y Fv(\))1821 2116 y Fs(j)p Fy(k)s Fs(j)1913 2086 y Ft(\015)p 1722 2154 333 4 v 1722 2230 a Fv(1)g(+)g Fs(j)p Fv(\003)1946 2242 y Fq(\000)2001 2230 y Fs(j)p Fy(t)1105 2402 y Fs(\024)1255 2346 y Fy(C)p 1203 2383 169 4 v 1203 2459 a(x)1250 2435 y Ft(\016)r Fq(\000)p Fu(1)1382 2402 y Fy(g)1422 2417 y Fu(\()p Ft(\014)s Fu(+)p Ft(\015)t Fu(\))p Ft(=)p Fu(2)1675 2402 y Fv(\(\003)1765 2414 y Fq(\000)1821 2402 y Fv(\))1863 2346 y(1)g(+)g Fs(j)p Fv(\003)2087 2358 y Fq(\000)2143 2346 y Fs(j)2166 2316 y Ft(\015)p 1863 2383 346 4 v 1869 2459 a Fv(1)g(+)g Fs(j)p Fv(\003)2093 2471 y Fq(\000)2149 2459 y Fs(j)p Fy(t)1105 2656 y Fs(\024)23 b Fy(C)28 b(J)1326 2668 y Ft(\014)s(;\015)t(;\016)1481 2656 y Fv(\()p Fy(x)p Fv(\))1628 2539 y Fn(\022)1755 2600 y Fs(j)p Fv(\003)1836 2612 y Fq(\000)1892 2600 y Fs(j)p Fy(t)p 1699 2637 303 4 v 1699 2713 a Fv(1)18 b(+)g Fs(j)p Fv(\003)1923 2725 y Fq(\000)1979 2713 y Fs(j)2012 2539 y Fn(\023)2073 2556 y FD(min)q Fu(\(1)p Ft(;)p Fu(\()p Ft(\014)s Fu(+)p Ft(\015)t Fu(\))p Ft(=)p Fu(2\))2573 2600 y Fv(1)g(+)g Fs(j)p Fv(\003)2797 2612 y Fq(\000)2853 2600 y Fs(j)2876 2570 y Ft(\015)p 2573 2637 346 4 v 2579 2713 a Fv(1)h(+)f Fs(j)p Fv(\003)2804 2725 y Fq(\000)2859 2713 y Fs(j)p Fy(t)1105 2912 y Fs(\024)23 b Fy(C)28 b(J)1326 2924 y Ft(\014)s(;\015)t(;\016)1481 2912 y Fv(\()p Fy(x)p Fv(\))1628 2795 y Fn(\022)1770 2856 y Fs(j)p Fv(\003)1851 2868 y Fq(\000)1907 2856 y Fs(j)p Fy(t)p 1699 2893 333 4 v 1699 2969 a Fv(1)18 b(+)g Fs(j)p Fv(\003)1923 2981 y Fq(\000)1979 2969 y Fs(j)p Fy(t)2042 2795 y Fn(\023)2103 2812 y FD(min)q Fu(\(1)p Ft(;)p Fu(\()p Ft(\014)s Fu(+)p Ft(\015)t Fu(\))p Ft(=)p Fu(2\))2603 2856 y Fv(1)g(+)g Fs(j)p Fv(\003)2827 2868 y Fq(\000)2883 2856 y Fs(j)2906 2826 y Ft(\015)p 2603 2893 346 4 v 2625 2969 a Fv(1)g(+)g Fs(j)p Fv(\003)2849 2981 y Fq(\000)2904 2969 y Fs(j)1105 3086 y(\024)23 b Fy(C)28 b(J)1326 3098 y Ft(\014)s(;\015)t(;\016)1481 3086 y Fv(\()p Fy(x)p Fv(\))c Fy(:)-9 3259 y Fz(F)o(or)e(inequality)f(\(E.7\),)h(we)g(ha)n(v)o(e)1180 3421 y Fy(F)1233 3433 y Ft(\013;\014)s(;p)1395 3421 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p 1180 3458 410 4 v 1192 3534 a Fy(w)1251 3546 y Ft(\013)p Fq(\000)p Fu(1)1384 3534 y Fv(\()p Fy(x;)g(k)s Fv(\))1623 3477 y Fs(\024)23 b Fy(C)k(T)1846 3489 y Fu(3)1883 3477 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(C)28 b(T)2315 3489 y Fu(4)2351 3477 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fz(,)-9 3693 y(where)933 3911 y Fy(T)982 3923 y Fu(3)1018 3911 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)f Fy(g)1364 3923 y Ft(\014)1408 3911 y Fv(\()p Fy(k)s Fv(\))p Fy(e)1557 3877 y Fu(\003)1602 3885 y Fc(\000)1652 3877 y Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(\(2)p Ft(p)p Fu(\))1998 3798 y Fn(Z)2081 3819 y Fu(\()p Ft(x)p Fu(+1\))p Ft(=)p Fu(2)2044 3987 y(1)2304 3911 y Fy(dx)2394 3923 y Fu(0)2495 3852 y Fs(j)p Fy(k)s Fs(j)2587 3799 y(p)p 2657 3799 85 4 v 2657 3852 a Fy(x)2704 3864 y Fu(0)p 2463 3892 311 4 v 2463 3968 a Fv(1)18 b(+)g Fy(k)2652 3944 y Fu(2)2689 3968 y Fy(x)2736 3939 y Fu(2)2736 3990 y(0)2807 3911 y Fz(,)933 4146 y Fy(T)982 4158 y Fu(4)1018 4146 y Fv(\()p Fy(x;)c(k)s Fv(\))24 b(=)f Fy(g)1364 4158 y Ft(\014)1408 4146 y Fv(\()p Fy(k)s Fv(\))1580 4090 y Fs(j)p Fy(k)s Fs(j)1672 4030 y(p)p 1741 4030 48 4 v 60 x Fy(x)p 1528 4127 311 4 v 1528 4203 a Fv(1)c(+)f Fy(k)1718 4179 y Fu(2)1755 4203 y Fy(x)1802 4179 y Fu(2)1863 4033 y Fn(Z)1946 4053 y Ft(x)1909 4222 y Fu(\()p Ft(x)p Fu(+1\))p Ft(=)p Fu(2)2133 4146 y Fy(dx)2223 4158 y Fu(0)2282 4146 y Fy(e)2321 4112 y Fu(\003)2366 4120 y Fc(\000)2415 4112 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)2569 4120 y Fo(0)2600 4112 y Fu(\))2654 4146 y Fy(:)-9 4379 y Fz(No)n(w)k(tak)o(e)f Fy(\014)27 b(<)c Fv(1)p Fy(=)p Fv(2)p Fz(,)d(then)532 4617 y Fy(T)581 4629 y Fu(3)617 4617 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(\024)f Fy(C)k(x)1056 4583 y Fq(\000)p Ft(\014)s(=)p Fu(2)1231 4561 y Fy(e)1270 4531 y Fu(\003)1315 4539 y Fc(\000)1364 4531 y Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(\(3)p Ft(p)p Fu(\))p 1231 4598 466 4 v 1346 4674 a Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1719 4504 y Fn(Z)1802 4525 y Fu(\()p Ft(x)p Fu(+1\))p Ft(=)p Fu(2)1766 4693 y(1)2026 4617 y Fy(dx)2116 4629 y Fu(0)2175 4617 y Fy(x)2222 4574 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2222 4640 y(0)2402 4617 y Fs(\024)k Fy(C)28 b(x)2623 4583 y Fu(\(1)p Fq(\000)p Ft(\014)s Fu(\))p Ft(=)p Fu(2)2893 4617 y Fy(w)2952 4629 y Fu(1)2990 4617 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fz(,)-9 4838 y(whereas)d(for)h Fy(\014)28 b Fs(\025)22 b Fv(1)p Fy(=)p Fv(2)p Fz(,)640 5076 y Fy(T)689 5088 y Fu(3)725 5076 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(\024)f Fy(g)1071 5091 y Ft(\014)s Fq(\000)p Fu(1)p Ft(=)p Fu(2)1267 5076 y Fv(\()p Fy(k)s Fv(\))1409 5020 y Fy(e)1448 4989 y Fu(\003)1493 4997 y Fc(\000)1542 4989 y Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(\(2)p Ft(p)p Fu(\))p 1409 5057 V 1524 5133 a Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1921 5076 y Fz(sup)1919 5148 y Ft(k)q Fq(\025)p Fu(0)2055 5001 y Fn(p)p 2138 5001 235 4 v 75 x Fv(1)g(+)g Fs(j)p Fy(k)s Fs(j)2386 4963 y Fn(Z)2469 4983 y Fq(1)2432 5151 y Fu(0)2518 5076 y Fy(d\030)2714 4947 y Fn(p)p 2797 4947 234 4 v 71 x Fy(\030)k Fv(+)c Fs(j)p Fy(k)s Fs(j)p 2633 5057 479 4 v 2633 5133 a Fv(1)g(+)g(\()p Fy(\030)23 b Fv(+)18 b Fs(j)p Fy(k)s Fs(j)p Fv(\))3074 5109 y Fu(2)943 5278 y Fs(\024)23 b Fy(C)k(x)1164 5244 y Fu(1)p Ft(=)p Fu(4)p Fq(\000)p Ft(\014)s(=)p Fu(2)1450 5278 y Fy(w)1509 5290 y Fu(1)1547 5278 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)-9 5451 y Fz(Combining)e(these)f(inequalities)f(gi)n(v)o(es)1211 5639 y Fy(T)1260 5651 y Fu(3)1297 5639 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(\024)e Fy(C)28 b(x)1736 5605 y Fq(\000)p FD(min)q Fu(\(1)p Ft(=)p Fu(2)p Ft(;)2062 5590 y Fu(~)2053 5605 y Ft(\014)s(=)p Fu(2\))2212 5639 y Fy(w)2271 5651 y Fu(1)2309 5639 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)1838 5909 y Fz(38)p eop %%Page: 39 39 bop -9 145 a Fz(F)o(or)22 b Fy(T)178 157 y Fu(4)214 145 y Fz(,)g(one)g(has) 324 378 y Fy(T)373 390 y Fu(4)410 378 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(\024)f Fy(C)6 b(g)821 390 y Ft(\014)865 378 y Fv(\()p Fy(k)s Fv(\))985 322 y Fs(j)p Fy(k)s Fs(j)22 b Fy(x)1146 292 y Fu(1)p Ft(=)p Fu(2)1272 322 y Fv(\()p Fy(x)d Fs(\000)f Fv(1\))p 985 359 542 4 v 1101 435 a(1)g(+)g Fy(k)1290 411 y Fu(2)1327 435 y Fy(x)1374 411 y Fu(2)1547 322 y Fv(1)g Fs(\000)g Fy(e)1729 292 y Fu(\003)1774 300 y Fc(\000)1823 292 y Fu(\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(4)p Ft(p)p 1547 359 557 4 v 1617 435 a Fs(j)p Fv(\003)1698 447 y Fq(\000)1754 435 y Fs(j)p Fv(\()p Fy(x)h Fs(\000)f Fv(1\))2136 378 y Fs(\024)23 b Fy(C)6 b(g)2329 390 y Ft(\014)2373 378 y Fv(\()p Fy(k)s Fv(\))2516 322 y Fs(j)p Fy(k)s Fs(j)22 b Fy(x)2677 292 y Fu(1)p Ft(=)p Fu(2)p 2494 359 311 4 v 2494 435 a Fv(1)c(+)g Fy(k)2683 411 y Fu(2)2720 435 y Fy(x)2767 411 y Fu(2)3008 322 y Fy(x)h Fs(\000)f Fv(1)p 2824 359 558 4 v 2824 435 a(1)g(+)g Fs(j)p Fv(\003)3048 447 y Fq(\000)3104 435 y Fs(j)p Fv(\()p Fy(x)h Fs(\000)f Fv(1\))3415 378 y Fz(,)-9 604 y(and)j(no)n(w)-6 b(,)24 b(we)d(use)g(that)346 837 y Fy(g)386 849 y Ft(\014)431 837 y Fv(\()p Fy(k)s Fv(\))574 781 y Fs(j)p Fy(k)s Fs(j)g Fy(x)734 751 y Fu(1)p Ft(=)p Fu(2)p 551 818 311 4 v 551 894 a Fv(1)d(+)g Fy(k)740 870 y Fu(2)777 894 y Fy(x)824 870 y Fu(2)1065 781 y Fy(x)h Fs(\000)f Fv(1)p 882 818 558 4 v 882 894 a(1)g(+)g Fs(j)p Fv(\003)1106 906 y Fq(\000)1162 894 y Fs(j)p Fv(\()p Fy(x)h Fs(\000)f Fv(1\))1473 837 y Fs(\024)k Fy(g)1600 849 y Ft(\014)1645 837 y Fv(\()p Fy(k)s Fv(\))1799 781 y Fs(j)p Fy(k)s Fs(j)p Fy(x)1938 751 y Fu(3)p Ft(=)p Fu(2)p 1765 818 311 4 v 1765 894 a Fv(1)c(+)g Fy(k)1954 870 y Fu(2)1991 894 y Fy(x)2038 870 y Fu(2)2251 781 y Fv(1)p 2096 818 352 4 v 2096 895 a(1)g(+)g Fs(j)p Fy(k)s Fs(j)2331 835 y(p)p 2400 835 48 4 v 60 x Fy(x)1473 1092 y Fs(\024)k Fy(x)1607 1058 y Fq(\000)p FD(min)r Fu(\(1)p Ft(=)p Fu(2)p Ft(;\014)s Fq(\000)p Fu(1)p Ft(=)p Fu(2\))2161 975 y Fn(\022)2280 1036 y Fs(j)p Fy(k)s Fs(j)p Fy(x)p 2232 1073 235 4 v 2232 1149 a Fv(1)c(+)g Fs(j)p Fy(k)s Fs(j)2477 975 y Fn(\023)2538 992 y FD(min)q Fu(\()p Ft(\014)s(;)p Fu(1\))2951 1036 y Fv(1)p 2830 1073 283 4 v 2830 1149 a(1)g(+)g Fs(j)p Fy(k)s Fs(j)p Fy(x)3123 1092 y(w)3182 1104 y Fu(1)3220 1092 y Fv(\()p Fy(x;)c(k)s Fv(\))1473 1293 y Fs(\024)22 b Fy(x)1607 1259 y Fq(\000)p FD(min)r Fu(\(1)p Ft(=)p Fu(2)p Ft(;\014)s Fq(\000)p Fu(1)p Ft(=)p Fu(2\))2147 1293 y Fy(w)2206 1305 y Fu(1)2244 1293 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(\024)e Fy(x)2596 1259 y Fq(\000)p FD(min)r Fu(\(1)p Ft(=)p Fu(2)p Ft(;)2923 1243 y Fu(~)2914 1259 y Ft(\014)r(=)p Fu(2\))3051 1293 y Fy(w)3110 1305 y Fu(1)3148 1293 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)-9 1476 y Fz(F)o(or)e(the)f(inequality)g (\(E.8\),)h(since)f Fy(x)1072 1488 y Fu(0)1133 1476 y Fs(\025)i Fy(x)p Fz(,)f(we)f(ha)n(v)o(e)1030 1712 y Fy(x)1077 1678 y Ft(\030)1114 1712 y Fy(G)1179 1724 y Ft(\013;\030)r(;p)1332 1712 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(\024)e Fy(w)1696 1724 y Ft(\013)1744 1712 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1952 1599 y Fn(Z)2036 1620 y Fq(1)1999 1788 y Ft(x)2084 1712 y Fy(dx)2174 1724 y Fu(0)2234 1712 y Fy(e)2273 1678 y Fu(\003)2318 1686 y Fo(+)2365 1678 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)2519 1686 y Fo(0)2550 1678 y Fu(\))2591 1656 y Fy(x)2638 1626 y Ft(\030)p 2590 1693 85 4 v 2590 1783 a Fy(x)2637 1743 y Ft(\030)2637 1805 y Fu(0)2708 1712 y Fy(:)-9 1954 y Fz(By)21 b(a)g(change)h(of)g(v)n (ariable,)f(we)g(ha)n(v)o(e)574 2091 y Fn(Z)657 2111 y Fq(1)620 2280 y Ft(x)706 2204 y Fy(dx)796 2216 y Fu(0)855 2204 y Fy(e)894 2170 y Fu(\003)939 2178 y Fo(+)986 2170 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)1140 2178 y Fo(0)1172 2170 y Fu(\))1212 2148 y Fy(x)1259 2118 y Ft(\030)p 1212 2185 V 1212 2275 a Fy(x)1259 2235 y Ft(\030)1259 2297 y Fu(0)1329 2204 y Fs(\024)1463 2148 y Fv(1)p 1427 2185 113 4 v 1427 2261 a(\003)1485 2273 y Fu(+)1563 2091 y Fn(Z)1646 2111 y Fq(1)1610 2280 y Fu(0)1695 2204 y Fy(dy)1817 2087 y Fn(\022)1878 2204 y Fv(1)d(+)2090 2148 y Fy(y)p 2031 2185 161 4 v 2031 2261 a Fv(\003)2089 2273 y Fu(+)2144 2261 y Fy(x)2201 2087 y Fn(\023)2263 2104 y Fq(\000)p Ft(\030)2365 2204 y Fy(e)2404 2170 y Fq(\000)p Ft(y)1329 2451 y Fs(\024)1463 2395 y Fv(1)p 1427 2432 113 4 v 1427 2508 a(\003)1485 2520 y Fu(+)1563 2451 y Fz(max)1724 2334 y Fn(\032)1786 2451 y Fv(1)p Fy(;)1865 2338 y Fn(Z)1947 2358 y Fq(1)1910 2526 y Fu(0)1996 2451 y Fy(dy)24 b Fv(\(1)18 b(+)g Fy(y)s Fv(\))2355 2417 y Fq(\000)p Ft(\030)2465 2451 y Fy(e)2504 2417 y Fq(\000)p Ft(y)2595 2334 y Fn(\033)2681 2451 y Fs(\024)k Fy(C)28 b(w)2914 2463 y Fu(1)2952 2451 y Fv(\(1)p Fy(;)14 b(k)s Fv(\))22 b Fy(:)-9 2678 y Fz(W)-7 b(e)21 b(conclude)h(the)f (proof)i(with)f(inequality)f(\(E.7\).)32 b(W)-7 b(e)21 b(ha)n(v)o(e)729 2906 y Fy(H)798 2918 y Ft(\013;\015)t(;p)958 2906 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(\024)g Fy(w)1322 2918 y Ft(\013)p Fq(\000)p Fu(1)1455 2906 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1692 2850 y(\()p Fs(j)p Fy(k)s Fs(j)p Fy(x)p Fv(\))1895 2820 y Ft(\015)p 1660 2887 311 4 v 1660 2963 a Fv(1)j(+)h Fy(k)1848 2939 y Fu(2)1886 2963 y Fy(x)1933 2939 y Fu(2)2028 2850 y Fv(1)p 1990 2887 117 4 v 1990 2904 a Fs(p)p 2059 2904 48 4 v 60 x Fy(x)2131 2793 y Fn(Z)2214 2814 y Fq(1)2177 2982 y Ft(x)2262 2906 y Fy(dx)2352 2918 y Fu(0)2412 2906 y Fy(e)2451 2872 y Fu(\003)2496 2880 y Fo(+)2542 2872 y Fu(\()p Ft(x)p Fq(\000)p Ft(x)2696 2880 y Fo(0)2728 2872 y Fu(\))2758 2849 y Fs(p)p 2827 2849 85 4 v 57 x Fy(x)2874 2918 y Fu(0)1175 3112 y Fs(\024)23 b Fy(w)1322 3124 y Ft(\013)p Fq(\000)p Fu(1)1455 3112 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1663 3020 y Fn(\020)1713 3112 y Fy(x)1760 3078 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1917 3112 y Fy(G)1982 3127 y Fu(0)p Ft(;)p Fq(\000)p Fu(1)p Ft(=)p Fu(2)p Ft(;)p Fu(1)2244 3112 y Fv(\()p Fy(x;)g(k)s Fv(\))2438 3020 y Fn(\021)2511 3112 y Fs(\024)23 b Fy(w)2658 3124 y Ft(\013)p Fq(\000)p Fu(1)2791 3112 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fy(:)-9 3315 y Fz(This)e(achie)n(v)o (es)f(the)h(proof.)p 855 3315 43 43 v -9 3414 a(Finally)-6 b(,)21 b(this)g(lemma)g(completes)g(information)h(we)f(ga)n(v)o(e)h(in)f (Section)g(C.3.)-9 3580 y FA(Lemma)f(E.7)42 b Fm(Let)414 3794 y Fy(P)467 3806 y Fu(1)505 3794 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e Fs(\000)p Fv(\(\003)965 3806 y Fu(+)1020 3794 y Fv(\()p Fy(k)s Fv(\))d(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1370 3676 y Fn(\024)1717 3737 y Fy(ik)p 1424 3774 662 4 v 1424 3850 a Fv(\001\()p Fy(k)s Fv(\)\(\003)1693 3862 y Fq(\000)1749 3850 y Fv(\()p Fy(k)s Fv(\))h(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))2095 3794 y Fy(e)2134 3759 y Fu(\003)2179 3767 y Fc(\000)2229 3759 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2514 3794 y Fs(\000)g Fy(i)35 b Fz(s)-5 b(i)g(g)g(n)r Fv(\()p Fy(k)s Fv(\))p Fy(e)2938 3759 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))3244 3676 y Fn(\025)3325 3794 y Fm(,)414 4026 y Fy(P)467 4038 y Fu(2)505 4026 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b(=)e(\(\003)900 4038 y Fu(+)955 4026 y Fv(\()p Fy(k)s Fv(\))d(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))1305 3909 y Fn(\024)1578 3970 y Fv(\003)1636 3982 y Fq(\000)1692 3970 y Fv(\()p Fy(k)s Fv(\))p 1359 4007 V 1359 4083 a(\001\()p Fy(k)s Fv(\)\(\003)1628 4095 y Fq(\000)1685 4083 y Fv(\()p Fy(k)s Fv(\))h(+)f Fs(j)p Fy(k)s Fs(j)p Fv(\))2031 4026 y Fy(e)2070 3992 y Fu(\003)2115 4000 y Fc(\000)2164 3992 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))2450 4026 y Fv(+)g Fy(e)2572 3992 y Fq(\000j)p Ft(k)q Fq(j)p Fu(\()p Ft(x)p Fq(\000)p Fu(1\))2878 3909 y Fn(\025)2959 4026 y Fy(:)-9 4258 y Fm(Then)k(ther)m(e)e(e)n(xists)h(a)g(constant)i Fy(C)k Fm(suc)o(h)22 b(that)f(for)h(all)f Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)1959 4270 y Fu(+)2014 4258 y Fm(,)e(we)g(have)1013 4441 y Fs(j)p Fy(P)1089 4453 y Fu(1)1127 4441 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b(e)1581 4407 y Fu(\003)1626 4415 y Fc(\000)1675 4407 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(2)2033 4441 y Fm(,)1013 4566 y Fs(j)p Fy(P)1089 4578 y Fu(2)1127 4566 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b(e)1581 4531 y Fu(\003)1626 4539 y Fc(\000)1675 4531 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(2)2033 4566 y Fm(,)942 4690 y Fs(j)p Fy(D)r(P)1089 4702 y Fu(1)1127 4690 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)1542 4626 y Fs(p)p 1611 4626 48 4 v 64 x Fy(x)q(e)1698 4656 y Fu(\003)1743 4664 y Fc(\000)1792 4656 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(2)2145 4690 y Fv(+)c Fy(C)27 b(xe)2400 4656 y Fq(\000j)p Ft(k)q Fq(j)p Ft(=)p Fu(2\()p Ft(x)p Fq(\000)p Fu(1\))2797 4690 y Fm(,)942 4815 y Fs(j)p Fy(D)r(P)1089 4827 y Fu(2)1127 4815 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b(e)1581 4780 y Fu(\003)1626 4788 y Fc(\000)1675 4780 y Fu(\()p Ft(k)q Fu(\)\()p Ft(x)p Fq(\000)p Fu(1\))p Ft(=)p Fu(2)2028 4815 y Fv(+)18 b Fy(C)28 b(xe)2284 4780 y Fq(\000j)p Ft(k)q Fq(j)p Ft(=)p Fu(2\()p Ft(x)p Fq(\000)p Fu(1\))2680 4815 y Fy(:)-9 4997 y Fm(Pr)l(oof)o(.)j Fz(This)21 b(is)g(elementary)-6 b(.)p 926 4997 43 43 v -9 5238 a Fd(E.2)102 b(Pr)n(oof)26 b(of)g(Lemma)f (3.10)-9 5395 y Fz(Using)19 b(De\256nition)g(E.3,)h(we)f(ha)n(v)o(e)g Fs(j)1072 5374 y Fv(^)1048 5395 y Fy(N)1115 5407 y Ft(i)1142 5395 y Fs(j)24 b(\024)e(j)1323 5374 y Fv(^)1299 5395 y Fy(N)1366 5407 y Ft(i;)p Fu(1)1446 5395 y Fs(j)13 b Fv(+)g Fs(j)1607 5374 y Fv(^)1583 5395 y Fy(N)1650 5407 y Ft(i;)p Fu(2)1730 5395 y Fs(j)g Fv(+)g Fs(j)p Fy(P)1920 5407 y Ft(i)1947 5395 y Fy(\034)c Fs(j)p Fz(.)31 b(The)18 b(follo)n(wing)i(proposition)h(bounds)g (the)d(contrib)n(ution)-9 5506 y(of)107 5485 y Fv(^)83 5506 y Fy(N)150 5518 y Ft(i;j)249 5506 y Fz(to)360 5485 y Fv(^)336 5506 y Fy(N)403 5518 y Ft(i)430 5506 y Fz(.)1838 5909 y(39)p eop %%Page: 40 40 bop -9 145 a FA(Pr)n(oposition)22 b(E.8)42 b Fm(F)-9 b(or)22 b(all)f Fy(\015)886 157 y Fu(2)946 145 y Fs(2)j Fv([0)p Fy(;)14 b Fv(1\))p Fm(,)20 b(ther)m(e)h(e)n(xists)g(a)g(constant)h Fy(C)28 b Fm(suc)o(h)22 b(that)f(for)h(all)e Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)g Fv(\012)3020 157 y Fu(+)3075 145 y Fm(,)d(we)g(have)332 253 y Fn(\014)332 303 y(\014)332 353 y(\014)384 328 y Fv(^)360 349 y Fy(N)427 361 y Fu(1)p Ft(;)p Fu(1)517 349 y Fv(\()p Fy(x;)14 b(k)s Fv(\))711 253 y Fn(\014)711 303 y(\014)711 353 y(\014)762 349 y Fs(\024)23 b Fy(C)k(Q)21 b(E)1084 364 y Ft(\013;)p Fu(1)p Ft(;)p Fu(1)p Ft(;)p Fu(\(1)p Fq(\000)p Ft(\015)t Fu(\))p Ft(=)p Fu(2)p Ft(;)p Fu(2)1552 349 y Fv(\()p Fy(x;)14 b(k)s Fv(\))20 b(+)e Fy(C)27 b(Q)21 b(F)2075 361 y Ft(\013;)p Fu(1)p Ft(;)p Fu(2)2228 349 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)261 436 y Fn(\014)261 486 y(\014)261 536 y(\014)289 531 y Fy(D)384 510 y Fv(^)360 531 y Fy(N)427 543 y Fu(1)p Ft(;)p Fu(1)517 531 y Fv(\()p Fy(x;)14 b(k)s Fv(\))711 436 y Fn(\014)711 486 y(\014)711 536 y(\014)762 531 y Fs(\024)23 b Fy(C)k(Q)21 b(g)1063 543 y Fq(\000)p Ft(\015)1157 531 y Fv(\()p Fy(k)s Fv(\))1281 464 y Fn(\000)1320 531 y Fy(E)1381 546 y Ft(\013;\015)t(;)p Fu(1)p Ft(;)p Fu(\(1)p Fq(\000)p Ft(\015)t Fu(\))p Ft(=)p Fu(2)p Ft(;)p Fu(2)1854 531 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(F)2203 543 y Ft(\013;\015)t(;)p Fu(2)2362 531 y Fv(\()p Fy(x;)c(k)s Fv(\))2556 464 y Fn(\001)2632 531 y Fm(,)332 619 y Fn(\014)332 668 y(\014)332 718 y(\014)384 693 y Fv(^)360 714 y Fy(N)427 726 y Fu(2)p Ft(;)p Fu(1)517 714 y Fv(\()p Fy(x;)g(k)s Fv(\))711 619 y Fn(\014)711 668 y(\014)711 718 y(\014)762 714 y Fs(\024)23 b Fy(C)k(Q)21 b(E)1084 726 y Ft(\013;)p Fu(2)p Ft(;)p Fu(0)p Ft(;)p Fu(1)p Ft(;)p Fu(2)1343 714 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(C)28 b(Q)21 b(F)1866 726 y Ft(\013;)p Fu(2)p Ft(;)p Fu(2)2019 714 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)261 801 y Fn(\014)261 851 y(\014)261 901 y(\014)289 897 y Fy(D)384 876 y Fv(^)360 897 y Fy(N)427 909 y Fu(2)p Ft(;)p Fu(1)517 897 y Fv(\()p Fy(x;)14 b(k)s Fv(\))711 801 y Fn(\014)711 851 y(\014)711 901 y(\014)762 897 y Fs(\024)23 b Fy(C)k(Q)21 b(E)1084 912 y Ft(\013;)p Fu(1)p Ft(;)p Fu(1)p Ft(;)p Fu(\(1)p Fq(\000)p Ft(\015)t Fu(\))p Ft(=)p Fu(2)p Ft(;)p Fu(2)1552 897 y Fv(\()p Fy(x;)14 b(k)s Fv(\))20 b(+)e Fy(C)27 b(Q)21 b(F)2075 909 y Ft(\013;)p Fu(1)p Ft(;)p Fu(2)2228 897 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)332 984 y Fn(\014)332 1034 y(\014)332 1083 y(\014)384 1058 y Fv(^)360 1079 y Fy(N)427 1091 y Fu(1)p Ft(;)p Fu(2)517 1079 y Fv(\()p Fy(x;)14 b(k)s Fv(\))711 984 y Fn(\014)711 1034 y(\014)711 1083 y(\014)762 1079 y Fs(\024)23 b Fy(C)k(Q)21 b(G)1088 1094 y Ft(\013;)p Fu(1)p Ft(=)p Fu(2)p Ft(;)p Fu(1)1309 1079 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fm(,)261 1167 y Fn(\014)261 1216 y(\014)261 1266 y(\014)289 1262 y Fy(D)384 1241 y Fv(^)360 1262 y Fy(N)427 1274 y Fu(1)p Ft(;)p Fu(2)517 1262 y Fv(\()p Fy(x;)14 b(k)s Fv(\))711 1167 y Fn(\014)711 1216 y(\014)711 1266 y(\014)762 1262 y Fs(\024)23 b Fy(C)k(Q)21 b(g)1063 1274 y Fq(\000)p Ft(\015)1157 1262 y Fv(\()p Fy(k)s Fv(\))p Fy(G)1332 1277 y Ft(\013;)p Fu(\(1)p Fq(\000)p Ft(\015)t Fu(\))p Ft(=)p Fu(2)p Ft(;)p Fu(1)1695 1262 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)332 1349 y Fn(\014)332 1399 y(\014)332 1449 y(\014)384 1424 y Fv(^)360 1445 y Fy(N)427 1457 y Fu(2)p Ft(;)p Fu(2)517 1445 y Fv(\()p Fy(x;)14 b(k)s Fv(\))711 1349 y Fn(\014)711 1399 y(\014)711 1449 y(\014)762 1445 y Fs(\024)23 b Fy(C)k(Q)21 b(G)1088 1460 y Ft(\013;)p Fu(1)p Ft(=)p Fu(2)p Ft(;)p Fu(1)1309 1445 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fm(,)261 1559 y Fn(\014)261 1609 y(\014)261 1659 y(\014)289 1654 y Fy(D)384 1633 y Fv(^)360 1654 y Fy(N)427 1666 y Fu(2)p Ft(;)p Fu(2)517 1654 y Fv(\()p Fy(x;)14 b(k)s Fv(\))711 1559 y Fn(\014)711 1609 y(\014)711 1659 y(\014)762 1654 y Fs(\024)23 b Fy(C)k(Q)21 b(G)1088 1669 y Ft(\013;)p Fu(\(1)p Fq(\000)p Ft(\015)t Fu(\))p Ft(=)p Fu(2)p Ft(;)p Fu(1)1451 1654 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(C)27 b(Q)2098 1538 y Fs(p)p 2167 1538 48 4 v 60 x Fy(x)p 1930 1635 452 4 v 1930 1711 a Fs(j)p Fy(k)s(x)p Fs(j)2069 1687 y Ft(\015)2104 1695 y Fo(2)2160 1711 y Fv(+)18 b Fs(j)p Fy(k)s(x)p Fs(j)2406 1654 y Fv(\()p Fy(H)2507 1666 y Ft(\013;\015)2605 1674 y Fo(2)2637 1666 y Ft(;)p Fu(1)2694 1654 y Fv(\()p Fy(x;)c(k)s Fv(\))19 b(+)f Fy(H)3059 1666 y Ft(\013;)p Fu(1)p Ft(;)p Fu(1)3212 1654 y Fv(\()p Fy(x;)c(k)s Fv(\)\))38 b Fy(:)-9 1882 y Fm(Pr)l(oof)o(.)31 b Fz(From)22 b(Condition)g(3.9,)h(there)e(e)o(xists)g(a)g(constant)g Fy(Q)g Fz(such)h(that)f(for)h(all)e Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)2709 1894 y Fu(+)2765 1882 y Fz(,)e(we)g(ha)n(v)o(e)930 2062 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(Q)e(x)1456 2027 y Fq(\000)p Fu(1)1566 2062 y Fy(w)1625 2074 y Ft(\013)1673 2062 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(Q)j Fs(j)p Fy(k)s Fs(j)g Fy(x)2216 2027 y Fu(1)p Ft(=)p Fu(2)2359 2062 y Fv(\026)-59 b Fy(w)2401 2074 y Fu(2)2439 2062 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fy(w)2692 2074 y Ft(\013)2741 2062 y Fv(\()p Fy(x;)g(k)s Fv(\))23 b Fz(,)930 2186 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(Q)e(x)1456 2152 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1633 2186 y Fy(w)1692 2198 y Ft(\013)1740 2186 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fz(,)930 2311 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(Q)e Fs(j)p Fy(k)s Fs(j)g Fy(x)1569 2276 y Fu(\()p Ft(\015)t Fq(\000)p Fu(1\))p Ft(=)p Fu(2)1837 2311 y Fy(w)1896 2323 y Ft(\013)p Fu(+1)2028 2311 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(Q)j Fs(j)p Fy(k)s Fs(j)g Fy(x)2571 2276 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2766 2311 y Fv(\026)-59 b Fy(w)2808 2323 y Fu(2)2846 2311 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fy(w)3099 2323 y Ft(\013)p Fq(\000)p Fu(1)3232 2311 y Fv(\()p Fy(x;)g(k)s Fv(\))24 b Fz(,)859 2435 y Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(Q)e(x)1456 2401 y Fu(\()p Ft(\015)t Fq(\000)p Fu(1\))p Ft(=)p Fu(2)1724 2435 y Fy(w)1783 2447 y Ft(\013)1831 2435 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(Q)j(x)2261 2401 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2434 2435 y Fv(\026)-59 b Fy(w)2476 2447 y Fu(2)2514 2435 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fy(w)2767 2447 y Ft(\013)p Fq(\000)p Fu(1)2901 2435 y Fv(\()p Fy(x;)g(k)s Fv(\))23 b Fz(,)289 2560 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)20 b Fv(+)e Fy(g)712 2572 y Fu(1)749 2560 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(Q)e Fs(j)p Fy(k)s Fs(j)g Fy(x)1569 2526 y Fu(\()p Ft(\015)t Fq(\000)p Fu(1\))p Ft(=)p Fu(2)1837 2560 y Fy(w)1896 2572 y Ft(\013)1944 2560 y Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b(+)f Fy(Q)j Fs(j)p Fy(k)s Fs(j)g Fy(x)2487 2526 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2682 2560 y Fv(\026)-59 b Fy(w)2724 2572 y Fu(2)2762 2560 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fy(w)3015 2572 y Ft(\013)p Fq(\000)p Fu(1)3148 2560 y Fv(\()p Fy(x;)g(k)s Fv(\))24 b Fz(,)289 2684 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)20 b Fv(+)e Fy(g)712 2696 y Fu(1)749 2684 y Fv(\()p Fy(k)s Fv(\))p Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(Q)e(x)1456 2650 y Fu(\()p Ft(\015)t Fq(\000)p Fu(1\))p Ft(=)p Fu(2)1724 2684 y Fy(w)1783 2696 y Ft(\013)1831 2684 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fz(,)477 2809 y Fs(j)6 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)19 b Fv(+)f Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(Q)e(x)1456 2775 y Fu(\()p Ft(\015)t Fq(\000)p Fu(1\))p Ft(=)p Fu(2)1703 2809 y Fy(w)1762 2821 y Ft(\013)1809 2809 y Fv(\()p Fy(x;)14 b(k)s Fv(\))20 b(+)e Fy(Q)i(x)2239 2775 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2434 2809 y Fv(\026)-59 b Fy(w)2476 2821 y Fu(2)2514 2809 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fy(w)2767 2821 y Ft(\013)p Fq(\000)p Fu(1)2901 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Fm(Pr)l(oof)o(.)32 b Fy(w)291 959 y Ft(\013)339 947 y Fz(,)22 b(and)h Fy(e)566 959 y Ft(\015)t(;\016)682 947 y Fz(are)e(e)n(v)o(en)h(functions)h(of)g Fy(k)s Fz(,)f(and)g(we)g(can)f (therefore)i(assume)f(without)g(loss)g(of)h(generality)e(that)g Fy(k)27 b Fs(\025)c Fv(0)p Fz(.)-9 1047 y(A)e(change)h(of)g(v)n(ariables)f (sho)n(ws)h(that)f(\(F)-7 b(.3\))23 b(follo)n(ws)f(from)912 1169 y Fn(Z)995 1190 y Fq(1)958 1358 y(\0001)1059 1282 y Fy(dk)1148 1248 y Fq(0)1408 1226 y Fy(x)1455 1196 y Fq(\000)p Ft(\015)t(=)p Fu(2)1635 1226 y Fv(+)c Fs(j)p Fy(k)1787 1196 y Fq(0)1811 1226 y Fs(j)1834 1196 y Fq(\000)p Ft(\015)p 1202 1263 932 4 v 1202 1339 a Fv(\(1)h(+)f Fs(j)p Fy(k)j Fs(\000)d Fy(k)1594 1315 y Fq(0)1617 1339 y Fs(j)p Fv(\))1672 1315 y Ft(\013)1741 1339 y Fv(\(1)g(+)g Fs(j)p Fy(k)1985 1315 y Fq(0)2009 1339 y Fs(j)p Fv(\))2064 1315 y Ft(\013)2107 1298 y Fc(0)2167 1282 y Fs(\024)2496 1226 y Fy(C)p 2264 1263 529 4 v 2264 1341 a Fv(1)g(+)g Fs(j)p Fy(k)s Fs(j)2499 1317 y Fp(min)p Fq(f)p 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3746 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1830 3690 y(1)p 1663 3727 376 4 v 1663 3803 a(\(1)19 b(+)f Fy(\016)s Fv(\))1911 3779 y Fu(1)p Fq(\000)p Ft(\015)2062 3746 y Fz(max)2223 3629 y Fn(\032)2285 3746 y Fv(1)p Fy(;)2587 3690 y Fv(1)p 2373 3727 470 4 v 2373 3803 a(\(1)g(+)g Fy(\016)s Fv(\))2620 3779 y Ft(\013)p Fq(\000)p Fu(1+)p Ft(\015)2852 3629 y Fn(\033)2950 3746 y Fy(;)960 3930 y Fs(\024)23 b Fy(C)k(x)1181 3895 y Ft(\015)t Fq(\000)p Fu(1)1331 3930 y Fy(w)1390 3942 y Ft(\013)1437 3930 y Fv(\()p Fy(x;)14 b(k)s Fv(\))22 b Fy(;)-9 4108 y Fz(as)f(claimed.)29 b(F)o(or)22 b(inequality)g(\(F)-7 b(.4\))23 b(we)e(use)g Fy(g)1379 4120 y Fq(\000)p Ft(\015)1473 4108 y Fv(\()p Fy(k)s Fv(\))j Fs(\024)f Fv(1)18 b(+)g Fs(j)p Fy(k)s Fs(j)1930 4078 y Fq(\000)p Ft(\015)2045 4108 y Fz(and)k(\256nd)164 4308 y Fv(\(\()p Fy(g)268 4320 y Fq(\000)p Ft(\015)362 4308 y Fy(w)421 4320 y Ft(\013)469 4308 y Fv(\))d Fs(\003)f Fy(e)619 4320 y Fu(0)p Ft(;\016)708 4308 y Fv(\))c(\()p Fy(x;)g(k)s Fv(\))24 b Fs(\024)e Fy(x)1106 4273 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1263 4308 y Fy(E)1324 4320 y Ft(\013;)p Fu(0)1424 4308 y Fv(\()p Fy(k)1502 4244 y Fs(p)p 1572 4244 48 4 v 1572 4308 a Fy(x;)14 b Fv(0)p Fy(;)1735 4244 y Fs(p)p 1804 4244 V 64 x Fy(x)p Fv(\(1)k(+)g Fy(\016)s(=x)p Fv(\))h(+)f Fy(x)2346 4246 y Fk(\015)s Fc(\000)p Fo(1)p 2346 4260 108 3 v 2386 4293 a(2)2468 4308 y Fy(E)2529 4320 y Ft(\013;\015)2634 4308 y Fv(\()p Fy(k)2712 4244 y Fs(p)p 2782 4244 48 4 v 2782 4308 a Fy(x;)c(k)2912 4244 y Fs(p)p 2981 4244 V 64 x Fy(x)q(;)3066 4244 y Fs(p)p 3135 4244 V 64 x Fy(x)p Fv(\(1)k(+)g Fy(\016)s(=x)p Fv(\)\))24 b Fy(:)-9 4486 y Fz(Therefore)e(by)g(Lemma)f(F)-7 b(.1)769 4712 y Fv(\(\()p Fy(g)873 4724 y Fq(\000)p Ft(\015)968 4712 y Fy(w)1027 4724 y Ft(\013)1075 4712 y Fv(\))18 b Fs(\003)g Fy(e)1224 4724 y Fu(0)p Ft(;\016)1313 4712 y Fv(\))c(\()p Fy(x;)g(k)s Fv(\))24 b Fs(\024)f Fy(C)k(x)1798 4678 y Ft(\015)t Fq(\000)p Fu(1)1926 4712 y Fy(w)1985 4724 y Ft(\013)2033 4712 y Fv(\()p Fy(x;)14 b(k)s Fv(\))g Fz(max)2402 4595 y Fn(\032)2464 4712 y Fv(1)p Fy(;)2679 4656 y Fv(1)p 2552 4693 295 4 v 2552 4769 a(\(1)19 b(+)f Fy(\016)s Fv(\))2800 4745 y Ft(\013)2857 4595 y Fn(\033)2955 4712 y Fy(;)1577 4896 y Fs(\024)23 b Fy(C)k(x)1798 4861 y Ft(\015)t Fq(\000)p Fu(1)1948 4896 y Fy(w)2007 4908 y Ft(\013)2054 4896 y Fv(\()p Fy(x;)14 b(k)s Fv(\))22 b Fy(;)-9 5074 y Fz(as)f(claimed.)p 425 5074 43 43 v -9 5174 a(The)g(follo)n(wing)h (lemma)f(is)g(v)o(ery)h(similar)e(to)h(Lemma)g(C.2.)-9 5336 y FA(Lemma)f(F)-9 b(.3)42 b Fm(Let)21 b Fy(h)i Fv(:)g(\012)745 5348 y Fu(+)823 5336 y Fs(!)g Fr(R)p Fm(.)31 b(Assume)21 b(that)h(ther)m(e)f (e)n(xist)f(constants)i Fy(H)28 b Fm(and)23 b Fy(\013)g Fs(\025)g Fv(0)e Fm(suc)o(h)g(that)h(for)f(all)g Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)3586 5348 y Fu(+)3641 5336 y Fm(,)1425 5515 y Fs(j)p Fy(h)p Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(H)28 b(w)1980 5527 y Ft(\013)2028 5515 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)1354 5639 y Fs(j)p Fy(D)r(h)p Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)e Fy(H)1921 5575 y Fs(p)p 1990 5575 48 4 v 64 x Fy(x)g(w)2118 5651 y Ft(\013)2166 5639 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)1838 5909 y Fz(42)p eop %%Page: 43 43 bop -9 145 a Fm(Then)22 b(ther)m(e)e(e)n(xists)h(a)g(constant)i Fy(C)k Fm(suc)o(h)22 b(that)f(for)h(all)f Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)1959 157 y Fu(+)2014 145 y Fm(,)e(we)g(have)984 237 y Fn(\014)984 287 y(\014)984 337 y(\014)1011 333 y Fy(h)p Fv(\()p Fy(x;)14 b(k)s Fv(\))20 b Fs(\000)e Fy(e)1395 299 y Fq(\000)p Ft(k)1483 274 y Fo(2)1515 299 y Ft(x)1557 333 y Fy(h)p Fv(\()p Fy(x;)c Fv(0\))1795 237 y Fn(\014)1795 287 y(\014)1795 337 y(\014)1846 333 y Fs(\024)23 b Fy(C)k(H)h Fs(j)p Fy(k)s Fs(j)2209 269 y(p)p 2278 269 48 4 v 64 x Fy(x)22 b(w)2406 345 y Ft(\013)p Fu(+1)2538 333 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)832 b Fz(\(F)-7 b(.7\))-9 517 y Fm(in)21 b(particular)-9 b(,)22 b(if)e Fy(h)h Fm(is)g(odd)i(in)e Fy(k)s Fm(,)g(we)g(have)1300 680 y Fs(j)p Fy(h)p Fv(\()p Fy(k)s Fv(\))p Fs(j)j(\024)e Fy(C)28 b(H)g Fs(j)p Fy(k)s Fs(j)1891 616 y(p)p 1960 616 V 64 x Fy(x)22 b(w)2088 692 y Ft(\013)p Fu(+1)2219 680 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)1149 b Fz(\(F)-7 b(.8\))-9 844 y Fm(Pr)l(oof)o(.)46 b Fz(As)27 b(is)e(easily)h(seen,)i(\(F)-7 b(.8\))28 b(for)f(an)f(odd)i (function)g(follo)n(ws)f(from)g(\(F)-7 b(.7\),)29 b(because)e(an)f(odd)i (function)f(with)f(bounded)-9 943 y(deri)n(v)n(ati)n(v)o(e)c(v)n(anishes)i (at)f Fy(k)29 b Fv(=)c(0)p Fz(.)37 b(T)-7 b(o)24 b(pro)o(v)o(e)h(\(F)-7 b(.7\),)26 b(we)d(\256rst)h(mak)o(e)g(the)f(follo)n(wing)i(observ)n(ations.) 38 b(There)24 b(e)o(xists)f(a)g(constant)-9 1043 y Fy(C)k Fz(such)22 b(that)f(for)h(all)e Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)f Fv(\012)985 1055 y Fu(+)1041 1043 y Fz(,)e Fv(0)i Fs(\024)f(j)p Fy(\030)t Fs(j)i(\024)e(j)p Fy(k)s Fs(j)p Fz(,)1053 1136 y Fn(\014)1053 1186 y(\014)1053 1236 y(\014)1080 1231 y Fy(@)1124 1243 y Ft(\030)1161 1231 y Fy(e)1200 1197 y Fq(\000)p Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))1424 1172 y Fo(2)1456 1197 y Ft(x)1498 1136 y Fn(\014)1498 1186 y(\014)1498 1236 y(\014)1549 1231 y Fs(\024)g Fv(2)p Fs(j)p Fy(k)f Fs(\000)d Fy(\030)t Fs(j)k Fy(x)f(e)2040 1197 y Fq(\000)p Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))2264 1172 y Fo(2)2297 1197 y Ft(x)2362 1231 y Fs(\024)i Fy(C)2536 1168 y Fs(p)p 2605 1168 V 63 x Fy(x)f(e)2713 1197 y Fq(\000)p Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))2937 1172 y Fo(2)2970 1197 y Ft(x=)p Fu(2)3102 1231 y Fz(,)648 1385 y Fv(\(1)c(+)g Fs(j)p Fy(k)s Fs(j)915 1325 y(p)p 984 1325 V 60 x Fy(x)q Fv(\))1064 1355 y Ft(\013)p 648 1422 464 4 v 651 1499 a Fv(\(1)g(+)g Fs(j)p Fy(\030)t Fs(j)912 1439 y(p)p 981 1439 48 4 v 60 x Fy(x)q Fv(\))1061 1475 y Ft(\013)1121 1441 y Fy(e)1160 1407 y Fq(\000)p Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))1384 1382 y Fo(2)1417 1407 y Ft(x=)p Fu(2)1549 1441 y Fs(\024)k Fy(C)28 b(e)1762 1407 y Fq(\000)p Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))1986 1382 y Fo(2)2019 1407 y Ft(x=)p Fu(4)2151 1441 y Fy(:)-9 1649 y Fz(Using)22 b(this)f(gi)n(v)o(es)731 1789 y Fn(\014)731 1839 y(\014)731 1889 y(\014)759 1885 y Fy(h)p Fv(\()p Fy(x;)14 b(k)s Fv(\))19 b Fs(\000)f Fy(e)1142 1850 y Fq(\000)p Ft(k)1230 1825 y Fo(2)1284 1850 y Ft(x)1326 1885 y Fy(h)p Fv(\()p Fy(x;)c Fv(0\))1564 1789 y Fn(\014)1564 1839 y(\014)1564 1889 y(\014)1615 1885 y Fs(\024)1702 1739 y Fn(\014)1702 1789 y(\014)1702 1839 y(\014)1702 1889 y(\014)1702 1939 y(\014)1730 1772 y(Z)1813 1792 y Ft(k)1776 1960 y Fu(0)1832 1885 y Fy(d\030)26 b(@)1981 1897 y Ft(\030)2031 1792 y Fn(\020)2081 1885 y Fy(e)2120 1850 y Fq(\000)p Fu(\()p Ft(k)q Fq(\000)p Ft(\030)r Fu(\))2344 1825 y Fo(2)2376 1850 y Ft(x)2418 1885 y Fy(h)p Fv(\()p Fy(x;)14 b(\030)t Fv(\))2654 1792 y Fn(\021)2705 1739 y(\014)2705 1789 y(\014)2705 1839 y(\014)2705 1889 y(\014)2705 1939 y(\014)1615 2160 y Fs(\024)22 b Fy(C)28 b(H)1886 2096 y Fs(p)p 1955 2096 V 64 x Fy(x)2016 2047 y Fn(Z)2099 2067 y Fq(j)p Ft(k)q Fq(j)2062 2235 y Fu(0)2158 2160 y Fy(d\030)2280 2104 y(e)2319 2073 y Fq(\000)p Fu(\()p Fq(j)p Ft(k)q Fq(j\000)p Ft(\030)r Fu(\))2583 2048 y Fo(2)2615 2073 y Ft(x=)p Fu(2)p 2273 2141 458 4 v 2273 2217 a Fv(\(1)18 b(+)g Fs(j)p Fy(\030)t Fs(j)2534 2157 y(p)p 2603 2157 48 4 v 60 x Fy(x)q Fv(\))2683 2193 y Ft(\013)1615 2407 y Fs(\024)k Fy(C)28 b(H)1886 2343 y Fs(p)p 1955 2343 V 64 x Fy(x)22 b(w)2083 2419 y Ft(\013)2130 2407 y Fv(\()p Fy(x;)14 b(k)s Fv(\))2338 2294 y Fn(Z)2422 2315 y Fq(j)p Ft(k)q Fq(j)2385 2483 y Fu(0)2481 2407 y Fy(d\030)25 b(e)2624 2373 y Fq(\000)p Fu(\()p Fq(j)p Ft(k)q Fq(j\000)p Ft(\030)r Fu(\))2888 2348 y Fo(2)2920 2373 y Ft(x=)p Fu(4)1615 2656 y Fs(\024)d Fy(C)28 b(H)g(w)1945 2668 y Ft(\013)1993 2656 y Fv(\()p Fy(x;)14 b(k)s Fv(\))2201 2543 y Fn(Z)2284 2563 y Fq(j)p Ft(k)q Fq(j)2360 2522 y(p)p 2415 2522 38 3 v 41 x Ft(x)2247 2732 y Fu(0)2435 2656 y Fy(d\030)26 b(e)2579 2622 y Fq(\000)p Ft(\030)2663 2597 y Fo(2)2695 2622 y Ft(=)p Fu(4)1615 2887 y Fs(\024)c Fy(C)28 b(H)g(w)1945 2899 y Ft(\013)1993 2887 y Fv(\()p Fy(x;)14 b(k)s Fv(\))2269 2831 y Fs(j)p Fy(k)s Fs(j)2361 2771 y(p)p 2430 2771 48 4 v 60 x Fy(x)p 2197 2868 352 4 v 2197 2945 a Fv(1)k(+)g Fs(j)p Fy(k)s Fs(j)2432 2885 y(p)p 2501 2885 48 4 v 60 x Fy(x)1615 3069 y Fs(\024)k Fy(C)28 b(H)g Fs(j)p Fy(k)s Fs(j)1978 3005 y(p)p 2047 3005 V 64 x Fy(x)22 b(w)2175 3081 y Ft(\013)p Fu(+1)2307 3069 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fy(:)-9 3233 y Fz(This)e(achie)n(v)o (es)f(the)h(proof.)p 855 3233 43 43 v -9 3332 a(The)g(ne)o(xt)g(lemma)g(gi)n (v)o(es)g(miscellaneous)g(properties)h(we)f(will)f(use)i(belo)n(w)-6 b(.)-9 3483 y FA(Lemma)20 b(F)-9 b(.4)42 b Fm(Let)19 b Fy(\013)k Fs(\025)g Fv(1)c Fm(and)i Fy(f)t(;)14 b(g)25 b Fv(:)e(\012)1191 3495 y Fu(+)1269 3483 y Fs(!)g Fr(R)p Fm(.)30 b(Ther)m(e)20 b(e)n(xists)e(a)i(constant)g Fy(C)26 b Fm(suc)o(h)19 b(that)h(for)f(all)g Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fs(2)g Fv(\012)3282 3495 y Fu(+)3337 3483 y Fm(,)c(we)e(have)i(for)-9 3583 y(the)h(multiplication)g (oper)o(ator)h(\(by)g Fy(k)s Fm(\))f Fy(K)27 b Fm(and)c(the)e(derivative)f Fy(D)1406 3746 y(K)6 b Fv(\()p Fy(f)26 b Fs(\003)18 b Fy(g)s Fv(\))23 b(=)g Fy(K)6 b(f)26 b Fs(\003)18 b Fy(g)j Fv(+)d Fy(f)27 b Fs(\003)18 b Fy(K)6 b(g)25 b Fm(,)1455 3871 y Fy(D)r(K)6 b Fv(\()p Fy(f)j Fv(\))23 b(=)g Fy(f)k Fv(+)18 b Fy(K)6 b(D)r Fv(\()p Fy(f)j Fv(\))23 b Fm(,)1293 4038 y Fs(j)p Fy(K)6 b(w)1452 4050 y Ft(\013)1499 4038 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)1881 3981 y Fy(C)p 1838 4018 152 4 v 1838 4096 a(x)1885 4072 y Fu(1)p Ft(=)p Fu(2)2000 4038 y Fy(w)2059 4050 y Ft(\013)p Fq(\000)p Fu(1)2191 4038 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)-9 4226 y Fm(Pr)l(oof)o(.)31 b Fz(This)21 b(is)g(elementary)-6 b(.)p 926 4226 43 43 v -9 4463 a Fd(F)-11 b(.2)102 b(Pr)n(oof)26 b(of)g(Pr)n(oposition)f(4.4)-9 4620 y Fz(The)e(proof)i(consists)e(only)h(in)f (applying)i(Proposition)f(F)-7 b(.2.)38 b(In)24 b(order)g(to)f(simplify)g (the)g(notation)h(we)f(omit)g(in)g(the)g(follo)n(wing)-9 4719 y(subsections)g(the)g(ar)n(guments)h Fy(x)f Fz(and)h Fy(k)s Fz(,)f(whene)n(v)o(er)g(no)h(confusion)h(is)d(possible.)36 b(W)-7 b(e)23 b(be)o(gin)g(with)g(tw)o(o)h(lemmas)e(which)h(gi)n(v)o(e)-9 4819 y(bounds)g(on)f Fv(\()p Fy(u;)14 b(v)s Fv(\))24 b Fs(2)f(B)g Fz(and)f(on)h(the)e(function)h Fr(u)1468 4831 y Ft(E)1545 4819 y Fz(of)g(Lemma)f(1.3.)-9 4970 y FA(Lemma)f(F)-9 b(.5)42 b Fm(Let)31 b Fr(v)38 b Fv(=)e(\()p Fy(u;)14 b(v)s Fv(\))37 b Fs(2)g(B)s Fm(,)c(then)f(ther)m(e)f(e)n(xists)f(a)i(constant)g Fy(C)37 b Fm(suc)o(h)32 b(that)g(for)f(all)g Fv(0)36 b Fs(\024)g Fy(\013)3121 4939 y Fq(0)3182 4970 y Fs(\024)g Fy(\013)p Fm(,)e(and)f(for)e (all)-9 5069 y Fv(\()p Fy(x;)14 b(k)s Fv(\))23 b Fs(2)h Fv(\012)347 5081 y Fu(+)402 5069 y Fm(,)d(we)g(have)1232 5233 y Fs(j)5 b Fv(^)-47 b Fy(u)p Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g(k)p Fr(v)q Fs(k)14 b Fy(w)1839 5245 y Ft(\013)1882 5229 y Fc(0)1909 5233 y Fv(\()p Fy(x;)g(k)s Fv(\))24 b Fm(,)1161 5380 y Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(u)p Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g(k)p Fr(v)q Fs(k)14 b Fy(x)1837 5314 y Fo(1)p Fc(\000)p Fk(\015)1941 5326 y Fo(1)p 1837 5332 137 3 v 1891 5366 a(2)2009 5380 y Fy(g)2049 5392 y Fq(\000)p Ft(\015)2136 5400 y Fo(1)2172 5380 y Fv(\()p Fy(k)s Fv(\))22 b Fy(w)2363 5392 y Ft(\013)2406 5376 y Fc(0)2433 5380 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fm(,)1236 5515 y Fs(j)s Fv(^)-45 b Fy(v)t Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g(k)p Fr(v)q Fs(k)14 b Fy(x)1827 5480 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2005 5515 y Fy(w)2064 5527 y Ft(\013)2107 5510 y Fc(0)2134 5515 y Fv(\()p Fy(x;)g(k)s Fv(\))24 b Fm(,)1089 5639 y Fs(j)p Fy(K)6 b(D)f Fv(^)-45 b Fy(v)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)23 b(\024)g Fy(C)41 b Fs(k)p Fr(v)q Fs(k)35 b Fy(x)1948 5605 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2126 5639 y Fy(w)2185 5651 y Ft(\013)2228 5635 y Fc(0)2251 5651 y Fq(\000)p Fu(1)2340 5639 y Fv(\()p Fy(x;)14 b(k)s Fv(\))24 b Fy(:)1028 b Fz(\(F)-7 b(.9\))1838 5909 y(43)p eop %%Page: 44 44 bop -9 145 a Fm(Pr)l(oof)o(.)37 b Fz(Only)25 b(inequality)e(\(F)-7 b(.9\))25 b(requires)f(a)f(proof,)k(the)c(others)h(follo)n(w)g(directly)f (from)i(the)e(de\256nition)h(of)g Fs(B)s Fz(.)37 b(F)o(or)24 b(\(F)-7 b(.9\),)26 b(we)-9 244 y(note)21 b(that)106 459 y Fs(j)p Fy(K)6 b(D)f Fv(^)-45 b Fy(v)s Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fs(j)24 b(\024)f Fy(w)708 471 y Ft(\013)p Fq(\000)p Fu(1)840 459 y Fv(\()p Fy(x;)14 b(k)s Fv(\))1167 403 y Fs(j)p Fy(k)1236 343 y Fs(p)p 1305 343 48 4 v 60 x Fy(x)p Fs(j)p 1044 440 452 4 v 1044 516 a(j)p Fy(k)s(x)p Fs(j)1183 492 y Ft(\015)1218 500 y Fo(2)1274 516 y Fv(+)k Fs(j)p Fy(k)s(x)p Fs(j)1530 459 y(\024)k Fy(w)1676 471 y Ft(\013)1719 455 y Fc(0)1743 471 y Fq(\000)p Fu(1)1832 459 y Fv(\()p Fy(x;)14 b(k)s Fv(\))p Fy(x)2073 424 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2273 459 y Fz(sup)2265 531 y Ft(\030)r Fq(2)p Fl(R)2551 403 y Fs(j)p Fy(\030)t Fs(j)p 2422 440 346 4 v 2422 516 a(j)p Fy(\030)t Fs(j)2508 492 y Ft(\015)2543 500 y Fo(2)2598 516 y Fv(+)k Fs(j)p Fy(\030)t Fs(j)2800 459 y(\024)23 b Fy(C)k(x)3021 424 y Fq(\000)p Fu(1)p 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Fs(k)p Fv(\))j Fs(k)p Fr(v)q Fs(k)g Fy(x)2199 5524 y Fk(\015)2230 5536 y Fo(1)2263 5524 y Fc(\000)p Fo(1)p 2199 5542 137 3 v 2253 5575 a(2)2371 5589 y Fy(w)2430 5601 y Ft(\013)2501 5589 y Fy(:)1043 b Fz(\(F)-7 b(.17\))1838 5909 y(44)p eop %%Page: 45 45 bop -9 145 a Fz(W)-7 b(e)21 b(ha)n(v)o(e)h Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(q)433 157 y Fu(0)470 145 y Fs(j)23 b(\024)g(j)p Fy(D)8 b Fv(^)-48 b Fy(u)18 b Fs(\003)j Fv(^)-45 b Fy(v)s Fs(j)19 b Fv(+)f Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(u)1134 157 y Ft(E)1208 145 y Fs(\003)21 b Fv(^)-45 b Fy(v)s Fs(j)18 b Fv(+)g Fs(j)p Fy(D)8 b Fv(^)-48 b Fy(u)19 b Fs(\003)i Fv(^)-46 b Fy(v)1695 157 y Ft(E)1752 145 y Fs(j)p Fz(,)21 b(and)h(since)597 322 y Fs(j)p Fv(\()p Fy(D)8 b Fv(^)-48 b Fy(u)18 b Fs(\003)j Fv(^)-45 b Fy(v)s Fv(\))q Fs(j)23 b(\024)f Fy(C)28 b Fs(k)p Fr(v)q Fs(k)1280 288 y Fu(2)1338 322 y Fy(x)1385 288 y Fq(\000)p Ft(\015)1472 296 y Fo(1)1505 288 y Ft(=)p Fu(2)1576 322 y Fv(\()p Fy(g)1648 334 y Fq(\000)p Ft(\015)1735 342 y Fo(1)1771 322 y 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b(\256rst)g(sho)n(w)i(that)1222 1953 y Fn(\014)1222 2003 y(\014)1222 2052 y(\014)1250 2026 y Fv(^)1249 2048 y Fy(h)1297 2060 y Fu(1)1334 1953 y Fn(\014)1334 2003 y(\014)1334 2052 y(\014)1385 2048 y Fs(\024)g Fy(C)k Fv(\()p Fy(")19 b Fv(+)f Fs(k)p Fr(v)q Fs(k)p Fv(\))j Fs(k)p Fr(v)q Fs(k)g Fy(x)2123 2014 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2301 2048 y Fy(w)2360 2060 y Ft(\013)p Fq(\000)p Fu(1)2516 2048 y Fy(:)1028 b Fz(\(F)-7 b(.19\))-9 2284 y(W)g(e)21 b(ha)n(v)o(e)302 2189 y Fn(\014)302 2239 y(\014)302 2289 y(\014)330 2262 y Fv(^)329 2284 y Fy(h)377 2296 y Fu(1)414 2189 y Fn(\014)414 2239 y(\014)414 2289 y(\014)465 2284 y Fs(\024)i(j)s Fv(^)-45 b Fy(v)21 b Fs(\003)d Fy(K)11 b Fv(^)-47 b Fy(u)o Fs(j)19 b Fv(+)f Fs(j)s Fv(^)-45 b Fy(v)22 b Fs(\003)c Fy(K)11 b Fv(^)-47 b Fy(u)1216 2296 y Ft(E)1271 2284 y Fs(j)18 b Fv(+)g Fs(j)s Fv(^)-45 b Fy(v)1458 2296 y Ft(E)1533 2284 y Fs(\003)18 b Fy(K)11 b Fv(^)-47 b Fy(u)o Fs(j)p Fz(,)22 b(and)f(since)743 2503 y Fs(j)p Fv(\()s(^)-45 b Fy(v)22 b Fs(\003)c 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4541 y Ft(\013)p Fu(+1)2057 4529 y Fs(\024)i Fy(C)k Fv(\()p Fy(")19 b Fv(+)f Fs(k)p Fr(v)q Fs(k)p Fv(\))j Fs(k)p Fr(v)q Fs(k)g Fy(x)2805 4464 y Fk(\015)2836 4476 y Fo(1)2869 4464 y Fc(\000)p Fo(1)p 2805 4482 V 2860 4515 a(2)2977 4529 y Fs(j)p Fy(k)s Fs(j)h Fy(w)3150 4541 y Ft(\013)3220 4529 y Fz(,)519 4677 y Fs(j)p Fy(K)6 b(D)i Fv(^)-48 b Fy(q)727 4689 y Fu(0)764 4677 y Fs(j)23 b(\024)g Fy(C)k Fv(\()p Fy(")18 b Fv(+)h Fs(k)p Fr(v)q Fs(k)p Fv(\))i Fs(k)p Fr(v)q Fs(k)g Fy(x)1558 4611 y Fk(\015)1589 4623 y Fo(1)1622 4611 y Fc(\000)p Fo(1)p 1558 4629 V 1612 4662 a(2)1730 4677 y Fs(j)p Fy(k)s Fs(j)g Fy(w)1902 4689 y Ft(\013)1973 4677 y Fy(:)-9 4854 y Fz(By)g(Lemma)g(F)-7 b(.3,)22 b(to)g(pro)o(v)o(e)g(\(F)-7 b(.20\),)24 b(it)c(is)h(suf)n(\256cient)g(to)h(pro)o(v)o(e)g(that)1278 5032 y Fs(j)1302 5010 y Fv(~)1301 5032 y Fy(h)p Fs(j)h(\024)g Fy(C)k Fv(\()p Fy(")19 b Fv(+)f Fs(k)p Fr(v)q Fs(k)p Fv(\))j Fs(k)p Fr(v)q Fs(k)g Fy(x)2133 4998 y Fq(\000)p Fu(1)2244 5032 y Fy(w)2303 5044 y Ft(\013)p 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Fq(\000)p Fu(1)2750 5660 y Fv(\)\))1838 5909 y Fz(45)p eop %%Page: 46 46 bop 991 182 a Fs(\024)23 b Fy(C)k Fs(k)p Fr(v)q Fs(k)21 b Fy(")g(x)1428 148 y Fq(\000)p Fu(1)1553 65 y Fn(\022)1614 182 y Fv(1)d(+)1959 126 y(1)p 1767 163 426 4 v 1767 239 a(\()p Fy(x)h Fv(+)f Fy(\016)s Fv(\))2020 215 y Fu(1)p Fq(\000)p Ft(\015)2140 223 y Fk(E)2202 65 y Fn(\023)2277 182 y Fy(w)2336 194 y Ft(\013)p Fq(\000)p Fu(1)2492 182 y Fs(\024)23 b Fy(C)k Fs(k)p Fr(v)q Fs(k)21 b Fy(")g(x)2929 148 y Fq(\000)p Fu(1)3040 182 y Fy(w)3099 194 y Ft(\013)p Fq(\000)p Fu(1)3255 182 y Fz(,)488 387 y Fs(j)p Fv(\()p Fy(K)9 b Fv(^)-45 b Fy(v)660 399 y Ft(E)734 387 y Fs(\003)18 b Fy(D)8 b Fv(^)-48 b Fy(u)p Fv(\))p Fs(j)23 b(\024)g Fy(C)k Fs(k)p Fr(v)q Fs(k)21 b Fy(")g(x)1428 353 y Fq(\000)1490 322 y Fo(1+)p Fk(\015)1592 334 y Fo(1)p 1490 340 135 3 v 1543 373 a(2)1660 387 y Fv(\()p Fy(g)1732 399 y Fq(\000)p Ft(\015)1819 407 y Fo(1)1855 387 y Fy(w)1914 399 y Ft(\013)1980 387 y Fs(\003)d Fy(e)2079 399 y Fu(0)p Ft(;\016)2168 387 y Fv(\))991 537 y Fs(\024)23 b Fy(C)k Fs(k)p Fr(v)q Fs(k)21 b Fy(")g(x)1438 472 y Fk(\015)1469 484 y Fo(1)1502 472 y Fc(\000)p Fo(3)p 1439 490 137 3 v 1493 523 a(2)1589 537 y Fy(w)1648 549 y Ft(\013)1719 537 y Fs(\024)i Fy(C)k Fs(k)p Fr(v)q Fs(k)21 b Fy(")g(x)2156 503 y Fq(\000)p Fu(1)2267 537 y Fy(w)2326 549 y Ft(\013)p Fq(\000)p Fu(1)2482 537 y Fz(,)-9 692 y(this)g(completes)f(the)h(proof)j(of)e(\(F)-7 b(.21\).)33 b(T)-7 b(o)22 b(pro)o(v)o(e)g(\(F)-7 b(.22\),)24 b(we)d(use)571 847 y Fy(D)643 825 y Fv(~)642 847 y Fy(h)i Fv(=)g Fy(D)r(K)9 b Fv(^)-45 b Fy(v)21 b Fs(\003)d Fy(D)8 b Fv(^)-48 b Fy(u)18 b Fv(+)g Fy(D)r(K)9 b Fv(^)-45 b Fy(v)21 b Fs(\003)d Fy(D)8 b Fv(^)-48 b Fy(u)1678 859 y Ft(E)1752 847 y Fv(+)18 b Fy(D)r(K)9 b Fv(^)-45 b Fy(v)2023 859 y Ft(E)2098 847 y Fs(\003)18 b Fy(D)8 b Fv(^)-48 b Fy(u)713 972 y Fv(=)23 b(\()s(^)-45 b Fy(v)22 b Fs(\003)c Fy(D)8 b Fv(^)-48 b Fy(u)18 b Fv(+)j(^)-45 b Fy(v)22 b Fs(\003)c Fy(D)8 b Fv(^)-48 b Fy(u)1416 984 y Ft(E)1471 972 y Fv(\))19 b(+)f(\()p Fy(K)6 b(D)f Fv(^)-45 b Fy(v)22 b Fs(\003)17 b Fy(D)8 b Fv(^)-48 b Fy(u)18 b Fv(+)h Fy(K)6 b(D)f Fv(^)-45 b Fy(v)21 b Fs(\003)d Fy(D)8 b Fv(^)-48 b Fy(u)2515 984 y Ft(E)2570 972 y Fv(\))19 b(+)f Fy(D)r(K)9 b Fv(^)-45 b Fy(v)2892 984 y Ft(E)2966 972 y Fs(\003)18 b Fy(D)8 b Fv(^)-48 b Fy(u)23 b Fz(,)-9 1127 y(and)e(therefore)497 1282 y Fs(j)p Fy(D)592 1260 y Fv(~)591 1282 y Fy(h)p Fs(j)i(\024)f(j)s Fv(^)-45 b Fy(v)22 b Fs(\003)c Fy(D)8 b Fv(^)-48 b Fy(u)p Fs(j)18 b Fv(+)g Fs(j)s Fv(^)-45 b Fy(v)22 b Fs(\003)c Fy(D)8 b Fv(^)-48 b Fy(u)1424 1294 y Ft(E)1480 1282 y Fs(j)18 b Fv(+)g Fs(j)p Fy(K)6 b(D)f Fv(^)-45 b Fy(v)21 b Fs(\003)d Fy(D)8 b Fv(^)-48 b Fy(u)p Fs(j)18 b Fv(+)g Fs(j)p Fy(K)6 b(D)f Fv(^)-45 b Fy(v)22 b Fs(\003)c Fy(D)8 b Fv(^)-48 b Fy(u)2551 1294 y Ft(E)2606 1282 y Fs(j)19 b Fv(+)f Fs(j)p Fy(D)r(K)9 b Fv(^)-45 b Fy(v)2942 1294 y Ft(E)3016 1282 y Fs(\003)18 b Fy(D)8 b Fv(^)-48 b Fy(u)p Fs(j)23 b Fy(:)-9 1437 y Fz(The)e(functions)26 b Fv(^)-46 b Fy(v)22 b Fs(\003)c Fy(D)8 b Fv(^)-48 b Fy(u)21 b Fz(and)k Fv(^)-45 b Fy(v)21 b Fs(\003)d Fy(D)8 b Fv(^)-48 b Fy(u)1127 1449 y Ft(E)1204 1437 y Fz(ha)n(v)o(e)22 b(already)f(been)h(bounded)i (before.)32 b(Namely)-6 b(,)969 1592 y Fs(j)t Fv(^)-46 b Fy(v)22 b Fs(\003)c Fy(D)8 b Fv(^)-48 b Fy(u)p Fs(j)18 b Fv(+)g Fs(j)s Fv(^)-45 b Fy(v)22 b Fs(\003)c Fy(D)8 b Fv(^)-48 b Fy(u)1621 1604 y Ft(E)1676 1592 y Fs(j)24 b(\024)e Fy(C)28 b Fv(\()p Fy(")18 b Fv(+)g Fs(k)p Fr(v)q Fs(k)p Fv(\))j Fs(k)p Fr(v)q Fs(k)g Fy(x)2460 1557 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2638 1592 y Fy(w)2697 1604 y Ft(\013)2768 1592 y Fy(:)-9 1746 y Fz(And)h(\256nally)-6 b(,)22 b(we)g(ha)n(v)o(e)438 1901 y Fs(j)p Fv(\()p Fy(K)6 b(D)f Fv(^)-45 b Fy(v)21 b Fs(\003)d Fy(D)8 b Fv(^)-48 b Fy(u)p Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fs(k)p Fr(v)q Fs(k)1269 1867 y Fu(2)1327 1901 y Fy(x)1374 1867 y Fq(\000)p Ft(\015)t(=)p Fu(2)1536 1901 y Fv(\()p Fy(g)1608 1913 y Fq(\000)p Ft(\015)1695 1921 y Fo(1)1731 1901 y Fy(w)1790 1913 y Ft(\013)1856 1901 y Fs(\003)18 b Fy(w)1975 1913 y Ft(\013)p Fq(\000)p Fu(1)2108 1901 y Fv(\))23 b Fs(\024)g Fy(C)k Fs(k)p Fr(v)q Fs(k)2472 1867 y Fu(2)2531 1901 y Fy(x)2578 1867 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2756 1901 y Fy(w)2815 1913 y Ft(\013)p Fq(\000)p Fu(1)2970 1901 y Fz(,)382 2036 y Fs(j)p Fv(\()p Fy(K)6 b(D)f Fv(^)-45 b Fy(v)21 b Fs(\003)d Fy(D)8 b Fv(^)-48 b Fy(u)825 2048 y Ft(E)881 2036 y Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fs(k)p Fr(v)q Fs(k)21 b Fy(")g(x)1397 2002 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1588 2036 y Fv(\()q Fy(x)g Fv(\()p Fy(e)1760 2048 y Fu(0)p Ft(;\016)1868 2036 y Fs(\003)d Fy(w)1987 2048 y Ft(\013)p Fq(\000)p Fu(1)2120 2036 y Fv(\))g(+)g(\()p Fy(e)2324 2048 y Fq(\000)p Ft(\015)2411 2056 y Fk(E)2459 2048 y Ft(;\016)2534 2036 y Fs(\003)g Fy(w)2653 2048 y Ft(\013)p Fq(\000)p Fu(1)2785 2036 y Fv(\)\))959 2214 y Fs(\024)23 b Fy(C)28 b Fs(k)p Fr(v)q Fs(k)21 b Fy(")g(x)1397 2179 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1588 2097 y Fn(\022)1649 2214 y Fv(1)d(+)1994 2157 y(1)p 1802 2195 426 4 v 1802 2271 a(\()p Fy(x)i Fv(+)e Fy(\016)s Fv(\))2056 2247 y Fu(1)p Fq(\000)p Ft(\015)2176 2255 y Fk(E)2238 2097 y Fn(\023)2313 2214 y Fy(w)2372 2226 y Ft(\013)p Fq(\000)p Fu(1)2528 2214 y Fs(\024)k Fy(C)28 b Fs(k)p Fr(v)q Fs(k)21 b Fy(")g(x)2965 2179 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)3143 2214 y Fy(w)3202 2226 y Ft(\013)p Fq(\000)p Fu(1)3357 2214 y Fz(,)385 2419 y Fs(j)p Fv(\()p Fy(D)r(K)9 b Fv(^)-45 b Fy(v)628 2431 y Ft(E)702 2419 y Fs(\003)18 b Fy(D)8 b Fv(^)-48 b Fy(u)p Fv(\))p Fs(j)23 b(\024)g Fy(C)28 b Fs(k)p Fr(v)q Fs(k)21 b Fy(")g(x)1407 2353 y Fo(1)p Fc(\000)p Fk(\015)1511 2365 y Fo(1)p 1407 2371 137 3 v 1461 2405 a(2)1579 2419 y Fv(\()p Fy(g)1651 2431 y Fq(\000)p Ft(\015)1738 2439 y Fo(1)1774 2419 y Fy(w)1833 2431 y Ft(\013)1899 2419 y Fs(\003)d Fy(e)1998 2431 y Fu(0)p Ft(;\016)2087 2419 y Fv(\))24 b Fs(\024)e Fy(C)28 b Fs(k)p Fr(v)q Fs(k)21 b Fy(")g(x)2590 2353 y Fk(\015)2621 2365 y Fo(1)2653 2353 y Fc(\000)p Fo(1)p 2590 2371 V 2644 2405 a(2)2762 2419 y Fy(w)2821 2431 y Ft(\013)2892 2419 y Fy(:)-9 2574 y Fz(This)g(completes)g (the)g(proof)i(of)f(\(F)-7 b(.22\),)24 b(so)d(that)g(the)g(proof)j(of)e (\(4.6\))h(and)f(\(4.7\))h(is)d(completed.)-9 2793 y FA(F)-9 b(.2.3)86 b(Bound)22 b(on)619 2771 y Fv(^)619 2793 y Fy(h)667 2805 y Fu(2)725 2793 y FA(and)g Fy(D)955 2771 y Fv(^)954 2793 y Fy(h)1002 2805 y Fu(2)-9 2950 y Fz(W)-7 b(e)21 b(\256rst)g(sho)n(w)i(that) 1208 3009 y Fn(\014)1208 3059 y(\014)1208 3109 y(\014)1236 3083 y Fv(^)1235 3105 y Fy(h)1283 3117 y Fu(2)1320 3009 y Fn(\014)1320 3059 y(\014)1320 3109 y(\014)1371 3105 y Fs(\024)g Fy(C)k Fv(\()p Fy(")19 b Fv(+)f Fs(k)p Fr(v)q Fs(k)p Fv(\))j Fs(k)p Fr(v)q Fs(k)g Fy(x)2109 3070 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2287 3105 y Fs(j)p Fy(k)s Fs(j)g Fy(w)2459 3117 y Ft(\013)2530 3105 y Fy(:)-9 3318 y Fz(Since)201 3222 y Fn(\014)201 3272 y(\014)201 3322 y(\014)229 3296 y Fv(^)228 3318 y Fy(h)276 3330 y Fu(2)313 3222 y Fn(\014)313 3272 y(\014)313 3322 y(\014)364 3318 y Fs(\024)i(j)p Fy(k)s Fs(j)14 b Fv(\()p Fs(j)s Fv(^)-45 b Fy(v)22 b Fs(\003)f Fv(^)-46 b Fy(v)t Fs(j)18 b Fv(+)g(2)c Fs(j)s Fv(^)-45 b Fy(v)21 b Fs(\003)g Fv(^)-45 b Fy(v)1142 3330 y Ft(E)1199 3318 y Fs(j)p Fv(\))p Fz(,)21 b(we)h(ha)n(v)o(e)876 3513 y Fs(j)p Fv(\()t(^)-45 b Fy(v)21 b Fs(\003)h Fv(^)-46 b Fy(v)t Fv(\))p Fs(j)23 b(\024)g Fy(C)k Fs(k)p Fr(v)q Fs(k)1484 3479 y Fu(2)1542 3513 y Fy(x)1589 3479 y Fq(\000)p Fu(1)1700 3513 y Fv(\()p Fy(w)1791 3525 y Ft(\013)1858 3513 y Fs(\003)18 b Fy(w)1977 3525 y Ft(\013)2024 3513 y Fv(\))24 b Fs(\024)e Fy(C)28 b Fs(k)p Fr(v)q Fs(k)2389 3479 y Fu(2)2447 3513 y Fy(x)2494 3479 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2672 3513 y Fy(w)2731 3525 y Ft(\013)2802 3513 y Fz(,)823 3638 y Fs(j)p Fv(\()t(^)-45 b Fy(v)21 b Fs(\003)g Fv(^)-45 b Fy(v)1040 3650 y Ft(E)1097 3638 y Fv(\))p Fs(j)23 b(\024)g Fy(C)k Fs(k)p Fr(v)q Fs(k)21 b Fy(")g(x)1612 3603 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)1769 3638 y Fv(\()p Fy(e)1840 3650 y Fu(0)p Ft(;\016)1947 3638 y Fs(\003)d Fy(w)2066 3650 y Ft(\013)2114 3638 y Fv(\))23 b Fs(\024)g Fy(C)28 b Fs(k)p 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y(we)21 b(ha)n(v)o(e)712 4589 y Fy(iD)813 4567 y Fv(^)812 4589 y Fy(h)860 4601 y Fu(2)920 4589 y Fv(=)i Fs(\000)1087 4532 y Fv(1)p 1083 4569 V 1083 4646 a Fy(\031)1156 4589 y Fv(\()q Fy(D)r(K)9 b Fv(^)-45 b Fy(v)21 b Fs(\003)g Fv(^)-45 b Fy(v)21 b Fv(+)d Fy(D)r(K)9 b Fv(^)-45 b Fy(v)22 b Fs(\003)f Fv(^)-45 b Fy(v)1912 4601 y Ft(E)1986 4589 y Fv(+)22 b(^)-46 b Fy(v)22 b Fs(\003)c Fy(D)r(K)9 b Fv(^)-45 b Fy(v)2379 4601 y Ft(E)2435 4589 y Fv(\))920 4788 y(=)1022 4732 y(1)p 1018 4769 V 1018 4845 a Fy(\031)1092 4788 y Fv(\()s(^)g Fy(v)22 b Fs(\003)f Fv(^)-45 b Fy(v)21 b Fv(+)h(^)-46 b Fy(v)22 b Fs(\003)f Fv(^)-45 b Fy(v)1552 4800 y Ft(E)1627 4788 y Fs(\000)18 b Fy(K)6 b(D)f Fv(^)-45 b Fy(v)21 b Fs(\003)g Fv(^)-45 b Fy(v)22 b Fs(\000)c Fy(K)6 b(D)f Fv(^)-45 b Fy(v)21 b Fs(\003)g Fv(^)-45 b Fy(v)2433 4800 y Ft(E)2508 4788 y Fs(\000)21 b Fv(^)-45 b Fy(v)21 b Fs(\003)d Fy(D)r(K)9 b Fv(^)-45 b Fy(v)2900 4800 y Ft(E)2956 4788 y Fv(\))37 b Fy(:)-9 4966 y Fz(Since)1282 5121 y Fs(j)q Fv(\()s(^)-45 b Fy(v)21 b Fs(\003)h Fv(^)-46 b Fy(v)t Fv(\))p Fs(j)18 b Fv(+)g Fs(j)q Fv(\()p Fy(K)6 b(D)f Fv(^)-45 b Fy(v)21 b Fs(\003)g Fv(^)-45 b Fy(v)s Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b Fs(k)p Fr(v)q Fs(k)2415 5087 y Fu(2)2473 5121 y Fy(x)2520 5087 y Fq(\000)p Fu(1)2631 5121 y Fv(\()p Fy(w)2722 5133 y Ft(\013)p Fq(\000)p Fu(1)2873 5121 y Fs(\003)18 b Fy(w)2992 5133 y Ft(\013)3040 5121 y Fv(\))2106 5246 y Fs(\024)k Fy(C)28 b Fs(k)p Fr(v)q Fs(k)2415 5211 y Fu(2)2473 5246 y Fy(x)2520 5211 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2698 5246 y Fy(w)2757 5258 y Ft(\013)p Fq(\000)p Fu(1)2913 5246 y Fz(,)599 5370 y Fs(j)p Fv(\()s(^)-45 b Fy(v)22 b Fs(\003)f Fv(^)-45 b Fy(v)816 5382 y Ft(E)872 5370 y Fv(\))p Fs(j)19 b Fv(+)f Fs(j)p Fv(\()p Fy(K)6 b(D)f Fv(^)-45 b Fy(v)21 b Fs(\003)g Fv(^)-45 b Fy(v)1393 5382 y Ft(E)1450 5370 y Fv(\))p Fs(j)18 b Fv(+)g Fs(j)p Fv(\()t(^)-45 b Fy(v)21 b Fs(\003)d Fy(D)r(K)9 b Fv(^)-45 b Fy(v)1971 5382 y Ft(E)2027 5370 y Fv(\))p Fs(j)24 b(\024)e Fy(C)28 b Fs(k)p Fr(v)q Fs(k)21 b Fy(")g(x)2543 5336 y Fq(\000)p Fu(1)p Ft(=)p Fu(2)2699 5370 y Fv(\()p Fy(e)2770 5382 y Fu(0)p Ft(;\016)2878 5370 y Fs(\003)d Fy(w)2997 5382 y Ft(\013)p Fq(\000)p Fu(1)3130 5370 y Fv(\))2106 5505 y Fs(\024)k Fy(C)28 b Fs(k)p Fr(v)q Fs(k)21 b Fy(")g(x)2543 5471 y Fq(\000)p Fu(3)p Ft(=)p Fu(2)2721 5505 y Fy(w)2780 5517 y Ft(\013)p Fq(\000)p Fu(1)2935 5505 y Fz(,)-9 5660 y(this)g(ends)h(the)f(proof)i(of)f(\(4.8\))h (and)f(\(4.9\))h(and)f(the)f(proof)i(of)f(Proposition)h(4.4.)1838 5909 y(46)p eop %%Page: 47 47 bop -9 145 a FB(Refer)n(ences)-9 332 y Fz([Bricmont)21 b(et)g(al.])41 b(Bricmont,)22 b(J.,)h(K)o(upiainen,)g(A.)f(&)g(Lin,)g(G.,)h(Renormalization) e(Group)j(and)f(Asymptotics)f(of)h(Solutions)596 432 y(of)f(Nonlinear)g(P)o (arabolic)f(Equations.)h(Comm.)g(Pure)g(Appl.)h(Math.)d FA(47)p Fz(,)j(no.)f(6,)g(893\261922)j(\(1994\))-9 598 y([Gallay])322 b(Gallay)-6 b(,)20 b(Th.:)29 b(A)19 b(Center)n(\261Stable)g(Manifold)g (Theorem)g(for)h(Dif)n(ferential)e(Equations)i(in)e(Banach)h(Spaces.)596 697 y(Comm.)j(Math.)f(Phys.)h FA(152)p Fz(,)h(no.)f(2,)g(249\261268)j (\(1993\))-9 863 y([Batchelor])218 b(Batchelor)m(,)20 b(G.)i(K.,)g(An)g (Introduction)i(to)d(Fluid)h(Dynamics.)g(Cambridge)g(Uni)n(v)o(ersity)f (Press)h(\(1999\))-9 1030 y([Smith])346 b(Smith,)22 b(D.)g(R.,)g(Estimates)e (at)h(In\256nity)h(for)h(Stationary)f(Solutions)g(of)h(the)e(Na)n(vier)n (\261Stok)o(es)j(Equations)e(in)596 1129 y(T)-7 b(w)o(o)22 b(Dimensions.)h(Arch.)f(Rational)e(Mech.)h(Anal.)h FA(20)g Fz(341\261372)j(\(1965\))-9 1295 y([Galdi])360 b(Galdi,)19 b(G.)g(P)-9 b(.,)19 b(An)g(Introduction)h(to)f(the)f(Mathematical)d(Theory)20 b(of)f(the)f(Na)n(vier)n(-Stok)o(es)i(Equations,)g(V)-11 b(ol.)596 1395 y(I+II.)22 b(Springer)h(T)m(racts)e(in)g(Natural)g(Philosophy)-6 b(,)24 b FA(38)p Fz(\261)p FA(39)p Fz(.)g(Springer)n(-V)-9 b(erlag,)24 b(Ne)n(w)d(Y)-9 b(ork,)23 b(\(1994\))-9 1561 y([Amick])318 b(Amick,)31 b(C.)d(J.,)j(On)e(the)g(Asymptotic)g(F)o(orm)g(of)g(Na)n(vier)n (-Stok)o(es)i(Flo)n(w)e(P)o(ast)f(a)g(Body)h(in)g(the)f(Plane.)h(J.)596 1661 y(Dif)n(ferential)21 b(Equations)h FA(91)p Fz(,)g(no.)h(1,)e(149\261167) k(\(1991\))-9 1827 y([W)-7 b(ayne])316 b(W)-7 b(ayne,)23 b(E.)f(C.,)g(In)m(v) n(ariant)h(Manifolds)f(for)h(P)o(arabolic)e(P)o(artial)g(Dif)n(ferential)g (Equations)i(on)g(Unbounded)596 1926 y(Domains.)f(Arch.)h(Rational)d(Mech.)h (Anal.)g FA(138)p Fz(,)i(no.)g(3,)f(279\261306)i(\(1997\))1838 5909 y(47)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0102190745538--