Content-Type: multipart/mixed; boundary="-------------0109242232131" This is a multi-part message in MIME format. ---------------0109242232131 Content-Type: text/plain; name="01-337.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-337.keywords" Matrix polyharmonic operator spectrum ---------------0109242232131 Content-Type: application/postscript; name="matrix.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="matrix.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.76 (MiKTeX 1.11) Copyright 1997 Radical Eye Software (www.radicaleye.com) %%Title: matrix.dvi %%CreationDate: Mon Sep 24 22:17:31 2001 %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: C:\TEXMF\MIKTEX\BIN\DVIPS.EXE matrix %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2001.09.24:2155 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 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Fq(\))39 b(quasip)s(erio)s(dically)d(\(see)41 b(\(2\)\))e(to)g Fp(R)2332 5281 y Fs(n)2379 5317 y Fq(,)j(w)m(e)e (obtain)f(a)g(complete)g(system)h(of)0 5437 y(eigenfunctions)32 b(of)g Fp(H)8 b Fq(.)1888 5686 y(2)p eop %%Page: 3 3 3 2 bop 146 107 a Fq(W)-8 b(e)33 b(use)h(the)f(follo)m(wing)d(represen) m(tation)j(of)f(the)h(p)s(oten)m(tial:)827 302 y(~)812 327 y Fp(V)869 342 y Fs(q)r(s)940 327 y Fq(\()p Fp(x)p Fq(\))28 b(=)1301 244 y Fm(X)1203 428 y Fs(j)t Fj(2)p Fs(Z)1336 409 y Fi(n)1377 428 y Fs(;j)t Fj(6)p Fn(=0)1536 327 y Fp(v)1583 342 y Fs(q)r(s;j)1722 327 y Fq(exp)18 b Fp(i)p Fq(\()q Fp(~)-50 b(p)2008 342 y Fs(j)2044 327 y Fq(\(0\))p Fp(;)17 b(x)p Fq(\))p Fp(;)82 b(q)t(;)17 b(s)27 b Fq(=)h(1)p Fp(;)17 b(:::;)g(m;)687 b Fq(\(3\))0 649 y(where)298 624 y(~)283 649 y Fp(V)340 664 y Fs(q)r(s)411 649 y Fq(\()p Fp(x)p Fq(\))34 b(are)g(matrix)e(elemen)m(ts)j(of)1586 624 y(~)1571 649 y Fp(V)22 b Fq(\()p Fp(x)p Fq(\),)35 b Fp(v)1890 664 y Fs(q)r(s;j)2012 649 y Fq(,)g Fp(j)h Fo(2)30 b Fp(Z)2320 613 y Fs(n)2367 649 y Fq(,)35 b(are)f(F)-8 b(ourier)32 b(co)s(e\016cien)m(ts)k(of)3554 624 y(~)3539 649 y Fp(V)3596 664 y Fs(q)r(s)3667 649 y Fq(\()p Fp(x)p Fq(\),)0 769 y(\()p Fo(\001)p Fp(;)17 b Fo(\001)p Fq(\))31 b(is)i(the)g(inner)f(pro)s(duct)h(in)e Fp(R)1277 733 y Fs(n)1324 769 y Fq(,)j Fp(~)-50 b(p)1433 784 y Fs(j)1470 769 y Fq(\(0\))32 b(are)g(v)m(ectors)i(of)f(the)g(dual)e(lattice:)1007 989 y Fp(~)-50 b(p)1055 1004 y Fs(j)1092 989 y Fq(\(0\))27 b(=)g(2)p Fp(\031)t Fq(\()p Fp(j)1533 1004 y Fn(1)1573 989 y Fp(a)1624 948 y Fj(\000)p Fn(1)1624 1014 y(1)1718 989 y Fp(;)17 b(j)1802 1004 y Fn(2)1842 989 y Fp(a)1893 948 y Fj(\000)p Fn(1)1893 1014 y(2)1987 989 y Fp(;)g(:::;)g(j)2196 1004 y Fs(n)2243 989 y Fp(a)2294 948 y Fj(\000)p Fn(1)2294 1014 y Fs(n)2389 989 y Fq(\))p Fp(;)49 b(j)34 b Fo(2)28 b Fp(Z)2745 948 y Fs(n)2792 989 y Fp(:)881 b Fq(\(4\))146 1292 y(In)38 b(the)f(case)671 1267 y(~)657 1292 y Fp(V)57 b Fq(=)35 b(0,)j(the)g(eigen)m(v)-5 b(alues)37 b(and)g(eigenfunctions)g (of)g(the)g(op)s(erators)g Fp(H)3319 1307 y Fn(0)3359 1292 y Fq(\()p Fp(t)p Fq(\),)h Fp(t)e Fo(2)g Fp(K)7 b Fq(,)0 1413 y(are)33 b(naturally)e(indexed)i(b)m(y)g(pairs)f(of)h (indices)f(\()p Fp(j;)17 b(q)t Fq(\),)32 b Fp(j)i Fo(2)28 b Fp(Z)2257 1376 y Fs(n)2304 1413 y Fq(,)33 b Fp(q)e Fq(=)d(1)p Fp(;)17 b(:::;)g(m)p Fq(:)692 1677 y Fp(\025)749 1636 y Fn(0)749 1702 y Fs(j;q)835 1677 y Fq(\()p Fp(t)p Fq(\))28 b(=)f Fp(p)1126 1636 y Fn(2)p Fs(l)1126 1702 y(j)1188 1677 y Fq(\()p Fp(t)p Fq(\))22 b(+)g Fp(v)1466 1692 y Fn(0)p Fs(q)1539 1677 y Fp(;)82 b( )1715 1636 y Fn(0)1711 1702 y Fs(j;q)1798 1677 y Fq(\()p Fp(t;)17 b(x)p Fq(\))27 b(=)2246 1610 y(1)p 2149 1654 243 4 v 2149 1745 a Fo(j)p Fp(Q)p Fo(j)2282 1717 y Fn(1)p Fs(=)p Fn(2)2398 1677 y Fp(~)-45 b(e)2479 1636 y Fn(0)2447 1702 y Fs(q)2535 1677 y Fq(exp)18 b Fp(i)p Fq(\()q Fp(~)-50 b(p)2821 1692 y Fs(j)2858 1677 y Fq(\()p Fp(t)p Fq(\))p Fp(;)17 b(x)p Fq(\))p Fp(;)0 1957 y Fq(here)36 b Fp(~)-45 b(e)296 1921 y Fn(0)263 1982 y Fs(q)375 1957 y Fq(are)39 b(v)m(ectors)i(of)e(the)h(canonical)e(basis)i(in)e Fp(R)2052 1921 y Fs(m)2119 1957 y Fq(:)57 b(\()l Fp(~)-45 b(e)2319 1921 y Fn(0)2286 1982 y Fs(q)2358 1957 y Fq(\))2396 1972 y Fs(s)2472 1957 y Fq(=)39 b Fp(\016)2630 1972 y Fs(q)r(s)2701 1957 y Fp(;)17 b(s)39 b Fq(=)h(1)p Fp(;)17 b(:::;)g(m)p Fq(,)41 b(and)f Fp(~)-50 b(p)3562 1972 y Fs(j)3599 1957 y Fq(\()p Fp(t)p Fq(\))39 b(=)1 2077 y Fp(~)-50 b(p)49 2092 y Fs(j)85 2077 y Fq(\(0\))28 b(+)f Fp(t)p Fq(,)43 b Fp(~)-49 b(p)495 2092 y Fs(j)531 2077 y Fq(\(0\))40 b(b)s(eing)g(giv)m(en)h(b)m(y)g(\(4\);)j Fp(p)1618 2041 y Fn(2)p Fs(l)1618 2102 y(j)1680 2077 y Fq(\()p Fp(t)p Fq(\))d(=)g Fo(j)q Fp(~)-50 b(p)2026 2092 y Fs(j)2062 2077 y Fq(\()p Fp(t)p Fq(\))p Fo(j)2201 2041 y Fn(2)p Fs(l)2262 2077 y Fq(,)43 b(and)d(the)h(sym)m(b)s(ol)f Fo(j)p Fp(Q)p Fo(j)g Fq(stands)i(for)e(the)0 2198 y(v)m(olume)32 b(of)g Fp(Q)p Fq(.)146 2401 y(The)i(op)s(erator)e Fp(H)8 b Fq(\()p Fp(t)p Fq(\))32 b(admits)f(the)i(matrix)e(represen)m(tation)j (in)d(the)i(basis)2936 2305 y Fm(n)2991 2401 y Fp( )3058 2365 y Fn(0)3054 2426 y Fs(j;q)3141 2305 y Fm(o)3196 2455 y Fs(j)t Fj(2)p Fs(Z)3329 2436 y Fi(n)3371 2455 y Fs(;q)r Fn(=1)p Fs(;:::;m)3679 2401 y Fq(:)1024 2663 y Fp(H)1105 2679 y Fn(\()p Fs(j;q)r Fn(\)\()p Fs(i;s)p Fn(\))1405 2663 y Fq(=)28 b Fp(p)1558 2622 y Fn(2)p Fs(l)1558 2688 y(j)1619 2663 y Fq(\()p Fp(t)p Fq(\))p Fp(\016)1773 2678 y Fs(j)t(i)1834 2663 y Fp(\016)1877 2678 y Fs(q)r(s)1970 2663 y Fq(+)22 b Fp(v)2115 2678 y Fs(oq)2187 2663 y Fp(\016)2230 2678 y Fs(j)t(i)2291 2663 y Fp(\016)2334 2678 y Fs(q)r(s)2427 2663 y Fq(+)g Fp(v)2572 2678 y Fs(q)r(s;j)t Fj(\000)p Fs(i)2774 2663 y Fp(:)146 2966 y Fq(In)42 b(ph)m(ysical)f(literature,)g (the)h(imp)s(ortan)m(t)d(concept)j(of)e(an)h(iso)s(energetic)g(surface) g(of)g(a)g(p)s(erio)s(dic)0 3087 y(op)s(erator)d(is)g(used)i(\(see)g (f.e.)62 b([Ki)o(,)38 b(Ma)q(]\).)61 b(It)39 b(is)f(said)g(that)h(a)f (p)s(oin)m(t)g Fp(t)h 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b(where)i Fp(s)p Fq(\()p Fo(\001)p Fq(\))e(is)g(the)h(area)g(of)f(a)g(set,)h (coun)m(ting)g(the)g(m)m(ultiplicit)m(y)c(of)j(the)h(spheres.)1888 5686 y(3)p eop %%Page: 4 4 4 3 bop 0 107 a Fr(2)161 b(The)53 b(main)i(results.)0 409 y Fq(F)-8 b(or)32 b(an)m(y)h Fp(t)28 b Fo(2)g Fp(S)576 424 y Fn(0)615 409 y Fq(\()p Fp(k)s Fq(\))33 b(there)g(is)f(a)h(pair)e (\()p Fp(j;)17 b(q)t Fq(\))33 b(suc)m(h)h(that)1545 629 y Fp(p)1594 588 y Fn(2)p Fs(l)1594 654 y(j)1656 629 y Fq(\()p Fp(t)p Fq(\))22 b(+)g Fp(v)1934 644 y Fs(oq)2034 629 y Fq(=)27 b Fp(k)2191 588 y Fn(2)p Fs(l)2253 629 y Fp(;)1420 b Fq(\(6\))0 849 y(i.e.,)32 b Fp(H)265 864 y Fn(0)304 849 y Fq(\()p Fp(t)p Fq(\))h(has)f(an)h(eigen)m(v)-5 b(alue)32 b(equal)g Fp(k)1534 813 y Fn(2)p Fs(l)1595 849 y Fq(.)44 b(The)33 b(m)m(ultiplicit)m(y)c(of)i(this)h(eigen)m(v)-5 b(alue)32 b(is)g(at)g(least)g(equal)0 969 y(to)44 b(the)h(m)m (ultiplicit)m(y)c Fp(m)931 984 y Fs(q)1014 969 y Fq(of)k Fp(v)1185 984 y Fn(0)p Fs(q)1303 969 y 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Fs(q)609 4308 y Fo(j)27 b Fp(<)h(k)822 4272 y Fn(2)p Fs(l)q Fj(\000)p Fs(n)p Fj(\000)p Fs(\016)1070 4308 y Fq(.)42 b(All)27 b(other)i(eigen)m(v)-5 b(alues)29 b(are)g(outside)f(the)i(circle.)41 b(Coun)m(ting)29 b(m)m(ultiplic-)0 4428 y(it)m(y)-8 b(,)32 b(w)m(e)i(ha)m(v)m(e)g(exactly)f Fp(m)g Fq(eigen)m(v)-5 b(alues)32 b(of)g Fp(H)1684 4443 y Fn(0)1724 4428 y Fq(\()p Fp(t)p Fq(\))g(inside)g(the)h(circle.)146 4632 y(In)j(the)f(scalar)g (situation)f Fp(m)e Fq(=)g Fp(m)1441 4647 y Fs(q)1512 4632 y Fq(=)f(1,)36 b(i.e.,)g(the)f(circle)g(has)g(one)h(simple)d (eigen)m(v)-5 b(alue)35 b(of)g Fp(H)3675 4647 y Fn(0)3714 4632 y Fq(\()p Fp(t)p Fq(\))0 4752 y(inside.)73 b(In)43 b(the)g(matrix)e(situation)g(w)m(e)j(can)f(ha)m(v)m(e)h(m)m(ultiple)c (eigen)m(v)-5 b(alues,)46 b(whic)m(h)d(mak)m(es)g(it)f(more)0 4873 y(di\016cult)32 b(to)g(describ)s(e)h(their)f(shift)g(under)i(the)f (p)s(erturbation)e Fp(\013)2401 4847 y Fq(~)2386 4873 y Fp(V)22 b Fq(.)146 5076 y(If)28 b(w)m(e)h(can)f(pro)m(v)m(e)h(the)f 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Fq(\)-)f(neigh)m(b)s(orho)s(o)s(d)1888 5686 y(5)p eop %%Page: 6 6 6 5 bop 0 107 a Fq(of)41 b Fp(\037)181 122 y Fn(0)220 107 y Fq(\()p Fp(k)s(;)17 b(\016)t Fq(\),)43 b(then)f(the)f(p)s (erturbation)f(series)i(con)m(v)m(erges)h(on)e Fp(C)2427 122 y Fn(0)2467 107 y Fq(,)i(since)e(the)h(distance)f(from)f Fp(C)3658 122 y Fn(0)3738 107 y Fq(to)0 227 y(the)d(unp)s(erturb)s(ed)g (eigen)m(v)-5 b(alues)36 b(is)g(greater)h(than)f Fo(k)1980 202 y Fq(~)1965 227 y Fp(V)21 b Fo(k)36 b Fq(for)g Fp(k)k Fq(large)35 b(enough.)55 b(When)37 b(2)p Fp(l)f Fo(\024)f Fp(n)h Fq(then,)0 348 y(generally)25 b(sp)s(eaking,)j(the)f(p)s (erturbation)f(series)h(div)m(erges)g(for)f Fp(t)i Fo(2)g Fp(\037)2544 363 y Fn(0)2583 348 y Fq(\()p Fp(k)s(;)17 b(\016)t Fq(\),)28 b(since)f(the)f(distance)h(from)0 468 y Fp(C)70 483 y Fn(0)145 468 y Fq(to)35 b(the)g(eigen)m(v)-5 b(alues)36 b(is)f(small)e(\(it)h(go)s(es)h(to)g(zero)h(as)g Fp(k)f Fo(!)d(1)p Fq(\).)51 b(W)-8 b(e)36 b(construct)h(a)e (\\nonsingular")0 588 y(set)30 b Fp(\037)210 603 y Fn(1)277 588 y Fo(\032)e Fp(\037)443 603 y Fn(0)483 588 y Fq(\()p Fp(k)s(;)17 b(\016)t Fq(\),)29 b(suc)m(h)i(that)d(it)g(pro)m(vides)i (the)g(con)m(v)m(ergence)h(of)e(the)g(series)h(in)e(the)i(case)g(2)p Fp(l)f Fo(\024)g Fp(n)g Fq(and)0 709 y(also)j(impro)m(v)m(es)g(the)h (con)m(v)m(ergence)j(in)31 b(the)i(case)h(2)p Fp(l)c(>)d(n)p Fq(.)0 984 y Fg(Lemma)74 b(3)49 b Ff(L)-5 b(et)37 b Fq(0)31 b Fp(<)f(\014)37 b(<)30 b Fq(1)p Ff(,)37 b(and)f Fq(0)30 b Fp(<)h Fq(2)p Fp(\016)j(<)d Fq(\()p Fp(n)24 b Fo(\000)f Fq(1\)\(1)g Fo(\000)h Fp(\014)6 b Fq(\))p Fp(:)36 b Ff(Then,)g(for)g (any)h(su\016ciently)g(lar)-5 b(ge)0 1104 y Fp(k)s Ff(,)35 b Fp(k)c(>)d(k)356 1119 y Fn(0)396 1104 y Fq(\()p Fp(\014)6 b(;)17 b(\016)n(;)g(a)675 1119 y Fn(1)714 1104 y Fp(;)g(:::a)890 1119 y Fs(n)937 1104 y Fq(\))p Ff(,)35 b(ther)-5 b(e)35 b(exists)g(a)g(set)g Fp(\037)1848 1119 y Fn(1)1888 1104 y Fq(\()p Fp(k)s(;)17 b(\014)6 b(;)17 b(\016)t Fq(\))27 b Fo(\032)h Fp(\037)2407 1119 y Fn(0)2447 1104 y Fq(\()p Fp(k)s(;)17 b(\016)t Fq(\))28 b Fo(\032)g Fp(S)2861 1119 y Fn(0)2901 1104 y Fq(\()p Fp(k)s Fq(\))p Ff(,)35 b(such)g(that)g(for)g (any)0 1225 y(p)-5 b(oint)35 b Fp(t)g Ff(in)f(the)h Fq(\()p Fp(k)690 1188 y Fj(\000)p Fs(n)p Fn(+1)p Fj(\000)p Fn(2)p Fs(\016)1006 1225 y Fq(\))p Ff(-neighb)-5 b(orho)g(o)g(d)33 b(of)i(this)f(set)h(the)g(fol)5 b(lowing)34 b(ine)-5 b(quality)34 b(holds:)214 1413 y Fq(min)211 1472 y Fs(z)s Fj(2)p Fs(C)344 1481 y Fh(0)954 1413 y Fq(min)472 1511 y Fp(i;)17 b(i)582 1475 y Fj(0)633 1511 y Fo(2)28 b Fp(Z)801 1475 y Fs(n)848 1511 y Fp(;)17 b Fq(0)27 b Fp(<)h Fo(j)p Fp(i)22 b Fo(\000)h Fp(i)1288 1475 y Fj(0)1311 1511 y Fo(j)k Fp(<)h(k)1524 1475 y Fs(\014)1571 1511 y Fp(;)472 1631 y(s;)17 b(s)608 1595 y Fj(0)658 1631 y Fq(=)28 b(1)p Fp(;)17 b(:::m)1668 1413 y Fo(j)27 b Fp(p)1772 1372 y Fn(2)p Fs(l)1772 1438 y(i)1833 1413 y Fq(\()p Fp(t)p Fq(\))22 b(+)g Fp(v)2111 1428 y Fn(0)p Fs(s)2206 1413 y Fo(\000)g Fp(z)33 b Fo(jj)27 b Fp(p)2515 1372 y Fn(2)p Fs(l)2515 1438 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Fs(k)r Fj(!1)2317 2231 y Fp(O)s Fq(\()p Fp(k)2487 2190 y Fj(\000)p Fs(\016)r(=)p Fn(8)2649 2231 y Fq(\))p Fp(:)938 b Fq(\(12\))146 2566 y(F)-8 b(urther)36 b(w)m(e)g(assume:)49 b(0)32 b Fp(<)h(\014)38 b(<)32 b(\014)1475 2581 y Fn(0)1514 2566 y Fq(,)k Fp(\014)1632 2581 y Fn(0)1704 2566 y Fq(=)c(min)o Fo(f)p Fq(1)p Fp(;)17 b Fq(\(4)p Fp(l)25 b Fo(\000)f Fp(n)h Fo(\000)f Fq(1\)\()p Fp(n)g Fo(\000)g Fq(1\))2939 2530 y Fj(\000)p Fn(1)3033 2566 y Fo(g)p Fq(,)36 b(and)g(0)c Fp(<)g(\016)k(<)c(\016)3758 2581 y Fn(0)3798 2566 y Fq(,)0 2687 y Fp(\016)43 2702 y Fn(0)111 2687 y Fq(=)225 2647 y Fn(1)p 225 2663 36 4 v 225 2721 a(4)287 2687 y Fq(min)o Fo(f)p Fq(\()p Fp(n)22 b Fo(\000)h Fq(1\)\(1)f Fo(\000)h Fp(\014)6 b Fq(\))p Fp(;)17 b Fq(4)p Fp(l)24 b Fo(\000)f Fp(n)f Fo(\000)h Fq(1)g Fo(\000)g Fp(\014)6 b Fq(\()p Fp(n)22 b Fo(\000)h Fq(1\))p Fo(g)p Fq(.)44 b(Ob)m(viously)-8 b(,)33 b(under)h(these)g(conditions)e Fp(\015)38 b Fq(is)0 2807 y(p)s(ositiv)m(e.)146 3010 y(The)c(set)f Fp(\037)560 3025 y 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y(\(9\))32 b(and)h(the)g(de\014nition)e(of)h Fp(C)1129 1871 y Fn(0)1201 1856 y Fq(yields)g(the)h(estimate:)447 2134 y Fo(k)p Fq(\()p Fp(H)616 2149 y Fn(0)677 2134 y Fo(\000)22 b Fp(z)t Fq(\))863 2093 y Fj(\000)p Fn(1)p Fs(=)p Fn(2)1029 2134 y Fo(k)27 b Fq(=)1210 2013 y Fm(\022)1458 2134 y Fq(min)1271 2194 y Fs(i)p Fj(2)p Fs(Z)1395 2175 y Fi(n)1437 2194 y Fs(;)32 b(s)p Fn(=1)p Fs(;)p Fn(2)p Fs(;:::m)1824 2134 y Fo(j)p Fp(p)1901 2093 y Fn(2)p Fs(l)1901 2159 y(i)1962 2134 y Fq(\()p Fp(t)p Fq(\))23 b(+)f Fp(v)2241 2149 y Fn(0)p Fs(s)2335 2134 y Fo(\000)h Fp(z)t Fo(j)2512 2013 y Fm(\023)2573 2036 y Fj(\000)p Fn(1)p Fs(=)p Fn(2)2766 2134 y Fq(=)k Fp(k)2923 2093 y Fj(\000)p Fn(\(2)p Fs(l)q Fj(\000)p Fs(n)p Fj(\000)p Fs(\016)r Fn(\))p Fs(=)p Fn(2)3351 2134 y Fp(:)274 b Fq(\(21\))0 2389 y(T)-8 b(o)33 b(pro)m(v)m(e)g(the)g (con)m(v)m(ergence)j(of)c(the)h(series)g(for)f(a)g(large)g Fp(k)s Fq(,)g(it)g(is)g(enough)h(to)f(c)m(hec)m(k)j(that)1440 2607 y Fo(k)p Fp(A)p Fo(k)28 b Fp(<)f(c)1786 2622 y Fs(V)1847 2607 y Fp(k)1901 2565 y Fj(\000)p Fs(\015)2001 2607 y Fp(;)49 b(z)32 b Fo(2)c Fp(C)2318 2622 y Fn(0)2358 2607 y Fp(;)1267 b Fq(\(22\))0 2824 y Fp(C)70 2839 y Fs(V)163 2824 y Fq(not)32 b(dep)s(ending)g(on)g Fp(k)s Fq(.)43 b(The)33 b(matrix)e(elemen)m(ts)h(of)g(the)g(op)s(erator)g Fp(A)g Fq(are)g(giv)m(en)g(b)m(y)h(the)g(form)m(ula:)842 3059 y Fp(A)915 3074 y Fn(\()p Fs(is)p Fn(\)\()p Fs(i)1077 3055 y Fd(0)1101 3074 y Fs(s)1134 3055 y Fd(0)1156 3074 y Fn(\))1215 3059 y Fq(=)1979 2991 y Fp(v)2026 3006 y Fs(ss)2092 2987 y Fd(0)2114 3006 y Fs(;i)p Fj(\000)p Fs(i)2237 2987 y Fd(0)p 1328 3036 1585 4 v 1328 3052 a Fm(r)p 1411 3052 1502 4 v 29 x(\020)1461 3177 y Fp(p)1510 3143 y Fn(2)p Fs(l)1510 3200 y(i)1571 3177 y Fq(\()p Fp(t)p Fq(\))23 b(+)f Fp(v)1850 3192 y Fn(0)p Fs(s)1944 3177 y Fo(\000)h Fp(z)2093 3081 y Fm(\021)16 b(\020)2209 3177 y Fp(p)2258 3143 y Fn(2)p Fs(l)2258 3203 y(i)2282 3184 y Fd(0)2319 3177 y Fq(\()p Fp(t)p Fq(\))23 b(+)f Fp(v)2598 3192 y Fn(0)p Fs(s)2666 3173 y Fd(0)2714 3177 y Fo(\000)h Fp(z)2863 3081 y Fm(\021)2923 3059 y Fp(;)702 b Fq(\(23\))0 3426 y(if)31 b Fp(i)d Fo(6)p Fq(=)g Fp(i)287 3390 y Fj(0)343 3426 y Fq(and)33 b(are)f(equal)h(to)f(zero)h(if)e Fp(i)d Fq(=)g Fp(i)1565 3390 y Fj(0)1588 3426 y Fq(.)44 b(W)-8 b(e)33 b(represen)m(t)h Fp(A)f Fq(in)f(the)h(form)e Fp(A)d Fq(=)f Fp(A)3142 3390 y Fj(0)3188 3426 y Fq(+)22 b Fp(A)3359 3390 y Fj(00)3402 3426 y Fq(,)32 b(where)969 3701 y Fp(A)1042 3659 y Fj(0)1042 3725 y Fn(\()p Fs(is)p Fn(\)\()p Fs(i)1204 3706 y Fd(0)1227 3725 y Fs(s)1260 3706 y Fd(0)1282 3725 y Fn(\))1342 3701 y Fq(=)1445 3554 y Fm(\()1554 3640 y Fp(A)1627 3655 y Fn(\()p Fs(is)p Fn(\)\()p Fs(i)1789 3636 y Fd(0)1812 3655 y Fs(s)1845 3636 y Fd(0)1867 3655 y Fn(\))1899 3640 y Fp(;)83 b Fq(if)31 b(0)d Fp(<)f Fo(j)p Fp(i)22 b Fo(\000)h Fp(i)2494 3603 y Fj(0)2517 3640 y Fo(j)28 b Fp(<)f(k)2730 3603 y Fs(\014)2778 3640 y Fp(;)1554 3760 y Fq(0)p Fp(;)379 b Fq(otherwise,)0 3975 y(and,)33 b(corresp)s(ondingly)-8 b(,)1059 4249 y Fp(A)1132 4208 y Fj(00)1132 4273 y Fn(\()p Fs(is)p Fn(\)\()p Fs(i)1294 4254 y Fd(0)1317 4273 y Fs(s)1350 4254 y Fd(0)1372 4273 y Fn(\))1432 4249 y Fq(=)1535 4103 y Fm(\()1644 4188 y Fq(0)p Fp(;)379 b Fq(if)31 b Fo(j)p Fp(i)22 b Fo(\000)h Fp(i)2404 4152 y Fj(0)2427 4188 y Fo(j)28 b Fp(<)f(k)2640 4152 y Fs(\014)2688 4188 y Fp(;)1644 4308 y(A)1717 4324 y Fn(\()p Fs(is)p Fn(\)\()p Fs(i)1879 4305 y Fd(0)1902 4324 y Fs(s)1935 4305 y Fd(0)1957 4324 y Fn(\))1989 4308 y Fp(;)83 b Fq(otherwise.)0 4528 y(Considering)31 b(the)h(estimate)e(\(11\),)h(w)m(e)i(see)f(that)f(the)h(denominator)e (on)h(the)h(righ)m(t-hand)e(side)i(of)f(\(23\))0 4649 y(can)i(b)s(e)g(b)s(ounded)g(from)e(b)s(elo)m(w)i(b)m(y)g Fp(k)1406 4613 y Fs(\015)1451 4649 y Fq(,)f(when)i(0)27 b Fp(<)h Fo(j)p Fp(i)22 b Fo(\000)h Fp(i)2161 4613 y Fj(0)2184 4649 y Fo(j)28 b Fp(<)f(k)2397 4613 y Fs(\014)2444 4649 y Fq(.)44 b(Therefore,)1387 4779 y Fm(\014)1387 4829 y(\014)1387 4879 y(\014)o Fp(A)1487 4837 y Fj(0)1487 4903 y Fn(\()p Fs(is)p Fn(\)\()p Fs(i)1649 4884 y Fd(0)1673 4903 y Fs(s)1706 4884 y Fd(0)1728 4903 y Fn(\))1760 4779 y Fm(\014)1760 4829 y(\014)1760 4879 y(\014)27 b Fp(<)g Fo(j)p Fp(v)1993 4894 y Fs(ss)2059 4875 y Fd(0)2081 4894 y Fs(;i)p Fj(\000)p Fs(i)2204 4875 y Fd(0)2230 4879 y Fo(j)p Fp(k)2312 4837 y Fj(\000)p Fs(\015)2411 4879 y Fp(:)0 5096 y Fq(T)-8 b(aking)48 b(in)m(to)h(accoun)m(t)g(that)g Fp(v)1207 5111 y Fs(ss)1273 5092 y Fd(0)1295 5111 y Fs(;i)p Fj(\000)p Fs(i)1418 5092 y Fd(0)1493 5096 y Fq(are)f(the)i(F)-8 b(ourier)47 b(co)s(e\016cien)m(ts)j(of)f(in\014nitely)e(di\013eren)m (tiable)0 5216 y(functions,)33 b(w)m(e)g(obtain)1280 5337 y(sup)1184 5416 y Fs(s;s)1270 5398 y Fd(0)1291 5416 y Fn(=1)p Fs(;:::m)1721 5254 y Fm(X)1539 5445 y Fs(i)p Fj(2)p Fs(Z)1663 5427 y Fi(n)1705 5445 y Fs(;)p Fj(j)p Fs(i)p Fj(\000)p Fs(i)1848 5427 y Fd(0)1869 5445 y Fj(j)p Fs(k)1401 1019 y Fi(\014)1459 929 y Fo(j)p Fp(v)1534 944 y Fs(ss)1600 925 y Fd(0)1621 944 y Fs(;i)p Fj(\000)p Fs(i)1744 925 y Fd(0)1770 929 y Fo(j)27 b Fp(<)h(c)1971 944 y Fs(V)2032 929 y Fp(k)2086 888 y Fn(2)p Fs(l)q Fj(\000)p Fs(n)p Fj(\000)p Fs(\016)r Fj(\000)p Fs(\015)2429 929 y Fp(;)81 b(c)2579 944 y Fs(V)2668 929 y Fq(=)28 b Fp(c)2814 944 y Fs(V)2874 929 y Fq(\()p Fp(\016)n(;)17 b(\014)6 b(;)17 b(\015)5 b Fq(\))p Fp(;)0 1240 y Fq(and,)33 b(therefore,)1356 1360 y Fo(k)p Fp(A)1479 1319 y Fj(00)1522 1360 y Fq(\()p Fp(z)t Fq(\))p Fo(k)28 b Fp(<)f(c)1870 1375 y Fs(V)1931 1360 y Fp(k)1985 1319 y Fj(\000)p Fs(\015)2085 1360 y Fp(;)49 b(z)32 b Fo(2)c Fp(C)2402 1375 y Fn(0)2442 1360 y Fp(:)1183 b Fq(\(25\))0 1534 y(Adding)32 b(\(24\))g(and)h(\(25\),)f(w)m(e)h(obtain)f(\(22\).)43 b(The)33 b(theorem)g(is)f(pro)m(v)m(ed.)p 1664 1567 990 4 v 146 1738 a(The)i(pro)s(of)e(of)g(Theorem)h(2.)p 146 1770 1055 4 v 30 w(The)f(op)s(erator)e Fp(H)1902 1753 y Fn(0)1973 1738 y Fq(has)h Fp(m)2230 1753 y Fs(q)2299 1738 y Fq(eigen)m(v)-5 b(alues)31 b(inside)f Fp(C)3147 1753 y Fn(0)3187 1738 y Fq(.)43 b(The)31 b(fact)g(that)0 1858 y(\()p Fp(H)8 b Fq(\()p Fp(t)p Fq(\))24 b Fo(\000)i Fp(z)t Fq(\))452 1822 y Fj(\000)p Fn(1)583 1858 y Fq(is)36 b(b)s(ounded)h(on)f Fp(C)1296 1873 y Fn(0)1372 1858 y Fq(for)g(all)e Fp(\013)j Fq(means)g(that)f(the)h(n)m(um)m(b)s(er)g(of)f (eigen)m(v)-5 b(alues)36 b(inside)g Fp(C)3786 1873 y Fn(0)0 1978 y Fq(do)s(es)f(not)f(c)m(hange)h(with)f(the)h(p)s (erturbation)f Fp(\013)1767 1953 y Fq(~)1752 1978 y Fp(V)21 b Fq(,)35 b Fp(\013)d Fo(2)f Fq([0)p Fp(;)17 b Fq(1].)48 b(Since)35 b(all)d(eigen)m(v)-5 b(alues)35 b(are)f(real,)g(they)0 2099 y(are)h(inside)f(the)h(in)m(terv)-5 b(al)33 b(\001)1049 2114 y Fn(0)1089 2099 y Fq(,)i(whic)m(h)g(is)f(cut)h(out)g(b)m(y)g(the) g(circle)f(on)g(the)h(real)f(axis.)50 b(In)m(tegrating)34 b(the)0 2219 y(resolv)m(en)m(t)g(o)m(v)m(er)f(the)g(con)m(tour,)g(w)m (e)h(get)f(the)g(corresp)s(onding)f(sp)s(ectral)g(pro)5 b(jection:)1223 2478 y Fp(E)h Fq(\()p Fp(t)p Fq(\))28 b(=)f Fo(\000)1677 2411 y Fq(1)p 1630 2455 141 4 v 1630 2546 a(2)p Fp(\031)t(i)1798 2361 y Fm(I)1844 2550 y Fs(C)1894 2559 y Fh(0)1933 2478 y Fq(\()p Fp(H)8 b Fq(\()p Fp(t)p Fq(\))22 b Fo(\000)h Fp(z)t Fq(\))2380 2437 y Fj(\000)p Fn(1)2475 2478 y Fp(dz)t(:)1050 b Fq(\(26\))0 2742 y(Substituting)46 b(the)g(series)h(for)f(the)h(resolv)m(en)m(t)h(\(13\))d(in)h(the)h (last)e(form)m(ula,)k(w)m(e)e(obtain)e(\(14\){\(16\).)0 2862 y(Considering)32 b(the)h(estimates)f(\(21\))g(and)h(\(22\),)f(w)m (e)i(get)e(\(17\).)146 3065 y(W)-8 b(e)33 b(pro)m(v)m(e)g(that,)f(in)f (fact,)h(the)g(p)s(erturbation)f(series)i(for)e(the)h(resolv)m(en)m(t)h (con)m(v)m(erges)i(on)c(a)h(smaller)0 3186 y(circle)37 b Fp(C)330 3201 y Fn(1)369 3186 y Fq(,)j(whic)m(h)e(has)g(the)h(radius) e Fp(k)1425 3150 y Fj(\000)p Fn(2)p Fs(l)q Fn(+1+)p Fs(\020)1760 3186 y Fq(and)h(the)g(same)g(cen)m(ter)h Fp(z)j Fq(=)36 b Fp(k)2926 3150 y Fn(2)p Fs(l)3025 3186 y Fq(as)i Fp(C)3220 3201 y Fn(0)3260 3186 y Fq(.)59 b(Let)38 b Fp(z)k Fo(2)37 b Fp(C)3786 3201 y Fn(1)0 3306 y Fq(and)c Fp(z)235 3321 y Fn(0)302 3306 y Fo(2)28 b Fp(C)466 3321 y Fn(0)505 3306 y Fq(.)44 b(F)-8 b(or)32 b(all)e Fp(i)e Fo(6)p Fq(=)g Fp(j)6 b Fq(:)1128 3526 y Fo(j)p Fp(p)1205 3485 y Fn(2)p Fs(l)1205 3551 y(i)1266 3526 y Fq(\()p Fp(t)p Fq(\))22 b(+)g Fp(v)1544 3541 y Fn(0)p Fs(s)1639 3526 y Fo(\000)g Fp(z)t Fo(j)29 b Fp(>)e Fo(j)p Fp(p)2024 3485 y Fn(2)p Fs(l)2024 3551 y(i)2085 3526 y Fq(\()p Fp(t)p Fq(\))22 b(+)g Fp(v)2363 3541 y Fn(0)p Fs(s)2458 3526 y Fo(\000)g Fp(z)2602 3541 y Fn(0)2642 3526 y Fo(j)p Fp(;)955 b Fq(\(27\))0 3746 y(since)28 b(the)f(circle)g Fp(C)716 3761 y Fn(1)782 3746 y Fq(is)g(smaller)e(than)i Fp(C)1498 3761 y Fn(0)1564 3746 y Fq(and,)i(therefore,)g(further)e(a)m(w)m(a)m(y)i(from)d(outside) h(eigen)m(v)-5 b(alues.)0 3866 y(Since)33 b(\(11\))f(holds)g(for)g Fp(z)910 3881 y Fn(0)950 3866 y Fq(,)g(w)m(e)i(easily)e(obtain:)361 4086 y Fo(j)p Fp(p)438 4045 y Fn(2)p Fs(l)438 4111 y(i)499 4086 y Fq(\()p Fp(t)p Fq(\))22 b(+)g Fp(v)777 4101 y Fn(0)p Fs(s)872 4086 y Fo(\000)g Fp(z)t Fo(jj)p Fp(p)1125 4045 y Fn(2)p Fs(l)1125 4111 y(i)1149 4092 y Fd(0)1187 4086 y Fq(\()p Fp(t)p Fq(\))g(+)g Fp(v)1465 4101 y Fn(0)p Fs(s)1533 4083 y Fd(0)1582 4086 y Fo(\000)g Fp(z)t Fo(j)28 b Fp(>)g(k)1944 4045 y Fn(2)p Fs(\015)2089 4086 y Fq(when)66 b Fp(i;)17 b(i)2486 4045 y Fj(0)2537 4086 y Fo(6)p Fq(=)28 b Fp(j;)49 b Fq(0)28 b Fp(<)f Fo(j)p Fp(i)c Fo(\000)f Fp(i)3153 4045 y Fj(0)3177 4086 y Fo(j)27 b Fp(<)h(k)3390 4045 y Fs(\014)3437 4086 y Fp(:)0 4306 y Fq(Supp)s(ose)34 b Fp(i)28 b Fq(=)f Fp(j;)17 b(i)664 4270 y Fj(0)716 4306 y Fo(6)p Fq(=)27 b Fp(j)6 b Fq(.)44 b(Then,)33 b(using)g(\(27\))f(for)g Fp(i)1861 4270 y Fj(0)1884 4306 y Fq(,)h(w)m(e)g(get:)1162 4526 y Fo(j)p Fp(p)1239 4485 y Fn(2)p Fs(l)1239 4551 y(j)1300 4526 y Fq(\()p Fp(t)p Fq(\))22 b(+)g Fp(v)1578 4541 y Fn(0)p Fs(s)1672 4526 y Fo(\000)h Fp(z)t Fo(jj)p Fp(p)1926 4485 y Fn(2)p Fs(l)1926 4551 y(i)1950 4532 y Fd(0)1987 4526 y Fq(\()p Fp(t)p Fq(\))g(+)f Fp(v)2266 4541 y Fn(0)p Fs(s)2334 4523 y Fd(0)2382 4526 y Fo(\000)h Fp(z)t Fo(j)28 b Fp(>)821 4754 y Fo(j)p Fp(p)898 4718 y Fn(2)p Fs(l)898 4779 y(j)959 4754 y Fq(\()p Fp(t)p Fq(\))23 b(+)f Fp(v)1238 4769 y Fn(0)p Fs(s)1332 4754 y Fo(\000)h Fp(z)t Fo(j)p 804 4805 723 4 v 804 4897 a(j)p Fp(p)881 4862 y Fn(2)p Fs(l)881 4920 y(j)942 4897 y Fq(\()p Fp(t)p Fq(\))f(+)g Fp(v)1220 4912 y Fn(0)p Fs(s)1314 4897 y Fo(\000)h Fp(z)1459 4912 y Fn(0)1499 4897 y Fo(j)1536 4828 y(j)p Fp(p)1613 4787 y Fn(2)p Fs(l)1613 4853 y(j)1674 4828 y Fq(\()p Fp(t)p Fq(\))g(+)f Fp(v)1953 4843 y Fn(0)p Fs(s)2047 4828 y Fo(\000)h Fp(z)2192 4843 y Fn(0)2231 4828 y Fo(jj)p Fp(p)2336 4787 y Fn(2)p Fs(l)2336 4853 y(i)2360 4834 y Fd(0)2397 4828 y Fq(\()p Fp(t)p Fq(\))f(+)g Fp(v)2675 4843 y Fn(0)p Fs(s)2743 4824 y Fd(0)2792 4828 y Fo(\000)h Fp(z)2937 4843 y Fn(0)2976 4828 y Fo(j)p Fp(:)0 5064 y Fq(Using)32 b(the)h(de\014nitions)f(of)g Fp(C)1095 5079 y Fn(0)1167 5064 y Fq(and)h Fp(C)1427 5079 y Fn(1)1466 5064 y Fq(,)g(w)m(e)g(obtain:)1100 5273 y Fo(j)p Fp(p)1177 5237 y Fn(2)p Fs(l)1177 5298 y(j)1238 5273 y Fq(\()p Fp(t)p Fq(\))22 b(+)g Fp(v)1516 5288 y Fn(0)p Fs(s)1610 5273 y Fo(\000)h Fp(z)t Fo(j)p 1082 5324 V 1082 5416 a(j)p Fp(p)1159 5382 y Fn(2)p Fs(l)1159 5439 y(j)1220 5416 y Fq(\()p Fp(t)p Fq(\))g(+)f Fp(v)1499 5431 y Fn(0)p Fs(s)1593 5416 y Fo(\000)h Fp(z)1738 5431 y Fn(0)1777 5416 y Fo(j)1843 5348 y Fq(=)1956 5280 y Fp(k)2010 5244 y Fj(\000)p Fn(2)p Fs(l)q Fn(+1+)p Fs(\020)p 1956 5324 351 4 v 1981 5416 a Fp(k)2035 5387 y Fn(2)p Fs(l)q Fj(\000)p Fs(n)p Fj(\000)p Fs(\016)2345 5348 y Fq(=)k Fp(k)2502 5306 y Fj(\000)p Fn(2)p Fs(\015)t Fn(+)p Fs(\016)2725 5348 y Fp(:)1888 5686 y Fq(9)p eop %%Page: 10 10 10 9 bop 0 107 a Fq(T)-8 b(aking)32 b(in)m(to)g(accoun)m(t)h(\(11\))f (for)g Fp(z)1283 122 y Fn(0)1323 107 y Fq(,)h(w)m(e)g(arriv)m(e)g(at)f (the)h(inequalit)m(y:)1089 327 y Fo(j)p Fp(p)1166 286 y Fn(2)p Fs(l)1166 351 y(j)1227 327 y Fq(\()p Fp(t)p Fq(\))22 b(+)g Fp(v)1505 342 y Fn(0)p Fs(s)1599 327 y Fo(\000)h Fp(z)t Fo(jj)p Fp(p)1853 286 y Fn(2)p Fs(l)1853 351 y(i)1877 333 y Fd(0)1914 327 y Fq(\()p Fp(t)p Fq(\))f(+)g Fp(v)2192 342 y 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1688 y Fp(k)3592 1652 y Fj(\000)p Fs(\016)3722 1688 y Fq(on)0 1809 y Fp(C)70 1824 y Fn(1)109 1809 y Fq(.)44 b(The)33 b(theorem)g(is)f(pro)m(v)m(ed.)p 180 1841 990 4 v 146 2012 a(The)i(pro)m(v)m(e)g(of)e(Theorem)h(3)p 146 2045 1036 4 v 37 w(is)j(analogous)g(to)h(that)g(of)g(Theorem)g(2.) 57 b(The)39 b(only)d(di\013erence)i(that)0 2133 y(w)m(e)33 b(replace)g(the)g(circle)e Fp(C)966 2148 y Fn(0)1038 2133 y Fq(b)m(y)i(smaller)d(circles)i Fp(C)1872 2148 y Fs(s)1941 2133 y Fq(whic)m(h)h(cut)g(out)f(the)h(in)m(terv)-5 b(als)32 b(\001)3201 2148 y Fs(s)3270 2133 y Fq(from)f(the)i(real)0 2253 y(axis.)146 2456 y(Let)39 b Fp(t)362 2471 y Fn(0)440 2456 y Fo(2)f Fp(\037)605 2471 y Fn(1)644 2456 y Fq(\()p Fp(k)s(;)17 b(\014)6 b(;)17 b(\016)t Fq(\))38 b(and)g Fp(t)g Fq(=)g Fp(t)1425 2471 y Fn(0)1491 2456 y Fq(+)26 b Fp(\034)9 b(~)-47 b(\027)7 b Fq(,)40 b(where)g Fo(j)p Fp(\034)11 b Fo(j)38 b Fp(<)f(k)2370 2420 y Fj(\000)p Fs(n)p Fn(+1)p Fj(\000)p Fn(2)p Fs(\016)2686 2456 y Fq(,)j(and)d Fp(~)-47 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y(complete)f(almost)f(orthogonal)f(system)k (in)e Fp(E)6 b Fq(\()p Fp(t)p Fq(\))p Fp(l)1870 483 y Fn(2)1909 468 y Fq(\()p Fp(Z)2021 432 y Fs(n)2068 468 y Fq(\))2106 432 y Fs(m)2173 468 y Fq(,)32 b(since)1483 688 y Fp(E)6 b Fq(\()p Fp(t)p Fq(\))28 b(=)g Fp(E)1876 703 y Fn(0)1937 688 y Fq(+)22 b Fp(O)s Fq(\()p Fp(k)2205 647 y Fj(\000)p Fs(\015)2304 688 y Fq(\))1310 b(\(32\))0 908 y(\(in)32 b(the)h(sense)h(of)e(b)s(ounded)i(op)s(erators\))e(and)h Fp(E)1811 923 y Fn(0)1846 908 y Fp(~)-45 b(e)1928 872 y Fn(0)1895 933 y Fs(j)t(r)1995 908 y Fq(=)24 b Fp(~)-45 b(e)2176 872 y Fn(0)2144 933 y Fs(j)t(r)2216 908 y Fq(,)32 b(and,)h(therefore,)1500 1137 y Fp(~)-45 b(e)1549 1152 y Fs(j)t(r)1647 1137 y Fq(=)23 b Fp(~)-45 b(e)1828 1096 y Fn(0)1795 1162 y Fs(j)t(r)1890 1137 y Fq(+)22 b Fp(O)s Fq(\()p Fp(k)2158 1096 y Fj(\000)p Fs(\015)2256 1137 y Fq(\))p Fp(:)1331 b Fq(\(33\))0 1357 y(Hence,)34 b(an)e(eigen)m(v)m (ector)i Fp(\036)1017 1372 y Fs(s)1054 1357 y Fq(\()p Fp(\034)11 b Fq(\))33 b(can)f(b)s(e)h(represen)m(ted)i(as)e(a)f(linear) f(com)m(bination)g(of)d Fp(~)-45 b(e)3229 1372 y Fs(j)t(r)3299 1357 y Fq(:)1451 1663 y Fp(\036)1509 1678 y Fs(s)1545 1663 y Fq(\()p Fp(\034)11 b Fq(\))28 b(=)1806 1544 y Fs(q)r Fn(+)p Fs(m)1957 1552 y Fi(q)1991 1544 y Fj(\000)p Fn(1)1884 1580 y Fm(X)1882 1754 y Fs(r)r Fn(=)p Fs(q)2098 1663 y Fp(c)2140 1622 y Fn(\()p Fs(s)p Fn(\))2140 1687 y Fs(r)2228 1663 y Fp(~)-45 b(e)2277 1678 y Fs(j)t(r)2347 1663 y Fp(:)0 1958 y Fq(Considering)32 b(that)g(function)g Fp(\036)1188 1973 y Fs(s)1225 1958 y Fq(\()p Fp(\034)11 b Fq(\))33 b(is)f(normalized,)f(w)m(e)i(obtain)1277 2142 y Fs(q)r Fn(+)p Fs(m)1428 2150 y Fi(q)1463 2142 y Fj(\000)p Fn(1)1355 2177 y Fm(X)1318 2362 y Fs(r)n(;r)1402 2344 y Fd(0)1423 2362 y Fn(=)p Fs(q)1569 2260 y Fp(c)1611 2219 y Fn(\()p Fs(s)p Fn(\))1611 2285 y Fs(r)1705 2260 y Fq(\026)-51 b Fp(c)1745 2210 y Fn(\()p Fs(s)p Fn(\))1745 2286 y Fs(r)1779 2267 y Fd(0)1864 2260 y Fp(<)24 b(~)-45 b(e)2013 2275 y Fs(j)t(r)2083 2260 y Fp(;)13 b(~)-45 b(e)2172 2275 y Fs(j)t(r)2239 2257 y Fd(0)2293 2260 y Fp(>)p Fq(=)27 b(1)p Fp(;)0 2566 y Fq(and,)33 b(b)m(y)g(\(33\),)1335 2651 y Fs(q)r Fn(+)p Fs(m)1486 2659 y Fi(q)1521 2651 y Fj(\000)p Fn(1)1413 2686 y Fm(X)1412 2860 y Fs(r)r Fn(=)p Fs(q)1628 2769 y Fo(j)p Fp(c)1698 2728 y Fn(\()p Fs(s)p Fn(\))1698 2794 y Fs(r)1789 2769 y Fo(j)1817 2728 y Fn(2)1884 2769 y Fq(=)27 b(1)22 b(+)g Fp(O)s Fq(\()p Fp(k)2326 2728 y Fj(\000)p Fs(\015)2425 2769 y Fq(\))p Fp(:)1162 b Fq(\(34\))0 3019 y(Using)32 b(again)f(\(33\),)h(w)m(e)i (obtain)1413 3139 y Fp(\036)1471 3154 y Fs(s)1507 3139 y Fq(\()p Fp(\034)11 b Fq(\))28 b(=)g Fp(\036)1826 3098 y Fn(0)1826 3164 y Fs(s)1865 3139 y Fq(\()p Fp(\034)11 b Fq(\))22 b(+)g Fp(\016)t(\036)2219 3154 y Fs(s)2256 3139 y Fq(\()p Fp(\034)11 b Fq(\))p Fp(;)1240 b Fq(\(35\))0 3313 y(where)34 b Fp(\036)340 3277 y Fn(0)340 3338 y Fs(s)379 3313 y Fq(\()p Fp(\034)11 b Fq(\))28 b(=)640 3247 y Fm(P)727 3273 y Fs(q)r Fn(+)p Fs(m)878 3281 y Fi(q)913 3273 y Fj(\000)p Fn(1)727 3338 y Fs(r)r Fn(=)p Fs(q)1024 3313 y Fp(c)1066 3277 y Fn(\()p Fs(s)p Fn(\))1066 3338 y Fs(r)1153 3313 y Fp(~)-45 b(e)1235 3277 y Fn(0)1202 3338 y Fs(j)t(r)1275 3313 y Fp(;)32 b Fq(and)h Fp(\016)t(\036)1629 3328 y Fs(s)1665 3313 y Fq(\()p Fp(\034)11 b Fq(\))33 b(satis\014es)g(the)g(estimate:)1520 3533 y Fo(k)p Fp(\016)t(\036)1675 3548 y Fs(s)1711 3533 y Fq(\()p Fp(\034)11 b Fq(\))p Fo(k)28 b Fp(<)g(c)2064 3548 y Fs(V)2125 3533 y Fp(k)2179 3492 y Fj(\000)p Fs(\015)2278 3533 y Fp(;)1347 b Fq(\(36\))0 3753 y Fo(k)22 b(\001)g(k)32 b Fq(b)s(eing)g(the)h(norm)f(in)f Fp(l)1032 3768 y Fn(2)1072 3753 y Fq(\()p Fp(Z)1184 3717 y Fs(n)1231 3753 y Fq(\))1269 3717 y Fs(m)1335 3753 y Fq(.)44 b(Substituting)31 b(\(35\))h(in)g(\(30\),)g(w)m(e)i(obtain)680 3961 y Fp(@)5 b(\025)793 3976 y Fs(s)831 3961 y Fq(\()p Fp(\034)11 b Fq(\))p 680 4005 280 4 v 765 4097 a Fp(@)5 b(\034)998 4028 y Fq(=)p Fp(<)1187 3961 y(@)g(H)1324 3976 y Fn(0)p 1187 4005 178 4 v 1221 4097 a Fp(@)g(\034)1374 4028 y(\036)1432 3987 y Fn(0)1432 4053 y Fs(s)1472 4028 y Fq(\()p Fp(\034)11 b Fq(\))p Fp(;)17 b(\036)1703 3987 y Fn(0)1703 4053 y Fs(s)1742 4028 y Fq(\()p Fp(\034)11 b Fq(\))28 b Fp(>)f Fq(+)h Fp(<)2219 3961 y(@)5 b(H)2356 3976 y Fn(0)p 2219 4005 V 2253 4097 a Fp(@)g(\034)2407 4028 y(\036)2465 3987 y Fn(0)2465 4053 y Fs(s)2504 4028 y Fq(\()p Fp(\034)11 b Fq(\))p Fp(;)17 b(\016)t(\036)2782 4043 y Fs(s)2818 4028 y Fq(\()p Fp(\034)11 b Fq(\))28 b Fp(>)g Fq(+)497 b(\(37\))852 4341 y Fp(<)966 4274 y(@)5 b(H)1103 4289 y Fn(0)p 966 4318 V 999 4410 a Fp(@)g(\034)1153 4341 y(\016)t(\036)1258 4356 y Fs(s)1294 4341 y Fq(\()p Fp(\034)11 b Fq(\))p Fp(;)17 b(\036)1525 4300 y Fn(0)1525 4366 y Fs(s)1564 4341 y Fq(\()p Fp(\034)11 b Fq(\))28 b Fp(>)g Fq(+)g Fp(<)2042 4274 y(@)5 b(H)2179 4289 y Fn(0)p 2042 4318 V 2075 4410 a Fp(@)g(\034)2229 4341 y(\016)t(\036)2334 4356 y Fs(s)2371 4341 y Fq(\()p Fp(\034)11 b Fq(\))p Fp(;)17 b(\016)t(\036)2649 4356 y Fs(s)2685 4341 y Fq(\()p Fp(\034)11 b Fq(\))28 b Fp(>)g(:)0 4541 y Fq(Ob)m(viously)-8 b(,)719 4738 y Fp(<)833 4671 y(@)5 b(H)970 4686 y Fn(0)p 833 4715 V 866 4807 a Fp(@)g(\034)1020 4738 y(\036)1078 4753 y Fs(s)1114 4738 y Fq(\()p Fp(\034)11 b Fq(\))1243 4697 y Fn(0)1283 4738 y Fp(;)17 b(\036)1385 4697 y Fn(0)1385 4763 y Fs(s)1424 4738 y Fq(\()p Fp(\034)11 b Fq(\))28 b Fp(>)p Fq(=)1761 4620 y Fs(q)r Fn(+)p Fs(m)1912 4628 y Fi(q)1946 4620 y Fj(\000)p Fn(1)1839 4655 y Fm(X)1801 4840 y Fs(r)n(;r)1885 4821 y Fd(0)1907 4840 y Fn(=)p Fs(q)2053 4738 y Fp(c)2095 4697 y Fn(\()p Fs(s)p Fn(\))2095 4763 y Fs(r)2189 4738 y Fq(\026)-51 b Fp(c)2229 4687 y Fn(\()p Fs(s)p Fn(\))2229 4764 y Fs(r)2263 4745 y Fd(0)2348 4738 y Fp(<)2461 4671 y(@)5 b(H)2598 4686 y Fn(0)p 2461 4715 V 2495 4807 a Fp(@)g(\034)2645 4738 y(~)-45 b(e)2726 4697 y Fn(0)2694 4763 y Fs(j)t(r)2766 4738 y Fp(;)13 b(~)-45 b(e)2887 4697 y Fn(0)2855 4763 y Fs(j)t(r)2922 4744 y Fd(0)2975 4738 y Fp(>)28 b(:)0 4998 y Fq(Substituting)k(\(31\),) g(w)m(e)h(get:)763 5301 y Fp(<)876 5234 y(@)5 b(H)1013 5249 y Fn(0)p 876 5278 V 909 5369 a Fp(@)g(\034)1063 5301 y(\036)1121 5260 y Fn(0)1121 5326 y Fs(s)1160 5301 y Fq(\()p Fp(\034)11 b Fq(\))p Fp(;)17 b(\036)1391 5260 y Fn(0)1391 5326 y Fs(s)1431 5301 y Fq(\()p Fp(\034)11 b Fq(\))28 b Fp(>)p Fq(=)f(2)p Fp(p)1865 5260 y Fn(2)p Fs(l)q Fj(\000)p Fn(2)1865 5326 y Fs(j)2016 5301 y Fq(\()p Fp(t)p Fq(\)\()q Fp(~)-50 b(p)2214 5316 y Fs(j)2251 5301 y Fq(\()p Fp(t)p Fq(\))p Fp(;)15 b(~)-47 b(\027)6 b Fq(\))2515 5182 y Fs(q)r Fn(+)p Fs(m)2666 5190 y Fi(q)2700 5182 y Fj(\000)p Fn(1)2592 5218 y Fm(X)2591 5392 y Fs(r)r Fn(=)p Fs(q)2807 5301 y Fo(j)p Fp(c)2877 5260 y Fn(\()p Fs(s)p Fn(\))2877 5326 y Fs(r)2968 5301 y Fo(j)2996 5260 y Fn(2)3035 5301 y Fp(:)1864 5686 y Fq(11)p eop %%Page: 12 12 12 11 bop 0 107 a Fq(T)-8 b(aking)32 b(in)m(to)g(accoun)m(t)h(\(34\),)f (w)m(e)i(obtain:)727 376 y Fp(<)841 309 y(@)5 b(H)978 324 y Fn(0)p 841 353 178 4 v 874 445 a Fp(@)g(\034)1028 376 y(\036)1086 335 y Fn(0)1086 401 y Fs(s)1125 376 y Fq(\()p Fp(\034)11 b Fq(\))p Fp(;)17 b(\036)1356 335 y Fn(0)1356 401 y Fs(s)1395 376 y Fq(\()p Fp(\034)11 b Fq(\))28 b Fp(>)p Fq(=)g(2)p Fp(p)1830 335 y Fn(2)p Fs(l)q Fj(\000)p Fn(2)1830 401 y Fs(j)1981 376 y Fq(\()p Fp(t)p Fq(\)\()q Fp(~)-50 b(p)2179 391 y Fs(j)2216 376 y Fq(\()p Fp(t)p Fq(\))p Fp(;)15 b(~)-47 b(\027)6 b Fq(\))2480 280 y Fm(\020)2529 376 y Fq(1)22 b(+)g Fp(O)s Fq(\()p Fp(k)2868 335 y Fj(\000)p Fs(\015)2966 376 y Fq(\))3004 280 y Fm(\021)3071 376 y Fp(:)554 b Fq(\(38\))0 638 y(Let)33 b(us)g(estimate)f Fo(k)751 599 y Fs(@)t(H)850 608 y Fh(0)p 750 615 134 4 v 777 672 a Fs(@)t(\034)894 638 y Fp(\016)t(\036)999 653 y Fs(s)1036 638 y Fq(\()p Fp(\034)11 b Fq(\))p Fo(k)p Fq(.)43 b(Using)32 b(\(31\),)g(is)g(easy)i(to)e(sho)m(w)i(that)724 797 y Fm(\015)724 846 y(\015)724 896 y(\015)724 946 y(\015)724 996 y(\015)780 879 y Fp(@)5 b(H)917 894 y Fn(0)p 780 923 178 4 v 813 1014 a Fp(@)g(\034)967 946 y(\016)t(\036)1072 961 y Fs(s)1108 797 y Fm(\015)1108 846 y(\015)1108 896 y(\015)1108 946 y(\015)1108 996 y(\015)1154 823 y Fn(2)1222 946 y Fq(=)27 b(4)1556 863 y Fm(X)1390 1047 y Fs(i)p Fj(2)p Fs(Z)1514 1028 y Fi(n)1557 1047 y Fs(;r)r Fn(=1)p Fs(;:::m)1858 946 y Fp(p)1907 905 y Fn(4)p Fs(l)q Fj(\000)p Fn(4)1907 971 y Fs(i)2058 946 y Fq(\()p Fp(t)p Fq(\)\()q Fp(~)-50 b(p)2256 961 y Fs(i)2284 946 y Fq(\()p Fp(t)p Fq(\))p Fp(;)15 b(~)-47 b(\027)6 b Fq(\))2531 905 y Fn(2)2571 946 y Fo(j)p Fq(\()p Fp(\016)t(\036)2742 961 y Fs(s)2778 946 y Fq(\()p Fp(\034)11 b Fq(\)\))2945 961 y Fs(ir)3007 946 y Fo(j)3035 905 y Fn(2)3074 946 y Fp(:)0 1245 y Fq(W)-8 b(e)33 b(estimate)f(the)h(terms)f(of)g(the)h(series)h(as)e(follo)m(ws:) 1158 1465 y Fp(p)1207 1424 y Fn(4)p Fs(l)q Fj(\000)p Fn(4)1207 1490 y Fs(i)1359 1465 y Fq(\()p Fp(t)p Fq(\)\()q Fp(~)-50 b(p)1557 1480 y Fs(i)1585 1465 y Fq(\()p Fp(t)p Fq(\))p Fp(;)15 b(~)-47 b(\027)6 b Fq(\))1832 1424 y Fn(2)1871 1465 y Fo(j)p Fq(\()p Fp(\016)t(\036)2042 1480 y Fs(s)2078 1465 y Fq(\))2116 1480 y Fs(ir)2178 1465 y Fo(j)2206 1424 y Fn(2)2273 1465 y Fo(\024)28 b Fp(X)2459 1480 y Fs(ir)2521 1465 y Fp(Y)2578 1480 y Fs(ir)2640 1465 y Fp(;)1366 1698 y(X)1447 1713 y Fs(ir)1537 1698 y Fq(=)g Fp(p)1690 1657 y Fn(4)p Fs(l)q Fj(\000)p Fn(2)1690 1723 y Fs(i)1841 1698 y Fq(\()p Fp(t)p Fq(\))p Fo(j)p Fq(\()p Fp(\016)t(\036)2123 1713 y Fs(s)2159 1698 y Fq(\))2197 1713 y Fs(ir)2259 1698 y Fo(j)2287 1657 y Fn(2)p Fj(\000)2387 1630 y Fh(1)p 2387 1642 31 4 v 2392 1683 a Fi(l)2431 1698 y Fp(;)1579 1886 y(Y)1636 1901 y Fs(ir)1726 1886 y Fq(=)f Fo(j)p Fq(\()p Fp(\016)t(\036)2000 1901 y Fs(s)2036 1886 y Fq(\))2074 1901 y Fs(ir)2136 1886 y Fo(j)2174 1818 y Fh(1)p 2174 1830 V 2179 1871 a Fi(l)2218 1886 y Fp(:)0 2060 y Fq(Using)32 b(H\177)-49 b(older's)33 b(inequalit)m(y)e(with)h Fp(p)c Fq(=)1567 2021 y Fn(2)p Fs(l)p 1522 2037 148 4 v 1522 2095 a Fn(2)p Fs(l)q Fj(\000)p Fn(1)1679 2060 y Fq(,)33 b Fp(q)e Fq(=)d(2)p Fp(l)r Fq(,)k(w)m(e)i (obtain:)982 2226 y Fm(\015)982 2276 y(\015)982 2326 y(\015)982 2375 y(\015)982 2425 y(\015)1038 2308 y Fp(@)5 b(H)1175 2323 y Fn(0)p 1038 2352 178 4 v 1071 2444 a Fp(@)g(\034)1225 2375 y(\016)t(\036)1330 2390 y Fs(s)1367 2375 y Fq(\()p Fp(\034)11 b Fq(\))1496 2226 y Fm(\015)1496 2276 y(\015)1496 2326 y(\015)1496 2375 y(\015)1496 2425 y(\015)1542 2252 y Fn(2)1609 2375 y Fo(\024)28 b(k)p Fp(H)1845 2390 y Fn(0)1884 2375 y Fp(\016)t(\036)1989 2390 y Fs(s)2026 2375 y Fq(\()p Fp(\034)11 b Fq(\))p Fo(k)2215 2299 y Fh(2)p Fi(l)p Fd(\000)p Fh(1)p 2215 2313 129 4 v 2269 2353 a Fi(l)2375 2375 y Fo(k)o Fp(\016)t(\036)2529 2390 y Fs(s)2566 2375 y Fq(\()p Fp(\034)g Fq(\))p Fo(k)2755 2300 y Fh(1)p 2755 2312 31 4 v 2760 2353 a Fi(l)2816 2375 y Fp(:)809 b Fq(\(39\))0 2658 y(It)48 b(is)g(easy)i(to)e(obtain)f (an)h(estimate)g(for)g Fo(k)o Fp(H)1773 2673 y Fn(0)1813 2658 y Fp(\016)t(\036)1918 2673 y Fs(s)1954 2658 y Fq(\()p Fp(\034)11 b Fq(\))p Fo(k)p Fq(.)91 b(In)49 b(fact,)j Fp(\036)2686 2673 y Fs(s)2723 2658 y Fq(\()p Fp(\034)11 b Fq(\))48 b(satis\014es)h(the)g(equation)0 2778 y Fp(H)8 b Fq(\()p Fp(t)p Fq(\))p Fp(\036)258 2793 y Fs(s)294 2778 y Fq(\()p Fp(\034)j Fq(\))28 b(=)g Fp(\025)612 2793 y Fs(s)648 2778 y Fp(\036)706 2793 y Fs(s)743 2778 y Fq(\()p Fp(\034)11 b Fq(\))22 b(and)g Fp(\036)1131 2742 y Fn(0)1131 2803 y Fs(s)1170 2778 y Fq(\()p Fp(\034)11 b Fq(\))22 b(satis\014es)g(the)h(equation)e Fp(H)2295 2793 y Fn(0)2334 2778 y Fq(\()p Fp(t)p Fq(\))p Fp(\036)2503 2742 y Fn(0)2503 2803 y Fs(s)2543 2778 y Fq(\()p Fp(\034)11 b Fq(\))28 b(=)f(\()p Fp(p)2890 2742 y Fn(2)p Fs(l)2890 2803 y(j)2951 2778 y Fq(\()p Fp(t)p Fq(\)+)p Fp(v)3185 2793 y Fn(0)p Fs(q)3259 2778 y Fq(\))p Fp(\036)3355 2742 y Fn(0)3355 2803 y Fs(s)3394 2778 y Fq(\()p Fp(\034)11 b Fq(\).Therefore,)0 2898 y(the)33 b(function)f Fp(\016)t(\036)655 2913 y Fs(s)724 2898 y Fq(satis\014es)h(the)g(equation)417 3118 y Fp(H)498 3133 y Fn(0)537 3118 y Fq(\()p Fp(t)p Fq(\))p Fp(\016)t(\036)753 3133 y Fs(s)790 3118 y Fq(\()p Fp(\034)11 b Fq(\))22 b(+)1054 3093 y(~)1039 3118 y Fp(V)g(\036)1176 3133 y Fs(s)1213 3118 y Fq(\()p Fp(\034)11 b Fq(\))28 b(=)f(\()p Fp(p)1560 3077 y Fn(2)p Fs(l)1560 3143 y(j)1621 3118 y Fq(\()p Fp(t)p Fq(\))c(+)f Fp(v)1900 3133 y Fn(0)p Fs(q)1973 3118 y Fq(\))p Fp(\016)t(\036)2116 3133 y Fs(s)2152 3118 y Fq(\()p Fp(\034)11 b Fq(\))23 b(+)f(\()p Fp(\025)2497 3133 y Fs(s)2556 3118 y Fo(\000)g Fp(p)2704 3077 y Fn(2)p Fs(l)2704 3143 y(j)2766 3118 y Fq(\()p Fp(t)p Fq(\))g Fo(\000)g Fp(v)3045 3133 y Fn(0)p Fs(q)3119 3118 y Fq(\))p Fp(\036)3215 3133 y Fs(s)3251 3118 y Fq(\()p Fp(\034)11 b Fq(\))p Fp(:)0 3338 y Fq(Hence,)810 3459 y Fp(H)891 3474 y Fn(0)931 3459 y Fq(\()p Fp(t)p Fq(\))p Fp(\016)t(\036)1147 3474 y Fs(s)1183 3459 y Fq(\()p Fp(\034)g Fq(\))28 b(=)g(\()p Fp(p)1531 3418 y Fn(2)p Fs(l)1531 3484 y(j)1592 3459 y Fq(\()p Fp(t)p Fq(\))22 b(+)g Fp(v)1870 3474 y Fn(0)p Fs(q)1944 3459 y Fq(\))p Fp(\016)t(\036)2087 3474 y Fs(s)2123 3459 y Fq(\()p Fp(\034)11 b Fq(\))23 b(+)f Fp(f)5 b(;)114 b Fo(k)p Fp(f)11 b Fo(k)27 b Fp(<)h(C)2927 3474 y Fs(V)2987 3459 y Fp(;)0 3633 y Fq(and)44 b Fo(k)p Fp(H)332 3648 y Fn(0)371 3633 y Fp(\016)t(\036)476 3648 y Fs(s)513 3633 y Fq(\()p Fp(\034)11 b Fq(\))p Fo(k)47 b Fp(<)h(k)917 3597 y Fn(2)p Fs(l)995 3633 y Fo(k)o Fp(\016)t(\036)1149 3648 y Fs(s)1186 3633 y Fq(\()p Fp(\034)11 b Fq(\))p Fo(k)30 b Fq(+)g Fp(C)1571 3648 y Fs(V)1632 3633 y Fp(:)44 b Fq(Noting)f(that)h(2)p Fp(l)32 b Fo(\000)e Fp(\015)53 b(>)47 b Fq(0)d(and)g(using)g(\(36\),)j(w)m(e)e(get:)0 3754 y Fo(k)p Fp(H)131 3769 y Fn(0)170 3754 y Fp(\016)t(\036)275 3769 y Fs(s)312 3754 y Fq(\()p Fp(\034)11 b Fq(\))p Fo(k)40 b Fp(<)f(C)716 3769 y Fs(V)777 3754 y Fp(k)831 3717 y Fn(2)p Fs(l)q Fj(\000)p Fs(\015)988 3754 y Fq(.)65 b(Substituting)39 b(the)h(last)f(estimate)g(in)g(\(39\))g(and)h(using)g(again)e(\(36\),)j (w)m(e)0 3874 y(obtain:)1351 3882 y Fm(\015)1351 3932 y(\015)1351 3982 y(\015)1351 4032 y(\015)1351 4082 y(\015)1407 3964 y Fp(@)5 b(H)1544 3979 y Fn(0)p 1407 4008 178 4 v 1440 4100 a Fp(@)g(\034)1594 4032 y(\016)t(\036)1699 4047 y Fs(s)1735 4032 y Fq(\()p Fp(\034)11 b Fq(\))1864 3882 y Fm(\015)1864 3932 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