Content-Type: multipart/mixed; boundary="-------------1001181527778" This is a multi-part message in MIME format. ---------------1001181527778 Content-Type: text/plain; name="10-12.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="10-12.keywords" Borel summability, Cauchy problem, Fourier transform, formal power series, Gevrey estimates ---------------1001181527778 Content-Type: application/postscript; name="idiv_gevrey.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="idiv_gevrey.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.96.1 Copyright 2007 Radical Eye Software %%Title: idiv_gevrey.dvi %%CreationDate: Thu Nov 26 14:04:39 2009 %%Pages: 22 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR17 CMR12 CMBX10 CMR10 MSBM10 CMSY10 CMMI10 CMBX12 %%+ CMEX10 CMMI8 CMSY8 CMR8 CMR6 CMMI6 CMSY6 CMTI10 MSBM7 LASY10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o idiv_gevrey.ps idiv_gevrey.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2009.11.26:1404 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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F509065E760E307175BE005EE05218EEBE42D1874F5F05A7753B37EF708BBD5C 03755827DD45CB18635DFACF6A2CE426B2293A60C22C139AA024A05F63BFA71E 9D6160AC68FEE98448B5C6E39C63707A8990A42FA009798A52CA7F69292ED893 6773B34EC41BCDEDAEF85222F8099926A3D4BB4D9C553F670C2D712838996E56 6CFB978CD65837B40FAF4936AF033AEFCB991F9C8092BC2B60CEAED447AE2ACC E510BC0214A428422FDB70FD2E718C672F0DD8923AF419CD2A0852113D8110EE 611281F679176A10FDF5406D0B689C8D8F0B75650AB0CD5AC92C86B4157E45FE AEEFD5C5CDED32D85DAFA0C0ED0F8596B71C902BDA09B364290E1806026AB081 385B155B907A83CCB5B6228CE2CC0817D386BAAB6EE9AF002BC3A8B131FD2482 3353A852A5001E836E8256FD29F4E2A1A995F9B12B578668A5731C9090D7F4A4 929B0AFF6D420242E937DC5CCBF1DF5AB114A79AC32548ED8E2B4079556DB2E9 8C3C7CF836B8FA811D733BADF0D1DB95C7710D8FF85051D13778413AAF72DDBB 62DA384CFE158792F2DFCED3CEFF26027F8FCBDEB4724C81B270905D5DAE1A2F 110310D0A5EF82D21374FC81C531F605B9411D6392086D7D0C8B5AB29BDD4138 246E84D462F1CB1FE38A771C1043981807DB7E00D574DD80FAA2B9E1DF61CEB1 320BA0D9495A282FE6687FAD3FE962671617740D3349A67B6E0DFE6874DEBC7B 22C986047880BB14AD03C376DFD6ACEA5C1AFE27145A64D1FB715FF8D3D77941 86F797F42B16E44A828D0698EF0EAA9E7F4B733CD24581A1F65188791F665931 E47B23EB79B65C4F7E0B8202347F7E506D81BF3BAC92416FB1BA6E6AE8DA33AF F7B8AEB5C1140BD591DCA9595903DA69888B80E5E8B71F974B2AB640AA45D562 988EE40054DA85B434346761E4C41FCAE09C75CA570B7D9EEAB126E07045E7E1 66D10B094520A16E4AED9FE8A0039BA06513D5DD06C51706A33C9B2D6EACDA6F BB9F4EA66D2BAD33422DE41BC5047FA9355677214B3AF92414286AC13C6196D1 61BAE0663BFC0FBAD6838EBCA62C280967C580311A4F4B723E0A4938C8505723 DF90F52DC5EA81CB57E19E3785F3D42C57260FBEF680E1027399C5A0B34C65CC B3EF8803E0771DF89BDBBF7876935F72C87D061345E3AADAA29D3F19FA94CB74 45BEEB8FF84E8EC24AACCAAF975C09AAB4510DD4F13702FF53A5AA68AF0D681F 2AFA02965E3E6C1DAEE9B1DF03C7C60503133445EECF57716C159CEF014AF378 F6757245E7DDE2533A372B57A1C128B41E2D33C20A49A087E3F722BF38F60ADE 3E5C1DBE3173E88DD0C8C363529C2BBBDFD76F916F3BFF38EAF2B461442AF0B2 E0061B008C36A46567610CDF6B0479E7259AC5E2EA8F8E9FDD64657911612180 BC01BA5A5B1B6E8917D5F89F811230FD2E568E72B94459A56D837839068CCD06 E1322DAD7413A2AFC56590A30C5495045021B58672CF2EAD09269073FAD1AD89 E01A5B61E25C2BBB853ABA164E206C29A6F87C3C9EC970C8A908953F3C43C4B7 0973993BE7B495DBC4434F9BDBEAFE672F904B799B160FCA2956CB6F4CDCACCB A702392CD746B2BFEAFCC234072F4DE09E83BD012B49663A66FA7B71FA29BD38 375DBC01425E41B9739C2586C9C0F5FA12E1A8DFB929B332B1E18D828246A782 AD879E6B1133AEA83BA1B6CB924722FB33AB74E9965ED3822D26E30AB4B3A457 09C20DB4FFD7DD61128D008A598BD8A311A76122A3D2552011B4A7B6877673C4 18286C2CC8AC101C0F1499F42363321A2BA196797DC4D06F9270AC1BF53DA182 8E6D50CAE5CDFA316AD8FA60A774BA7CE2D4A31D86E026704508F4AF72BEB02C 26EFEB1AE2091AD5FBD7056AE450D03CB95D356855ABE2FB64A8135D5FBB3A70 E81C4609303454DEEFA2FACFB015726B8EFC3DEE16B49013921F668C22A164E3 029B37DF547B86B88DA4448758FD178109EF5ECF8F2C0036C1BF381F35E3108E 859EB999280DD03428627B0774DF86400180C77BD82BD8764576FC88910DDB58 A2278C6C899BF2BA2C1FC28DAD7A3D0C46255B45700381ADD6CC070F37DD8226 E47B2C123BF404A672BB2C86CA2E1D9C608D57DECB1C7D514AFA627657B7934B C0541B7F1F8AF9B38ABF8207FBDAA834E98885B27C7012DBADAB8BFF96C37915 A733676B5FFE85B0C1359541302BB6FFE43F951A31B0A6148C10E61A546326B0 4CCC37CC29BD8DCCE92E708F02BD2AD79DC87CED7234E6DEC327031671CC046E A5087D785AA0D9DCCD718E3A5D91AF5413D09F7B2AB4A5A4570DDD9B4DB16988 1041C72C749475D667F0490F5DC57EE8190573FF88DECCB0ED40EC1340D74B61 3D59B83E626C745161039E409863A02597BF4FD4977770CF28E003475A483702 A1BA0945610D50915C6E3A6C03DC048D594B02281EAD38AF9F27EE70FCE12261 4F06C0410687C1547E89C25520D34029F5F7998F80C8AC05EC2EA7C2C19B23EF 9DB8F0D685F7CBF861FEDBA30568FAF2D21C7CA4AE1E7B1C43837A3A0ECD6EB2 E878DD491BF473F9888B38922D6969C28822E97185D1F02B5F196AD9AEB89B7D AC0A2940ED79F19B518AC7FB015260A4337F0C1C438CDEC04D65CC0D13ED74A3 83F1F14116A6235445138B760CE813F8BB37F44FA4DB3C63DC85C7F0CCB07ED0 923508BF8F049DFC07969C934E4426F4FA7B0AD9DC6C09671AFD359DF47391E8 BD3E5DBE4911B4C9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: LASY10 %!PS-AdobeFont-1.1: LASY10 1.001 %%CreationDate: 1992 Oct 23 20:19:17 %%RevisionDate: 2001 Jun 05 20:19:17 % Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.001) readonly def /Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def /FullName (LASY10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /LASY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 50 /a50 put readonly def /FontBBox{-19 -192 944 683}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB0B4E70F212EC976D65099D84E0D37A7A771C3101D6AD26A0513378F 21EC3643079EECE0C9AB54B4772E5DCA82D0D4ACC7F42FB493AA04A3BF4A1BD6 06ECE186315DBE9CFDCB1A0303E8D3E83027CD3AFA8F0BD466A8E8CA0E7164CF 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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 59 /comma put dup 61 /slash put dup 106 /j put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 68 /D put dup 69 /E put dup 72 /H put dup 76 /L put dup 80 /P put dup 82 /R put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 120 /x put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 65 /A put dup 66 /B put dup 67 /C put dup 70 /F put dup 72 /H put dup 91 /union put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 70 /F put dup 75 /K put dup 82 /R put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 61 /equal put dup 100 /d put dup 101 /e put dup 103 /g put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 91 /union put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 18 /theta put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 39 /phi1 put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 68 /D put dup 72 /H put dup 77 /M put dup 78 /N put dup 82 /R put dup 84 /T put dup 88 /X put dup 97 /a put dup 104 /h put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec 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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 14 /delta put dup 18 /theta put dup 20 /kappa put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 32 /psi put dup 39 /phi1 put dup 44 /arrowhookleft put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 22 /macron put dup 33 /exclam put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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y(data)d(and)f(the)h(co)s(e\016cien)m(t)h Fo(b)p Fp(\()p Fo(t;)15 b(z)t Fp(\).)61 b(In)36 b(fact,)j(one)e(can)g(sho)m(w) f(that)h(if)g(the)g(initial)g(condition)h Fo(u)p Fp(\(0)p Fo(;)15 b(z)t Fp(\))38 b(and)75 2490 y(the)g(co)s(e\016cien)m(t)j Fo(b)p Fp(\()p Fo(t;)15 b(z)t Fp(\))39 b(are)g(holomorphic)f(on)g(ev)m (ery)i(strips)d Fo(H)2352 2503 y Fk(~)2342 2520 y Fm(\014)2389 2490 y Fp(,)j(for)2614 2466 y(~)2601 2490 y Fo(\014)k(>)38 b Fp(0,)j(the)e(solution)g Fo(u)p Fp(\()p Fo(t;)15 b(z)t Fp(\))39 b(will)75 2603 y(b)s(e)32 b(an)h(en)m(tire)h(function)f(in)g (the)g(v)-5 b(ariable)34 b Fo(z)t Fp(.)49 b(Notice)35 b(that)f(there)f(are)g(v)m(ery)h(few)f(results)f(in)h(the)g(literature) 75 2716 y(concerning)e(en)m(tire)g(solutions)g(for)f(non-linear)h(Cauc) m(h)m(y)f(problems,)g(see)h(for)f(instance)h([7)q(].)216 2828 y(The)23 b(metho)s(dology)h(w)m(e)g(follo)m(w)g(is)f(similar)h(to) g(the)f(one)h(p)s(erformed)e(in)h(the)g(pap)s(er)f([11)q(])i(b)m(y)f (O.)g(Costin)g(and)75 2941 y(S.)35 b(T)-8 b(an)m(v)m(eer)37 b(for)e(the)g(Na)m(vier-Stok)m(es)j(equations.)57 b(W)-8 b(e)36 b(use)f(a)h(F)-8 b(ourier)36 b(transform)e(in)h(the)h(space)g(v) -5 b(ariable)75 3054 y Fo(z)42 b Fp(and)c(a)g(mo)s(di\014ed)f(Borel)i (transform)f(in)f(the)i(time)f(v)-5 b(ariable)39 b Fo(t)f Fp(to)h(transform)e(the)h(problem)g(\(2\))h(in)m(to)g(a)75 3167 y(di\013eren)m(tial)f(equation)g(with)e(constan)m(t)j(co)s (e\016cien)m(ts)f(dep)s(ending)e(on)g(a)h(real)h(parameter,)h(see)f (\(14\).)61 b(This)75 3280 y(di\013eren)m(tial)36 b(equation)f(can)g(b) s(e)f(rewritten)h(in)m(to)g(an)g(in)m(tegral)h(\014xed)e(p)s(oin)m(t)h (equation)g(\(24\).)55 b(The)34 b(strategy)75 3393 y(will)39 b(consist)f(in)g(constructing)h(a)g(suitable)g(Banac)m(h)g(space)g(in)f (whic)m(h)g(\(24\))h(has)f(a)h(unique)e(solution)i(and)75 3506 y(then)34 b(apply)f(a)h(mo)s(di\014ed)f(Laplace)i(transform)f(in)f (time)i Fo(t)e Fp(and)h(the)g(F)-8 b(ourier)34 b(in)m(v)m(erse)h(in)f (space)g Fo(z)k Fp(to)d(get)g(a)75 3619 y(holomorphic)28 b(solution)h Fo(u)p Fp(\()p Fo(t;)15 b(z)t Fp(\))29 b(of)g(the)f (original)i(problem)d(\(2\))i(with)f(the)h(required)e(asymptotic)j (expansion)75 3732 y(at)h Fo(t)25 b Fp(=)g(0.)216 3845 y(In)j(the)g(\014rst)g(part)g(of)h(the)f(pap)s(er,)g(w)m(e)h(in)m(tro)s (duce)g(Banac)m(h)g(spaces)g Fo(E)2599 3863 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2846 3845 y Fp(of)g(functions)f Fo(h)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))30 b(whic)m(h)75 3958 y(are)f(exp)s(onen)m(tially)h(decreasing)f(in)g(the)g(v)-5 b(ariable)29 b Fo(k)f Fi(2)d Fj(R)k Fp(and)f(exp)s(onen)m(tially)h (increasing)h(functions)e(in)g(the)75 4070 y(v)-5 b(ariable)38 b Fo(\034)46 b Fi(2)36 b Fj(R)p Fp(.)60 b(Similar)37 b(Banac)m(h)h(spaces)g(ha)m(v)m(e)g(b)s(een)e(in)m(tro)s(duced)h(in)f (sev)m(eral)j(recen)m(t)f(w)m(orks,)h(see)e([2)q(],)75 4183 y([10)q(],)31 b([11)q(],)g([12)q(].)216 4296 y(In)23 b(a)h(second)g(part,)h(w)m(e)f(construct)g(solutions)g Fo(h)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))25 b(of)f(the)g(\014xed)f (p)s(oin)m(t)h(equation)g(\(24\))h(in)f(the)f(Banac)m(h)75 4409 y(spaces)34 b(in)m(tro)s(duced)g(in)g(the)g(\014rst)f(part.)52 b(T)-8 b(o)34 b(ensure)g(the)g(holomorph)m(y)g(of)g(the)g(solutions)h (near)f(the)g(origin)75 4522 y(in)c Fj(C)h Fp(with)f(resp)s(ect)h(to)g Fo(\034)10 b Fp(,)31 b(w)m(e)g(use)g(the)g(crucial)g(prop)s(ert)m(y)f (of)h(functions)f(in)h Fo(E)2798 4541 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))3047 4522 y Fp(to)h(ha)m(v)m(e)g(a)f(decreasing)75 4635 y(rate)26 b(in)f(the)h(v)-5 b(ariable)26 b Fo(k)j Fp(dep)s(ending)24 b(on)h(the)h(p)s(osition)f(of)h Fo(\034)10 b Fp(.)39 b(Notice)27 b(that)f(a)g(similar)g(phenomenon)e(of)i(space)75 4748 y(regularit)m(y)36 b(that)g(dep)s(ends)d(on)i(the)h(p)s(osition)f (of)g(the)h(time)f(v)-5 b(ariable)36 b(has)f(b)s(een)g(observ)m(ed)g (for)g(the)g(Na)m(vier)75 4861 y(Stok)m(es)28 b(equations,)h(see)f([2)q (],)g([15)r(].)40 b(W)-8 b(e)28 b(sho)m(w)f(that)h(the)g(constructed)g 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Fo(u)p Fp(\()p Fo(t;)15 b(z)t Fp(\))33 b(of)g(the)f(problem)f(\(2\))j(as)e(the)g(F)-8 b(ourier)33 b(in)m(v)m(erse)g(transform)f(of)g Fo(X)2608 5553 y Fm(h)2653 5538 y Fp(\()p Fo(t;)15 b(k)s Fp(\))33 b(with)f(resp)s(ect)g(to)h Fo(k)s Fp(,)g(where)75 5651 y Fo(h)e Fp(solv)m(es)g(\(24\).)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 3809 100 a Fp(4)75 399 y Fq(2)135 b(W)-11 b(eigh)l(ted)46 b(Banac)l(h)e(spaces)75 601 y Fp(Let)32 b Fo(')c Fp(:)g Fj(R)445 615 y Fk(+)531 601 y Fi(!)f Fj(R)715 615 y Fk(+)805 601 y Fp(b)s(e)k(a)h(decreasing)h (function)e(on)h Fj(R)2000 615 y Fk(+)2090 601 y Fp(whic)m(h)f(is)h (deriv)-5 b(able)32 b(on)g(an)f(in)m(terv)-5 b(al)33 b(\(0)p Fo(;)15 b(\032)p Fp(\),)34 b(with)75 714 y Fo(\032)29 b(>)g Fp(0,)k(con)m(tin)m(uous)h(on)e Fj(R)1006 728 y Fk(+)1097 714 y Fp(and)g(satis\014es)h(the)g(follo)m(wing)h(prop)s (erties)e(:)45 b(there)33 b(exists)g Fo(M)3234 728 y Fk(0)3303 714 y Fo(>)c Fp(0)k(suc)m(h)f(that)75 827 y Fo(')134 794 y Fl(0)158 827 y Fp(\()p Fo(x)p 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Fd(such)f(that)440 2439 y Fi(jj)p Fo(h)p Fp(\()p Fo(t;)15 b(k)s Fp(\))p Fi(jj)785 2402 y Fc(R)785 2466 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))1031 2439 y Fp(=)25 b(sup)1135 2518 y Fm(t)p Fl(2)p Fc(R)1279 2439 y Fp(sup)1281 2518 y Fm(k)r Fl(2)p Fc(R)1416 2439 y Fp(\(1)c(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1743 2402 y Fm(\026)1791 2439 y Fp(\(1)h(+)f Fi(j)p Fo(t)p Fi(j)2066 2402 y Fk(2)2105 2439 y Fp(\))15 b(exp)q(\()p Fo(\014)5 b Fi(j)p Fo(k)s Fi(j)21 b(\000)f Fo(\027)6 b Fi(j)p Fo(t)p Fi(j)20 b Fp(+)g Fi(j)p Fo(k)s Fi(j)p Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\)\))p Fi(j)p Fo(h)p Fp(\()p Fo(t;)15 b(k)s Fp(\))p Fi(j)75 2673 y Fd(is)33 b(\014nite.)75 2843 y Fg(Remark)g Fp(:)44 b(Similar)33 b(Banac)m(h)g(spaces)g(ha)m(v)m (e)g(b)s(een)f(in)m(tro)s(duced)g(in)g(sev)m(eral)h(w)m(orks)g(lik)m(e) g([2)q(],)g([11)q(],)h([12)q(].)46 b(As)75 2956 y(in)30 b([11)q(],)h(w)m(e)g(in)m(tro)s(duce)f(the)h(follo)m(wing)h(v)m(ector)g (space.)75 3126 y Fg(De\014nition)j(2)46 b Fd(L)-5 b(et)46 b Fo(\014)5 b(;)15 b(\026)49 b Fi(2)g Fj(R)p Fd(.)81 b(We)45 b(denote)h(by)g Fo(E)1996 3144 y Fk(\()p Fm(\014)s(;\026)p Fk(\))2206 3126 y Fd(the)g(ve)-5 b(ctor)46 b(sp)-5 b(ac)g(e)47 b(of)f(c)-5 b(ontinuous)47 b(functions)75 3239 y Fo(h)26 b Fp(:)f Fj(R)g Fi(!)g Fj(C)32 b Fd(such)h(that)1127 3352 y Fi(jj)p Fo(h)p Fp(\()p Fo(k)s Fp(\))p Fi(jj)1399 3370 y Fk(\()p Fm(\014)s(;\026)p Fk(\))1590 3352 y Fp(=)25 b(sup)1688 3431 y Fm(k)r Fl(2)p Fc(R)1823 3352 y Fp(\(1)c(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))2150 3314 y Fm(\026)2213 3352 y Fp(exp\()p Fo(\014)5 b Fi(j)p Fo(k)s Fi(j)p Fp(\))p Fi(j)p Fo(h)p Fp(\()p Fo(k)s Fp(\))p Fi(j)75 3559 y Fd(is)33 b(\014nite.)216 3729 y Fp(In)e(the)h(follo)m(wing)h(lemmas)f(w)m(e)h (consider)e(m)m(ultiplication)j(and)d(in)m(tegral)j(op)s(erators)d (whic)m(h)h(are)g(sho)m(wn)75 3842 y(to)f(b)s(e)f(con)m(tin)m(uous)h (on)f(the)h(spaces)g Fo(E)1394 3860 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))1612 3842 y Fp(.)75 4033 y Fg(Lemma)k(1)45 b Fd(L)-5 b(et)28 b Fo(K)k Fp(:)26 b Fj(R)f Fi(!)g Fj(R)i Fd(b)-5 b(e)28 b(a)g(c)-5 b(ontinuous)28 b(function)g(such)g(that)h(ther)-5 b(e)28 b(exists)g Fo(M)3077 4047 y Fk(1)3142 4033 y Fi(\025)d Fp(0)j Fd(with)g Fi(j)p Fo(K)7 b Fp(\()p Fo(k)s Fp(\))p Fi(j)27 b(\024)75 4146 y Fo(M)163 4160 y Fk(0)203 4146 y Fi(j)p Fo(k)s Fi(j)12 b Fp(+)g Fo(M)486 4160 y Fk(1)526 4146 y Fd(,)29 b(for)h(al)5 b(l)29 b Fo(k)g Fi(2)c Fj(R)p Fd(.)40 b(L)-5 b(et)29 b Fo(p)p Fp(\()p Fo(t;)15 b(k)s Fp(\))26 b(:)f Fj(C)12 b Fi(\002)g Fj(R)25 b Fi(!)g Fj(C)k Fd(b)-5 b(e)28 b(a)i(p)-5 b(olynomial)32 b(with)d(r)-5 b(esp)g(e)g(ct)31 b(to)e Fo(t)g Fd(and)h(b)-5 b(ounde)g(d)75 4259 y(c)g(ontinuous)39 b(with)g(r)-5 b(esp)g(e)g(ct)39 b(to)g Fo(k)s Fd(.)58 b(Then,)40 b(ther)-5 b(e)38 b(exists)h Fo(C)2116 4273 y Fk(1)2190 4259 y Fo(>)c Fp(0)j Fd(\(dep)-5 b(ending)40 b(on)e Fo(p)p Fd(,)h Fo(\032)p Fd(,)g Fo(M)3295 4273 y Fk(1)3373 4259 y Fd(and)g Fo(\027)6 b Fd(\))37 b(such)75 4372 y(that)75 4555 y Fp(\(5\))635 b Fi(jj)p Fo(p)p Fp(\()p Fo(t;)15 b(k)s Fp(\))g(exp)r(\()p Fo(iK)7 b Fp(\()p Fo(k)s Fp(\))p Fo(t)p Fp(\))p Fo(f)j Fp(\()p Fo(k)s Fp(\))p Fi(jj)1833 4573 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2079 4555 y Fi(\024)25 b Fo(C)2240 4569 y Fk(1)2279 4555 y Fi(jj)p Fo(f)10 b Fp(\()p Fo(k)s Fp(\))p Fi(jj)2554 4573 y Fk(\()p Fm(\014)s Fk(+)p Fm(M)2747 4582 y Fh(0)2783 4573 y Fm(\032)p Fk(+)p Fm(')p Fk(\(0\))p Fm(;\026)p Fk(\))75 4738 y Fd(for)33 b(al)5 b(l)33 b Fo(f)i Fi(2)25 b Fo(E)583 4756 y Fk(\()p Fm(\014)s Fk(+)p Fm(M)776 4765 y Fh(0)811 4756 y Fm(\032)p Fk(+)p Fm(')p Fk(\(0\))p Fm(;\026)p Fk(\))1131 4738 y Fd(.)75 4972 y Fg(Pro)s(of)31 b Fp(W)-8 b(e)32 b(write)158 5180 y Fi(jj)p Fo(p)p Fp(\()p Fo(t;)15 b(k)s Fp(\))g(exp)q(\()p Fo(iK)7 b Fp(\()p Fo(k)s Fp(\))p Fo(t)p Fp(\))p Fo(f)j Fp(\()p Fo(k)s Fp(\))p Fi(jj)1165 5199 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))854 5327 y Fp(=)219 b(sup)950 5415 y Fm(t)p Fl(2)1038 5398 y Fk(\026)1022 5415 y Fm(D)r Fk(\(0)p Fm(;\032)p Fk(\))p Fl([)p Fc(R)p Fm(;k)r Fl(2)p Fc(R)1476 5327 y Fp(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1803 5290 y Fm(\026)1866 5327 y Fp(exp\(\()p Fo(\014)26 b Fp(+)20 b Fo(M)2331 5341 y Fk(0)2370 5327 y Fo(\032)h Fp(+)e Fo(')p Fp(\(0\)\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))p Fi(j)p Fo(f)10 b Fp(\()p Fo(k)s Fp(\))p Fi(j)481 5554 y(\002)20 b Fp(\(1)h(+)f Fi(j)p Fo(t)p Fi(j)847 5517 y Fk(2)887 5554 y Fp(\))15 b(exp\()p Fi(\000)p Fo(\027)6 b Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(p)p Fp(\()p Fo(t;)15 b(k)s Fp(\))p Fi(jj)g Fp(exp)r(\()p Fo(itK)7 b Fp(\()p Fo(k)s Fp(\)\))p Fi(j)15 b Fp(exp)r(\()p Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(k)s Fi(j)23 b(\000)d Fp(\()p Fo(M)2924 5568 y Fk(0)2964 5554 y Fo(\032)g Fp(+)g Fo(')p Fp(\(0\)\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))2746 5705 y Fi(\024)2863 5682 y Fp(~)2842 5705 y Fo(C)2907 5719 y Fk(1)2947 5705 y Fi(jj)p Fo(f)10 b Fp(\()p Fo(k)s Fp(\))p Fi(jj)3222 5724 y Fk(\()p Fm(\014)s Fk(+)p Fm(M)3415 5733 y Fh(0)3451 5724 y Fm(\032)p Fk(+)p Fm(')p Fk(\(0\))p Fm(;\026)p Fk(\))p eop end %%Page: 5 5 TeXDict begin 5 4 bop 3809 100 a Fp(5)75 399 y(where)96 579 y(~)75 602 y Fo(C)140 616 y Fk(1)205 602 y Fp(=)143 b(sup)301 690 y Fm(t)p Fl(2)389 673 y Fk(\026)373 690 y Fm(D)r Fk(\(0)p Fm(;\032)p Fk(\))p Fl([)p Fc(R)689 602 y Fp(sup)691 681 y Fm(k)r Fl(2)p Fc(R)826 602 y Fp(\(1)20 b(+)g Fi(j)p Fo(t)p Fi(j)1100 565 y Fk(2)1140 602 y Fp(\))15 b(exp\()p Fi(\000)p Fo(\027)6 b Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(p)p Fp(\()p Fo(t;)15 b(k)s Fp(\))p Fi(j)21 b(\002)f(j)15 b Fp(exp\()p Fo(itK)7 b Fp(\()p Fo(k)s Fp(\)\))p Fi(j)15 b Fp(exp)r(\()p Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(k)s Fi(j)22 b(\000)d Fp(\()p Fo(M)3285 616 y Fk(0)3325 602 y Fo(\032)h Fp(+)f Fo(')p Fp(\(0\)\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))p Fo(:)75 878 y Fp(F)-8 b(rom)31 b(the)f(fact)i(that)f Fo(')f Fp(is)h(decreasing) g(and)f(the)g(estimates)i(on)f Fo(K)7 b Fp(,)30 b(w)m(e)h(get)g(that) 158 1106 y Fo(C)223 1120 y Fk(1)p Fm(;)p Fk(1)343 1106 y Fp(=)172 b(sup)439 1194 y Fm(t)p Fl(2)527 1177 y Fk(\026)511 1194 y Fm(D)r Fk(\(0)p Fm(;\032)p Fk(\))p Fm(;k)r Fl(2)p Fc(R)870 1106 y Fp(\(1)21 b(+)f Fi(j)p Fo(t)p Fi(j)1145 1068 y Fk(2)1185 1106 y Fp(\))15 b(exp\()p Fi(\000)p Fo(\027)6 b Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(p)p Fp(\()p Fo(t;)15 b(k)s Fp(\))p Fi(jj)g Fp(exp)r(\()p Fo(itK)7 b Fp(\()p Fo(k)s Fp(\)\))p Fi(j)15 b Fp(exp)r(\()p Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(k)s Fi(j)23 b(\000)d Fp(\()p Fo(M)3222 1120 y Fk(0)3262 1106 y Fo(\032)g Fp(+)g Fo(')p Fp(\(0\)\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))284 1333 y Fi(\024)172 b Fp(sup)379 1421 y Fm(t)p Fl(2)468 1404 y Fk(\026)451 1421 y Fm(D)s Fk(\(0)p Fm(;\032)p Fk(\))p Fm(;k)r Fl(2)p Fc(R)811 1333 y Fp(\(1)21 b(+)f Fi(j)p Fo(t)p Fi(j)1086 1295 y Fk(2)1126 1333 y Fp(\))15 b(exp\()p Fi(\000)p Fo(\027)6 b Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(p)p Fp(\()p Fo(t;)15 b(k)s Fp(\))p Fi(j)22 b(\002)e Fp(exp\()p Fi(j)p Fo(K)7 b Fp(\()p Fo(k)s Fp(\))p Fi(jj)p Fo(t)p Fi(j)22 b Fp(+)e Fo(')p Fp(\(0\))p Fi(j)p Fo(k)s Fi(j)i(\000)e Fp(\()p Fo(M)3126 1347 y Fk(0)3166 1333 y Fo(\032)g Fp(+)g Fo(')p Fp(\(0\)\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))1517 1560 y Fi(\024)25 b Fo(C)1678 1574 y Fk(1)p Fm(;)p Fk(1)p Fm(;)p Fk(1)1852 1560 y Fp(=)172 b(sup)1948 1648 y Fm(t)p Fl(2)2037 1631 y Fk(\026)2020 1648 y Fm(D)r Fk(\(0)p Fm(;\032)p Fk(\))p Fm(;k)r Fl(2)p Fc(R)2380 1560 y Fp(\(1)21 b(+)e Fi(j)p Fo(t)p Fi(j)2654 1522 y Fk(2)2694 1560 y Fp(\))c(exp)q(\()p Fi(\000)p Fo(\027)6 b Fi(j)p Fo(t)p Fi(j)20 b Fp(+)g Fo(M)3323 1574 y Fk(1)3362 1560 y Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(p)p Fp(\()p Fo(t;)15 b(k)s Fp(\))p Fi(j)75 1840 y Fp(whic)m(h)33 b(is)h(\014nite)f(b)s(ecause)g Fo(p)p Fp(\()p Fo(t;)15 b(k)s Fp(\))34 b(is)g(b)s(ounded)d(with)i(resp) s(ect)h(to)g Fo(k)f Fi(2)d Fj(R)p Fp(.)49 b(On)33 b(the)g(other)h (hand,)f(from)g(the)75 1953 y(fact)e(that)g Fo(K)7 b Fp(\()p Fo(k)s Fp(\))31 b(is)g(real)g(v)-5 b(alued)30 b(and)g Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))c(=)f(0)31 b(for)f Fi(j)p Fo(t)p Fi(j)c Fo(>)f(\032)p Fp(,)30 b(w)m(e)h(get)158 2182 y Fo(C)223 2196 y Fk(1)p Fm(;)p Fk(2)343 2182 y Fp(=)329 b(sup)439 2264 y Fm(t)p Fl(2)p Fk(\()p Fl(\0001)p Fm(;\032)p Fk(\))p Fl([)p Fk(\()p Fm(\032;)p Fk(+)p Fl(1)p Fk(\))p Fm(;k)r Fl(2)p Fc(R)1183 2182 y Fp(\(1)21 b(+)f Fi(j)p Fo(t)p Fi(j)1458 2144 y Fk(2)1498 2182 y Fp(\))15 b(exp\()p Fi(\000)p Fo(\027)6 b Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(p)p Fp(\()p Fo(t;)15 b(k)s Fp(\))p Fi(jj)g Fp(exp)s(\()p Fo(itK)7 b Fp(\()p Fo(k)s Fp(\)\))p Fi(j)1296 2402 y(\002)19 b Fp(exp\()p Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(k)s Fi(j)k(\000)d Fp(\()p Fo(M)2109 2416 y Fk(0)2149 2402 y Fo(\032)g Fp(+)g Fo(')p Fp(\(0\)\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))1525 2550 y Fi(\024)25 b Fo(C)1686 2564 y Fk(1)p Fm(;)p Fk(2)p Fm(;)p Fk(1)1861 2550 y Fp(=)329 b(sup)1957 2633 y Fm(t)p Fl(2)p Fk(\()p Fl(\0001)p Fm(;\032)p Fk(\))p Fl([)p Fk(\()p Fm(\032;)p Fk(+)p Fl(1)p Fk(\))p Fm(;k)r Fl(2)p Fc(R)2702 2550 y Fp(\(1)21 b(+)f Fi(j)p Fo(t)p Fi(j)2977 2512 y Fk(2)3016 2550 y Fp(\))15 b(exp)q(\()p Fi(\000)p Fo(\027)6 b Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(p)p Fp(\()p Fo(t;)15 b(k)s Fp(\))p Fi(j)75 2825 y Fp(whic)m(h)24 b(is)g(\014nite)g(from)g(the)g(fact)h(that)f Fo(p)p Fp(\()p Fo(t;)15 b(k)s Fp(\))25 b(is)f(p)s(olynomial)h(in)f Fo(t)f Fp(and)h(b)s(ounded)e(in)i Fo(k)s Fp(.)38 b(F)-8 b(rom)25 b(the)f(inequalit)m(y)96 2915 y(~)75 2938 y Fo(C)140 2952 y Fk(1)205 2938 y Fi(\024)h Fo(C)366 2952 y Fk(1)430 2938 y Fp(=)g(max\()p Fo(C)795 2952 y Fk(1)p Fm(;)p Fk(1)p Fm(;)p Fk(1)945 2938 y Fo(;)15 b(C)1050 2952 y Fk(1)p Fm(;)p Fk(2)p Fm(;)p Fk(1)1200 2938 y Fp(\),)30 b(w)m(e)h(get)h(the)e (inequalit)m(y)i(\(5\).)1462 b Fb(2)75 3200 y Fg(Lemma)35 b(2)45 b Fd(Ther)-5 b(e)34 b(exists)f Fo(C)1119 3214 y Fk(2)1183 3200 y Fo(>)25 b Fp(0)33 b Fd(\(dep)-5 b(ending)34 b(on)f Fo(\032)p Fd(,)g Fo(\027)6 b Fd(\),)32 b(such)h(that)75 3460 y Fp(\(6\))968 b Fi(jj)1223 3336 y Fn(Z)1315 3362 y Fm(t)1275 3542 y Fk(0)1360 3460 y Fo(f)10 b Fp(\()p Fo(s;)15 b(k)s Fp(\))p Fo(ds)p Fi(jj)1758 3478 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2002 3460 y Fi(\024)25 b Fo(C)2163 3474 y Fk(2)2203 3460 y Fi(jj)p Fo(f)10 b Fp(\()p Fo(t;)15 b(k)s Fp(\))p Fi(jj)2551 3478 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))75 3704 y Fd(for)33 b(al)5 b(l)33 b Fo(f)i Fi(2)25 b Fo(E)583 3723 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))802 3704 y Fd(.)75 3966 y Fg(Pro)s(of)31 b Fp(W)-8 b(e)32 b(write)158 4228 y Fi(jj)223 4104 y Fn(Z)315 4130 y Fm(t)274 4310 y Fk(0)359 4228 y Fo(f)10 b Fp(\()p Fo(s;)15 b(k)s Fp(\))p Fo(ds)p Fi(jj)757 4246 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))778 4424 y Fp(=)220 b(sup)874 4511 y Fm(t)p Fl(2)963 4495 y Fk(\026)946 4511 y Fm(D)r Fk(\(0)p Fm(;\032)p Fk(\))p Fl([)p Fc(R)p Fm(;k)r Fl(2)p Fc(R)1400 4424 y Fp(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1727 4386 y Fm(\026)1775 4424 y Fp(\(1)h(+)f Fi(j)p Fo(t)p Fi(j)2050 4386 y Fk(2)2089 4424 y Fp(\))15 b(exp)q(\()p Fo(\014)5 b Fi(j)p Fo(k)s Fi(j)21 b(\000)f Fo(\027)6 b Fi(j)p Fo(t)p Fi(j)20 b Fp(+)g Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))900 4709 y Fi(\002)f(j)1030 4585 y Fn(Z)1122 4611 y Fm(t)1081 4791 y Fk(0)1151 4709 y Fi(f)p Fp(\(1)j(+)e Fi(j)p Fo(s)p Fi(j)1482 4671 y Fk(2)1521 4709 y Fp(\)\(1)i(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))1884 4671 y Fm(\026)1946 4709 y Fp(exp\()p Fo(\014)5 b Fi(j)p Fo(k)s Fi(j)22 b(\000)e Fo(\027)6 b Fi(j)p Fo(s)p Fi(j)20 b Fp(+)g Fo(')p Fp(\()p Fi(j)p Fo(s)p Fi(j)p Fp(\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))p Fo(f)10 b Fp(\()p Fo(s;)15 b(k)s Fp(\))p Fi(gA)p Fp(\()p Fo(t;)g(s;)g(k)s Fp(\))p Fo(ds)p Fi(j)75 4953 y Fp(where)347 5182 y Fi(A)p Fp(\()p Fo(t;)g(s;)g(k)s Fp(\))26 b(=)f(\(1)c(+)f Fi(j)p Fo(t)g Fi(\000)g Fo(s)p Fi(j)1247 5144 y Fk(2)1287 5182 y Fp(\))15 b(exp\()p Fi(\000)p Fo(\027)6 b Fi(j)p Fo(t)20 b Fi(\000)g Fo(s)p Fi(j)p Fp(\))g Fi(\002)2026 5120 y Fp(exp\()p Fi(\000)p Fo(\014)5 b Fi(j)p Fo(k)s Fi(j)21 b Fp(+)f Fo(\027)6 b Fp(\()p Fi(j)p Fo(t)21 b Fi(\000)e Fo(s)p Fi(j)i Fp(+)e Fi(j)p Fo(s)p Fi(j)p Fp(\))i Fi(\000)f Fo(')p Fp(\()p Fi(j)p Fo(s)p Fi(j)p Fp(\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))p 2026 5161 1547 4 v 2180 5244 a(\(1)h(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))2507 5218 y Fm(\026)2555 5244 y Fp(\(1)h(+)f Fi(j)p Fo(t)g Fi(\000)g Fo(s)p Fi(j)2984 5218 y Fk(2)3023 5244 y Fp(\)\(1)i(+)d Fi(j)p Fo(s)p Fi(j)3343 5218 y Fk(2)3383 5244 y Fp(\))75 5434 y(Using)31 b(the)f(parametrisation)i Fo(s)25 b Fp(=)g Fo(th)p Fp(,)30 b Fo(h)c Fi(2)f Fp([0)p Fo(;)15 b Fp(1],)32 b(w)m(e)f(get)h(that)668 5694 y Fi(jj)733 5570 y Fn(Z)825 5597 y Fm(t)785 5776 y Fk(0)870 5694 y Fo(f)10 b Fp(\()p Fo(s;)15 b(k)s Fp(\))p Fo(ds)p Fi(jj)1268 5713 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))1513 5694 y Fi(\024)24 b Fp(sup)1610 5773 y Fm(x)p Fl(2)p Fc(R)1745 5694 y Fp(\(1)d(+)f Fi(j)p Fo(x)p Fi(j)2039 5657 y Fk(2)2079 5694 y Fp(\))15 b(exp)q(\()p Fi(\000)p Fo(\027)6 b Fi(j)p Fo(x)p Fi(j)p Fp(\))2584 5671 y(~)2563 5694 y Fo(C)2628 5708 y Fk(2)2668 5694 y Fi(jj)p Fo(f)k Fp(\()p Fo(t;)15 b(k)s Fp(\))p Fi(jj)3016 5713 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))3236 5694 y Fo(;)p eop end %%Page: 6 6 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2853 y Fm(t)p Fl(2)2298 2836 y Fk(\026)2282 2853 y Fm(D)r Fk(\(0)p Fm(;\032)p Fk(\))p Fl([)p Fc(R)p Fm(;k)r Fl(2)p Fc(R)2736 2765 y Fp(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))3063 2728 y Fm(\026)3111 2765 y Fp(\(1)h(+)f Fi(j)p Fo(t)p Fi(j)3386 2728 y Fk(2)3425 2765 y Fp(\))90 3050 y(exp\()p Fo(\014)5 b Fi(j)p Fo(k)s Fi(j)s(\000)s Fo(\027)h Fi(j)p Fo(t)p Fi(j)s Fp(+)s Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))s Fi(\002)s(j)1172 2926 y Fn(Z)1268 2953 y Fm(t)1227 3132 y Fk(0)1297 3050 y Fi(f)p Fp(\(1)s(+)s Fi(j)p Fo(k)s Fi(j)p Fp(\))1634 3013 y Fm(\026)1683 3050 y Fp(\(1)s(+)s Fi(j)p Fo(s)p Fi(j)1933 3013 y Fk(2)1974 3050 y Fp(\))15 b(exp\()p Fo(\014)5 b Fi(j)p Fo(k)s Fi(j)s(\000)s Fo(\027)h Fi(j)p Fo(s)p Fi(j)s Fp(+)s Fo(')p Fp(\()p Fi(j)p Fo(s)p Fi(j)p Fp(\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))p Fo(f)k Fp(\()p Fo(s;)15 b(k)s Fp(\))p Fi(gB)s Fp(\()p Fo(t;)g(s;)g(k)s Fp(\))p Fo(ds)p Fi(j)75 3271 y Fp(where)158 3476 y Fi(B)s Fp(\()p Fo(t;)g(s;)g(k)s Fp(\))26 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y(0)1961 5565 y Fp(exp\()p Fi(j)p Fo(K)7 b Fp(\()p Fo(k)s Fp(\))p Fi(jj)p Fo(t)p Fi(j)p Fp(\(1)23 b Fi(\000)d Fo(h)p Fp(\))h(+)f(\()p Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))i Fi(\000)e Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fo(h)p Fp(\)\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))p 1961 5606 1701 4 v 2256 5689 a(\(1)h(+)f Fi(j)p Fo(t)p Fi(j)2531 5663 y Fk(2)2571 5689 y Fo(h)2623 5663 y Fk(2)2662 5689 y Fp(\)\(1)i(+)e Fi(j)p Fo(t)p Fi(j)2973 5663 y Fk(2)3013 5689 y Fp(\(1)h Fi(\000)e Fo(h)p Fp(\))3291 5663 y Fk(2)3332 5689 y Fp(\))3672 5627 y Fo(dh)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 3809 100 a Fp(7)75 399 y(F)-8 b(rom)31 b(the)f(h)m(yp)s(othesis)g(on)h Fo(')1104 366 y Fl(0)1127 399 y Fp(,)g(w)m(e)g(get)g(b)m(y)g(the)f(mean)h(v)-5 b(alue)30 b(theorem)h(that)1328 579 y Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))22 b Fi(\000)e Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fo(h)p Fp(\))27 b Fi(\024)e(\000)p Fo(M)2199 593 y Fk(0)2238 579 y Fi(j)p Fo(t)p Fi(j)p Fp(\(1)d Fi(\000)e Fo(h)p Fp(\))75 760 y(for)30 b(all)h Fi(j)p Fo(t)p Fi(j)26 b(\024)f Fo(\032)p Fp(,)31 b Fo(h)25 b Fi(2)g Fp([0)p Fo(;)15 b Fp(1].)42 b(So)31 b(that)g(from)f(the)g (estimates)i(on)e Fo(K)7 b Fp(,)31 b(w)m(e)g(get)598 999 y Fo(C)663 1013 y Fk(3)p Fm(;)p Fk(1)782 999 y Fi(\024)25 b Fo(C)943 1013 y Fk(3)p Fm(;)p Fk(1)p Fm(;)p Fk(1)1118 999 y Fp(=)95 b(sup)1214 1086 y Fm(t)p Fl(2)1302 1069 y Fk(\026)1286 1086 y Fm(D)r Fk(\(0)p Fm(;\032)p Fk(\))1507 999 y Fi(j)p Fo(t)p Fi(j)p Fp(\(1)21 b(+)f Fi(j)p Fo(t)p Fi(j)1865 961 y Fk(2)1905 999 y Fp(\))1955 875 y Fn(Z)2047 901 y Fk(1)2006 1081 y(0)2317 937 y Fp(exp)o(\()p Fo(M)2578 951 y Fk(1)2618 937 y Fi(j)p Fo(t)p Fi(j)p Fp(\(1)i Fi(\000)e Fo(h)p Fp(\)\))p 2111 978 1111 4 v 2111 1061 a(\(1)h(+)f Fi(j)p Fo(t)p Fi(j)2386 1035 y Fk(2)2426 1061 y Fo(h)2478 1035 y Fk(2)2518 1061 y Fp(\)\(1)h(+)f Fi(j)p Fo(t)p Fi(j)2828 1035 y Fk(2)2868 1061 y Fp(\(1)h Fi(\000)f Fo(h)p Fp(\))3147 1035 y Fk(2)3187 1061 y Fp(\))3232 999 y Fo(dh)75 1256 y Fp(whic)m(h)28 b(is)h(\014nite.)40 b(Moreo)m(v)m(er,)31 b(from)e(the)f(fact)i(that)f Fo(K)7 b Fp(\()p Fo(k)s Fp(\))29 b(is)g(real)g(v)-5 b(alued)29 b(and)f(that)h Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))e(=)e(0,)k (for)g Fi(j)p Fo(t)p Fi(j)c Fo(>)g(\032)p Fp(,)75 1369 y(w)m(e)31 b(ha)m(v)m(e)158 1610 y Fo(C)223 1624 y Fk(3)p Fm(;)p Fk(2)343 1610 y Fp(=)356 b(sup)439 1693 y Fm(t)p Fl(2)p Fk(\()p Fl(\0001)p Fm(;)p Fl(\000)p Fm(\032)p Fk(\))p Fl([)p Fk(\()p Fm(\032;)p Fk(+)p Fl(1)p Fk(\))p Fm(;k)r Fl(2)p Fc(R)1253 1610 y Fi(j)p Fo(t)p Fi(j)p Fp(\(1)22 b(+)e Fi(j)p Fo(t)p Fi(j)1612 1573 y Fk(2)1651 1610 y Fp(\))1701 1486 y Fn(Z)1793 1513 y Fk(1)1752 1693 y(0)1857 1549 y Fi(j)15 b Fp(exp)q(\()p Fo(it)p Fp(\(1)21 b Fi(\000)f Fo(h)p Fp(\))p Fo(K)7 b Fp(\()p Fo(k)s Fp(\)\))p Fi(j)15 b Fp(exp)r(\(\()p Fi(\000)p Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fo(h)p Fp(\))23 b(+)d Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\)\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))p 1857 1589 1882 4 v 2242 1672 a(\(1)h(+)f Fi(j)p Fo(t)p Fi(j)2517 1646 y Fk(2)2557 1672 y Fo(h)2609 1646 y Fk(2)2649 1672 y Fp(\)\(1)h(+)f Fi(j)p Fo(t)p Fi(j)2959 1646 y Fk(2)2999 1672 y Fp(\(1)h Fi(\000)f Fo(h)p Fp(\))3278 1646 y Fk(2)3318 1672 y Fp(\))3748 1610 y Fo(dh)854 1892 y Fi(\024)25 b Fo(C)1015 1906 y Fk(3)p Fm(;)p Fk(2)p Fm(;)p Fk(1)1189 1892 y Fp(=)280 b(sup)1285 1975 y Fm(t)p Fl(2)p Fk(\()p Fl(\0001)p Fm(;)p Fl(\000)p Fm(\032)p Fk(\))p Fl([)p Fk(\()p Fm(\032;)p Fk(+)p Fl(1)p Fk(\))1947 1892 y Fi(j)p Fo(t)p Fi(j)p Fp(\(1)21 b(+)f Fi(j)p Fo(t)p Fi(j)2305 1854 y Fk(2)2345 1892 y Fp(\))2395 1768 y Fn(Z)2486 1794 y Fk(1)2446 1974 y(0)3084 1830 y Fp(1)p 2551 1871 1111 4 v 2551 1954 a(\(1)h(+)f Fi(j)p Fo(t)p Fi(j)2826 1928 y Fk(2)2866 1954 y Fo(h)2918 1928 y Fk(2)2957 1954 y Fp(\)\(1)i(+)e Fi(j)p Fo(t)p Fi(j)3268 1928 y Fk(2)3307 1954 y Fp(\(1)h Fi(\000)f Fo(h)p Fp(\))3586 1928 y Fk(2)3626 1954 y Fp(\))3672 1892 y Fo(dh)75 2160 y Fp(This)28 b(latter)i(in)m (tegral)g Fo(C)921 2174 y Fk(3)p Fm(;)p Fk(2)p Fm(;)p Fk(1)1099 2160 y Fp(is)f(\014nite)g(from)f(the)h(corollary)h(4.9)g(of)f ([10)q(].)40 b(F)-8 b(rom)30 b(the)e(inequalit)m(y)3449 2137 y(~)3429 2160 y Fo(C)3494 2174 y Fk(3)3558 2160 y Fi(\024)d Fo(C)3726 2127 y Fl(0)3719 2184 y Fk(3)3784 2160 y Fp(=)75 2273 y(max\()p Fo(C)344 2287 y Fk(3)p Fm(;)p Fk(1)p Fm(;)p Fk(1)494 2273 y Fo(;)15 b(C)599 2287 y Fk(3)p Fm(;)p Fk(2)p Fm(;)p Fk(1)748 2273 y Fp(\),)31 b(the)g(inequalit)m(y)h(\(7\))f(follo)m(ws.)1900 b Fb(2)75 2504 y Fg(Lemma)35 b(4)45 b Fd(L)-5 b(et)36 b Fo(s)746 2518 y Fk(1)785 2504 y Fo(;)15 b(s)868 2518 y Fk(2)939 2504 y Fi(\025)30 b Fp(0)36 b Fd(b)-5 b(e)36 b(non)g(ne)-5 b(gative)36 b(inte)-5 b(gers.)51 b(Assume)36 b(that)h Fo(\026)30 b(>)h Fp(max\()p Fo(s)3124 2518 y Fk(1)3186 2504 y Fp(+)22 b(1)p Fo(;)15 b(s)3407 2518 y Fk(2)3469 2504 y Fp(+)22 b(1\))p Fd(.)52 b(We)75 2616 y(cho)-5 b(ose)34 b Fo(s)406 2630 y Fk(1)p Fm(;)p Fk(2)525 2616 y Fi(\024)25 b Fp(min\()p Fi(\000)p Fo(s)922 2630 y Fk(1)961 2616 y Fo(;)15 b Fi(\000)p Fo(s)1115 2630 y Fk(2)1154 2616 y Fp(\))p Fd(.)42 b(Then,)33 b(ther)-5 b(e)33 b(exists)g Fo(C)2064 2630 y Fk(4)2129 2616 y Fo(>)25 b Fp(0)33 b Fd(\(dep)-5 b(ending)34 b(on)f Fo(\032)p Fd(,)f Fo(\026)p Fd(,)h Fo(s)3159 2630 y Fk(1)3198 2616 y Fd(,)f Fo(s)3301 2630 y Fk(2)3340 2616 y Fd(\))h(such)g(that)75 2882 y Fp(\(8\))84 b Fi(jj)p Fp(\(1)22 b(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))652 2845 y Fm(s)685 2854 y Fh(1)p Ff(;)p Fh(2)789 2758 y Fn(Z)879 2785 y Fm(t)839 2965 y Fk(0)924 2758 y Fn(Z)1015 2785 y Fk(+)p Fl(1)975 2965 y(\0001)1145 2882 y Fp(\()p Fo(k)j Fi(\000)d Fo(k)1388 2896 y Fk(1)1428 2882 y Fp(\))1463 2845 y Fm(s)1496 2854 y Fh(1)1535 2882 y Fo(k)1585 2842 y Fm(s)1618 2851 y Fh(2)1582 2908 y Fk(1)1656 2882 y Fo(f)10 b Fp(\()p Fo(t)20 b Fi(\000)g Fo(s;)15 b(k)23 b Fi(\000)d Fo(k)2181 2896 y Fk(1)2221 2882 y Fp(\))p Fo(g)s Fp(\()p Fo(s;)15 b(k)2467 2896 y Fk(1)2508 2882 y Fp(\))p Fo(dk)2637 2896 y Fk(1)2678 2882 y Fo(ds)p Fi(jj)2818 2901 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2442 3084 y Fi(\024)25 b Fo(C)2603 3098 y Fk(4)2643 3084 y Fi(jj)p Fo(f)10 b Fp(\()p Fo(t;)15 b(k)s Fp(\))p Fi(jj)2991 3102 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))3211 3084 y Fi(jj)p Fo(g)s Fp(\()p Fo(t;)g(k)s Fp(\))p Fi(jj)3550 3102 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))75 3264 y Fd(for)33 b(al)5 b(l)33 b Fo(f)5 b(;)15 b(g)29 b Fi(2)c Fo(E)665 3283 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))884 3264 y Fd(.)75 3495 y Fg(Pro)s(of)31 b Fp(W)-8 b(e)32 b(write)158 3734 y Fi(jj)p Fp(\(1)22 b(+)d Fi(j)p Fo(k)s Fi(j)p Fp(\))535 3697 y Fm(s)568 3706 y Fh(1)p Ff(;)p Fh(2)672 3611 y Fn(Z)763 3637 y Fm(t)723 3817 y Fk(0)808 3611 y Fn(Z)899 3637 y Fk(+)p Fl(1)859 3817 y(\0001)1029 3734 y Fp(\()p Fo(k)k Fi(\000)d Fo(k)1272 3748 y Fk(1)1312 3734 y Fp(\))1347 3697 y Fm(s)1380 3706 y Fh(1)1419 3734 y Fo(k)1469 3694 y Fm(s)1502 3703 y Fh(2)1466 3760 y Fk(1)1540 3734 y Fo(f)10 b Fp(\()p Fo(t)20 b Fi(\000)g Fo(s;)15 b(k)23 b Fi(\000)d Fo(k)2065 3748 y Fk(1)2105 3734 y Fp(\))p Fo(g)s Fp(\()p Fo(s;)15 b(k)2351 3748 y Fk(1)2392 3734 y Fp(\))p Fo(dk)2521 3748 y Fk(1)2562 3734 y Fo(ds)p Fi(jj)2702 3753 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))644 3937 y Fp(=)220 b(sup)740 4025 y Fm(t)p Fl(2)829 4008 y Fk(\026)812 4025 y Fm(D)r Fk(\(0)p Fm(;\032)p Fk(\))p Fl([)p Fc(R)p Fm(;k)r Fl(2)p Fc(R)1267 3937 y Fp(\(1)21 b(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))1593 3900 y Fm(\026)p Fk(+)p Fm(s)1723 3909 y Fh(1)p Ff(;)p Fh(2)1813 3937 y Fp(\(1)i(+)e Fi(j)p Fo(t)p Fi(j)2087 3900 y Fk(2)2127 3937 y Fp(\))i Fi(\002)f Fp(exp\()p Fo(\014)5 b Fi(j)p Fo(k)s Fi(j)21 b(\000)f Fo(\027)6 b Fi(j)p Fo(t)p Fi(j)20 b Fp(+)g Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))86 4222 y Fi(\002)11 b(j)208 4098 y Fn(Z)298 4125 y Fm(t)258 4305 y Fk(0)343 4098 y Fn(Z)434 4125 y Fk(+)p Fl(1)393 4305 y(\0001)563 4222 y Fi(f)p Fp(\(1)g(+)g Fi(j)p Fo(k)i Fi(\000)e Fo(k)995 4236 y Fk(1)1036 4222 y Fi(j)p Fp(\))1096 4185 y Fm(\026)1143 4222 y Fp(\(1)g(+)g Fi(j)p Fo(t)g Fi(\000)g Fo(s)p Fi(j)1535 4185 y Fk(2)1573 4222 y Fp(\))k(exp\()p Fo(\014)5 b Fi(j)p Fo(k)13 b Fi(\000)e Fo(k)2067 4236 y Fk(1)2108 4222 y Fi(j)g(\000)g Fo(\027)6 b Fi(j)p Fo(t)11 b Fi(\000)g Fo(s)p Fi(j)g Fp(+)g Fo(')p Fp(\()p Fi(j)p Fo(t)g Fi(\000)g Fo(s)p Fi(j)p Fp(\))p Fi(j)p Fo(k)i Fi(\000)e Fo(k)3151 4236 y Fk(1)3187 4222 y Fi(j)p Fp(\))p Fo(f)f Fp(\()p Fo(t)h Fi(\000)g Fo(s;)k(k)e Fi(\000)e Fo(k)3735 4236 y Fk(1)3774 4222 y Fp(\))p Fi(g)680 4425 y(\002)20 b(f)p Fp(\(1)h(+)f Fi(j)p Fo(k)1080 4439 y Fk(1)1120 4425 y Fi(j)p Fp(\))1180 4387 y Fm(\026)1228 4425 y Fp(\(1)h(+)e Fi(j)p Fo(s)p Fi(j)1512 4387 y Fk(2)1552 4425 y Fp(\))c(exp\()p Fo(\014)5 b Fi(j)p Fo(k)1904 4439 y Fk(1)1945 4425 y Fi(j)20 b(\000)g Fo(\027)6 b Fi(j)p Fo(s)p Fi(j)20 b Fp(+)g Fo(')p Fp(\()p Fi(j)p Fo(s)p Fi(j)p Fp(\))p Fi(j)p Fo(k)2630 4439 y Fk(1)2672 4425 y Fi(j)p Fp(\))p Fo(g)s Fp(\()p Fo(s;)15 b(k)2943 4439 y Fk(1)2984 4425 y Fp(\))p Fi(gC)5 b Fp(\()p Fo(t;)15 b(s;)g(k)s(;)g(k)3445 4439 y Fk(1)3487 4425 y Fp(\))p Fo(dk)3616 4439 y Fk(1)3656 4425 y Fo(ds)p Fi(j)75 4606 y Fp(where)158 4811 y Fi(C)5 b Fp(\()p Fo(t;)15 b(s;)g(k)s(;)g(k)539 4825 y Fk(1)580 4811 y Fp(\))26 b(=)f(\()p Fo(k)e Fi(\000)d Fo(k)980 4825 y Fk(1)1020 4811 y Fp(\))1055 4773 y Fm(s)1088 4782 y Fh(1)1127 4811 y Fo(k)1177 4770 y Fm(s)1210 4779 y Fh(2)1174 4837 y Fk(1)1248 4811 y Fi(\002)899 4946 y Fp(exp\()p Fi(\000)p Fo(\014)5 b Fi(j)p Fo(k)24 b Fi(\000)c Fo(k)1434 4960 y Fk(1)1474 4946 y Fi(j)g(\000)g Fo(\014)5 b Fi(j)p Fo(k)1738 4960 y Fk(1)1779 4946 y Fi(j)20 b Fp(+)g Fo(\027)6 b Fp(\()p Fi(j)p Fo(t)20 b Fi(\000)g Fo(s)p Fi(j)g Fp(+)g Fi(j)p Fo(s)p Fi(j)p Fp(\))h Fi(\000)f Fo(')p Fp(\()p Fi(j)p Fo(s)p Fi(j)p Fp(\))p Fi(j)p Fo(k)2883 4960 y Fk(1)2924 4946 y Fi(j)g(\000)g Fo(')p Fp(\()p Fi(j)p Fo(t)h Fi(\000)f Fo(s)p Fi(j)p Fp(\))p Fi(j)p Fo(k)k Fi(\000)c Fo(k)3661 4960 y Fk(1)3701 4946 y Fi(j)p Fp(\))p 899 4987 2863 4 v 1407 5070 a(\(1)h(+)f Fi(j)p Fo(k)j Fi(\000)d Fo(k)1832 5084 y Fk(1)1872 5070 y Fi(j)p Fp(\))1932 5044 y Fm(\026)1979 5070 y Fp(\(1)h(+)f Fi(j)p Fo(k)2243 5084 y Fk(1)2283 5070 y Fi(j)p Fp(\))2343 5044 y Fm(\026)2391 5070 y Fp(\(1)h(+)e Fi(j)p Fo(t)i Fi(\000)f Fo(s)p Fi(j)2820 5044 y Fk(2)2859 5070 y Fp(\)\(1)h(+)f Fi(j)p Fo(s)p Fi(j)3179 5044 y Fk(2)3219 5070 y Fp(\))75 5237 y(Using)31 b(the)f(parametrization)i Fo(s)25 b Fp(=)g Fo(th)p Fp(,)31 b Fo(h)25 b Fi(2)g Fp([0)p Fo(;)15 b Fp(1],)33 b(w)m(e)e(get)75 5499 y(\(9\))84 b Fi(jj)p Fp(\(1)22 b(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))652 5461 y Fm(s)685 5470 y Fh(1)p Ff(;)p Fh(2)789 5375 y Fn(Z)879 5401 y Fm(t)839 5581 y Fk(0)924 5375 y Fn(Z)1015 5401 y Fk(+)p Fl(1)975 5581 y(\0001)1145 5499 y Fp(\()p Fo(k)j Fi(\000)d Fo(k)1388 5513 y Fk(1)1428 5499 y Fp(\))1463 5461 y Fm(s)1496 5470 y Fh(1)1535 5499 y Fo(k)1585 5458 y Fm(s)1618 5467 y Fh(2)1582 5525 y Fk(1)1656 5499 y Fo(f)10 b Fp(\()p Fo(t)20 b Fi(\000)g Fo(s;)15 b(k)23 b Fi(\000)d Fo(k)2181 5513 y Fk(1)2221 5499 y Fp(\))p Fo(g)s Fp(\()p Fo(s;)15 b(k)2467 5513 y Fk(1)2508 5499 y Fp(\))p Fo(dk)2637 5513 y Fk(1)2678 5499 y Fo(ds)p Fi(jj)2818 5517 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2442 5705 y Fi(\024)2559 5682 y Fp(~)2538 5705 y Fo(C)2603 5719 y Fk(4)2643 5705 y Fi(jj)p Fo(f)10 b Fp(\()p Fo(t;)15 b(k)s Fp(\))p Fi(jj)2991 5724 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))3211 5705 y Fi(jj)p Fo(g)s Fp(\()p Fo(t;)g(k)s Fp(\))p Fi(jj)3550 5724 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))p eop end %%Page: 8 8 TeXDict begin 8 7 bop 3809 100 a Fp(8)75 399 y(where)179 588 y(~)158 611 y Fo(C)223 625 y Fk(4)288 611 y Fp(=)219 b(sup)384 699 y Fm(t)p Fl(2)472 682 y Fk(\026)456 699 y Fm(D)r Fk(\(0)p Fm(;\032)p Fk(\))p Fl([)p Fc(R)p Fm(;k)r Fl(2)p Fc(R)910 611 y Fp(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1237 573 y Fm(\026)p Fk(+)p Fm(s)1367 582 y Fh(1)p Ff(;)p Fh(2)1456 611 y Fp(\(1)h(+)f Fi(j)p Fo(t)p Fi(j)1731 573 y Fk(2)1771 611 y Fp(\))p Fi(j)p Fo(t)p Fi(j)951 898 y(\002)1042 774 y Fn(Z)1133 800 y Fk(1)1093 980 y(0)1188 774 y Fn(Z)1278 800 y Fk(+)p Fl(1)1238 980 y(\0001)1621 836 y Fp(\(1)h(+)f Fi(j)p Fo(k)j Fi(\000)d Fo(k)2046 850 y Fk(1)2086 836 y Fi(j)p Fp(\))2146 803 y Fm(s)2179 812 y Fh(1)2218 836 y Fp(\(1)h(+)f Fi(j)p Fo(k)2482 850 y Fk(1)2522 836 y Fi(j)p Fp(\))2582 803 y Fm(s)2615 812 y Fh(2)2669 836 y Fp(exp\()p Fo(d)p Fp(\()p Fo(t;)15 b(h;)g(k)s(;)g(k)3227 850 y Fk(1)3269 836 y Fp(\)\))p 1433 877 2095 4 v 1433 960 a(\(1)21 b(+)f Fi(j)p Fo(k)k Fi(\000)c Fo(k)1859 974 y Fk(1)1898 960 y Fi(j)p Fp(\))1958 934 y Fm(\026)2006 960 y Fp(\(1)h(+)f Fi(j)p Fo(k)2270 974 y Fk(1)2310 960 y Fi(j)p Fp(\))2370 934 y Fm(\026)2417 960 y Fp(\(1)h(+)f Fi(j)p Fo(t)p Fi(j)2692 934 y Fk(2)2731 960 y Fp(\(1)h Fi(\000)f Fo(h)p Fp(\))3010 934 y Fk(2)3051 960 y Fp(\)\(1)h(+)f Fi(j)p Fo(t)p Fi(j)3361 934 y Fk(2)3401 960 y Fo(h)3453 934 y Fk(2)3492 960 y Fp(\))3538 898 y Fo(dk)3632 912 y Fk(1)3672 898 y Fo(dh)75 1133 y Fp(where)158 1346 y Fo(d)p Fp(\()p Fo(t;)15 b(h;)g(k)s(;)g(k)542 1360 y Fk(1)584 1346 y Fp(\))26 b(=)f Fo(\014)5 b Fi(j)p Fo(k)s Fi(j)21 b(\000)f Fo(\014)5 b Fi(j)p Fo(k)24 b Fi(\000)19 b Fo(k)1298 1360 y Fk(1)1338 1346 y 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y Fi(j)75 2046 y Fp(for)38 b(all)g Fi(j)p Fo(t)p Fi(j)g(\025)g Fp(0,)i Fo(k)s(;)15 b(k)832 2060 y Fk(1)910 2046 y Fi(2)37 b Fj(R)p Fp(.)63 b(Indeed,)39 b(if)f Fo(k)i Fp(=)e(0)g(this)g (inequalit)m(y)h(is)f(satis\014ed.)63 b(Let)38 b Fo(k)j Fi(6)p Fp(=)c(0.)64 b(F)-8 b(rom)38 b(the)75 2159 y(triangular)31 b(inequalit)m(y)g Fi(j)p Fo(k)s Fi(j)26 b(\024)f(j)p Fo(k)f Fi(\000)c Fo(k)1377 2173 y Fk(1)1417 2159 y Fi(j)g Fp(+)g Fi(j)p Fo(k)1625 2173 y Fk(1)1665 2159 y Fi(j)p Fp(,)31 b(w)m(e)g(get)g(that)1300 2394 y Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))c Fi(\024)1645 2333 y Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(k)e Fi(\000)20 b Fo(k)2092 2347 y Fk(1)2132 2333 y Fi(j)g Fp(+)g Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))p Fi(j)p Fo(k)2552 2347 y Fk(1)2594 2333 y Fi(j)p 1645 2373 974 4 v 2082 2456 a(j)p Fo(k)s Fi(j)75 2630 y Fp(and)i(the)h(fact)h(that)f Fo(')g Fp(is)g(decreasing)g(giv)m(es)h Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))j Fi(\024)e Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fo(h)p Fp(\))g(and)d Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\))k Fi(\024)f Fo(')p Fp(\()p Fi(j)p Fo(t)p Fi(j)p Fp(\(1)5 b Fi(\000)g Fo(h)p Fp(\)\).)41 b(This)22 b(yields)h(\(10\).)75 2743 y(F)-8 b(rom)35 b(\(10\))i(and)d(the)i(triangular)f(inequalit)m(y)-8 b(,)38 b(w)m(e)e(get)g(that)f Fo(d)p Fp(\()p Fo(t;)15 b(h;)g(k)s(;)g(k)2612 2757 y Fk(1)2654 2743 y Fp(\))34 b Fi(\024)e Fp(0,)37 b(for)e(all)h Fo(t)c Fi(2)h Fj(C)p Fp(,)j Fo(h)d Fi(2)g Fp([0)p Fo(;)15 b Fp(1])75 2856 y(and)30 b Fo(k)s(;)15 b(k)389 2870 y Fk(1)454 2856 y Fi(2)25 b Fj(R)p Fp(.)41 b(So)30 b(that)179 3099 y(~)158 3122 y Fo(C)223 3136 y Fk(4)288 3122 y Fi(\024)25 b Fo(C)449 3136 y Fk(4)513 3122 y Fp(=)143 b(sup)609 3210 y Fm(t)p Fl(2)698 3193 y Fk(\026)681 3210 y Fm(D)s Fk(\(0)p Fm(;\032)p Fk(\))p Fl([)p Fc(R)983 3122 y Fp(\(1)21 b(+)e Fi(j)p Fo(t)p Fi(j)1257 3084 y Fk(2)1297 3122 y Fp(\))p Fi(j)p Fo(t)p Fi(j)1430 2998 y Fn(Z)1522 3024 y Fk(1)1482 3204 y(0)2119 3060 y Fp(1)p 1587 3101 1111 4 v 1587 3184 a(\(1)i(+)e Fi(j)p Fo(t)p Fi(j)1861 3158 y Fk(2)1901 3184 y Fp(\(1)i Fi(\000)f Fo(h)p Fp(\))2180 3158 y Fk(2)2220 3184 y Fp(\)\(1)i(+)e Fi(j)p Fo(t)p Fi(j)2531 3158 y Fk(2)2570 3184 y Fo(h)2622 3158 y Fk(2)2662 3184 y Fp(\))2708 3122 y Fo(dh)1340 3406 y Fi(\002)g Fp(sup)1432 3485 y Fm(k)r Fl(2)p Fc(R)1568 3406 y Fp(\(1)h(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))1894 3368 y Fm(\026)p Fk(+)p Fm(s)2024 3377 y Fh(1)p Ff(;)p Fh(2)2129 3282 y Fn(Z)2220 3309 y Fk(+)p Fl(1)2179 3488 y(\0001)2966 3345 y Fp(1)p 2374 3385 1228 4 v 2374 3468 a(\(1)i(+)f Fi(j)p Fo(k)k Fi(\000)c Fo(k)2800 3482 y Fk(1)2840 3468 y Fi(j)p Fp(\))2900 3442 y Fm(\026)p Fl(\000)p Fm(s)3030 3451 y Fh(1)3069 3468 y Fp(\(1)h(+)f Fi(j)p Fo(k)3333 3482 y Fk(1)3373 3468 y Fi(j)p Fp(\))3433 3442 y Fm(\026)p Fl(\000)p Fm(s)3563 3451 y Fh(2)3612 3406 y Fo(dk)3706 3420 y Fk(1)3746 3406 y Fo(:)75 3649 y Fp(W)-8 b(e)32 b(no)m(w)e(pro)s(ceed)g(as)h(in)f(Lemma)g(2.2)i(of)e([11)q(].)41 b(W)-8 b(e)32 b(write)342 3893 y(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))669 3855 y Fm(\026)p Fk(+)p Fm(s)799 3864 y Fh(1)p Ff(;)p Fh(2)903 3769 y Fn(Z)994 3795 y Fk(+)p Fl(1)954 3975 y(\0001)1740 3831 y Fp(1)p 1149 3872 V 1149 3955 a(\(1)h(+)f Fi(j)p Fo(k)j Fi(\000)d Fo(k)1574 3969 y Fk(1)1614 3955 y Fi(j)p Fp(\))1674 3929 y Fm(\026)p Fl(\000)p Fm(s)1804 3938 y Fh(1)1843 3955 y Fp(\(1)h(+)f Fi(j)p Fo(k)2107 3969 y Fk(1)2147 3955 y Fi(j)p Fp(\))2207 3929 y Fm(\026)p Fl(\000)p Fm(s)2337 3938 y Fh(2)2387 3893 y Fo(dk)2481 3907 y Fk(1)2546 3893 y Fp(=)25 b(\(1)c(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))2969 3855 y Fm(\026)p Fk(+)p Fm(s)3099 3864 y Fh(1)p Ff(;)p Fh(2)3188 3893 y Fp(\()p Fi(A)3296 3908 y Fm(k)3359 3893 y Fp(+)g Fi(B)3510 3908 y Fm(k)3552 3893 y Fp(\))75 4128 y(where)964 4260 y Fi(A)1037 4275 y Fm(k)1104 4260 y Fp(=)1200 4136 y Fn(Z)1251 4342 y Fl(j)p Fm(k)1308 4351 y Fh(1)1342 4342 y Fl(j\024j)p Fm(k)r Fl(j)p Fm(=)p Fk(2)2185 4198 y Fp(1)p 1594 4239 V 1594 4322 a(\(1)h(+)f Fi(j)p Fo(k)j Fi(\000)d Fo(k)2019 4336 y Fk(1)2059 4322 y Fi(j)p Fp(\))2119 4296 y Fm(\026)p Fl(\000)p Fm(s)2249 4305 y Fh(1)2288 4322 y Fp(\(1)h(+)f Fi(j)p Fo(k)2552 4336 y Fk(1)2592 4322 y Fi(j)p Fp(\))2652 4296 y Fm(\026)p Fl(\000)p Fm(s)2782 4305 y Fh(2)2832 4260 y Fo(dk)2926 4274 y Fk(1)75 4476 y Fp(and)958 4608 y Fi(B)1018 4623 y Fm(k)1085 4608 y Fp(=)1181 4484 y Fn(Z)1232 4690 y Fl(j)p Fm(k)1289 4699 y Fh(1)1323 4690 y Fl(j)p Fm(>)p Fl(j)p Fm(k)r Fl(j)p Fm(=)p Fk(2)2166 4547 y Fp(1)p 1575 4587 V 1575 4670 a(\(1)h(+)f Fi(j)p Fo(k)j Fi(\000)d Fo(k)2000 4684 y Fk(1)2040 4670 y Fi(j)p Fp(\))2100 4644 y Fm(\026)p Fl(\000)p Fm(s)2230 4653 y Fh(1)2269 4670 y Fp(\(1)h(+)f Fi(j)p Fo(k)2533 4684 y Fk(1)2573 4670 y Fi(j)p Fp(\))2633 4644 y Fm(\026)p Fl(\000)p Fm(s)2763 4653 y Fh(2)2813 4608 y Fo(dk)2907 4622 y Fk(1)2947 4608 y Fo(;)75 4840 y Fp(for)30 b(all)h Fo(k)e Fi(2)c Fj(R)p Fp(.)40 b(F)-8 b(rom)31 b(the)f(inequalit)m(y)i(1)21 b(+)f Fi(j)p Fo(k)j Fi(\000)d Fo(k)1840 4854 y Fk(1)1880 4840 y Fi(j)26 b(\025)2036 4804 y Fk(1)p 2036 4819 36 4 v 2036 4871 a(2)2082 4840 y Fp(\()p Fi(j)p Fo(k)s Fi(j)21 b Fp(+)f(1\),)31 b(when)f Fi(j)p Fo(k)2775 4854 y Fk(1)2815 4840 y Fi(j)25 b(\024)g(j)p Fo(k)s Fi(j)p Fo(=)p Fp(2,)32 b(w)m(e)f(get)75 5091 y(\(11\))872 b Fi(A)1180 5106 y Fm(k)1247 5091 y Fi(\024)1494 5030 y Fp(2)1539 4997 y Fm(\026)p Fl(\000)p Fm(s)1669 5006 y Fh(1)p 1353 5070 497 4 v 1353 5154 a Fp(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1680 5127 y Fm(\026)p Fl(\000)p Fm(s)1810 5136 y Fh(1)1875 4968 y Fn(Z)1966 4994 y Fk(+)p Fl(1)1925 5174 y(\0001)2364 5030 y Fp(1)p 2120 5070 534 4 v 2120 5154 a(\(1)h(+)f Fi(j)p Fo(k)2384 5168 y Fk(1)2424 5154 y Fi(j)p Fp(\))2484 5127 y Fm(\026)p Fl(\000)p Fm(s)2614 5136 y Fh(2)2664 5091 y Fo(dk)2758 5105 y Fk(1)2798 5091 y Fo(;)75 5332 y Fp(for)36 b(all)h Fo(k)h Fi(2)d Fj(R)p Fp(.)58 b(So)36 b(that)h Fi(A)1090 5347 y Fm(k)1169 5332 y Fp(is)f(\014nite)g(pro)m(vided)g(that)h Fo(\026)e(>)f(s)2325 5346 y Fk(2)2389 5332 y Fp(+)23 b(1.)59 b(F)-8 b(rom)37 b(the)f(inequalit)m(y)i(1)25 b(+)e Fi(j)p 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5742 y Fh(2)2818 5573 y Fn(Z)2909 5599 y Fk(+)p Fl(1)2868 5779 y(\0001)3307 5635 y Fp(1)p 3063 5676 534 4 v 3063 5759 a(\(1)h(+)f Fi(j)p Fo(k)3327 5773 y Fk(1)3367 5759 y Fi(j)p Fp(\))3427 5733 y Fm(\026)p Fl(\000)p Fm(s)3557 5742 y Fh(1)3607 5697 y Fo(dk)3701 5711 y Fk(1)3741 5697 y Fo(;)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 3809 100 a Fp(9)75 399 y(for)28 b(all)i Fo(k)e Fi(2)d Fj(R)p Fp(.)40 b(So)29 b(that)g Fi(B)1009 414 y Fm(k)1080 399 y Fp(is)g(\014nite)f(if)h Fo(\026)c(>)g(s)1701 413 y Fk(1)1757 399 y Fp(+)16 b(1.)41 b(F)-8 b(rom)29 b(the)g(inequalities)h(\(11\),)h(\(12\),)f(w)m(e)g (deduce)e(that)1424 596 y(sup)1425 675 y Fm(k)r Fl(2)p Fc(R)1560 596 y Fp(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1887 559 y Fm(\026)p Fk(+)p Fm(s)2017 568 y Fh(1)p Ff(;)p Fh(2)2106 596 y Fp(\()p Fi(A)2214 611 y Fm(k)2277 596 y Fp(+)g Fi(B)2428 611 y Fm(k)2471 596 y Fp(\))75 849 y(is)28 b(\014nite)g(if)f Fo(s)517 863 y Fk(1)p Fm(;)p Fk(2)636 849 y Fi(\024)e Fp(min\()p Fi(\000)p Fo(s)1033 863 y 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b(whose)f(line)g(n)m(um) m(b)s(er)e(2)p Fo(n)h Fp(is)h(\()p Fi(\000)p Fo(a)2444 5754 y Fk(0)2484 5740 y Fp(\()p Fo(ik)s Fp(\))p Fo(;)15 b Fi(\001)g(\001)g(\001)33 b Fo(;)15 b Fi(\000)p Fo(a)2972 5754 y Fk(2)p Fm(n)p Fl(\000)p Fk(1)3144 5740 y Fp(\()p Fo(ik)s Fp(\)\))36 b(and)e(with)h(all)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 3764 100 a Fp(10)75 399 y(other)34 b(co)s(e\016cien)m(ts)i(b)s(e)e(equal)g(to)h(0)g(and)e Fo(J)9 b Fp(\()p Fo(k)s Fp(\))35 b(is)f(the)h(upp)s(er)d(triangular)i (Jordan)f(\(2)p Fo(n;)15 b Fp(2)p Fo(n)p Fp(\))p Fi(\000)p Fp(matrix)36 b(with)75 511 y(eigen)m(v)-5 b(alue)32 b(equal)f(to)g(0.) 216 624 y(W)-8 b(e)25 b(denote)g(b)m(y)f Fj(F)g Fp(=)h Fj(C)p Fp(\()p Fo(k)s(;)15 b Fp(\()p Fo(K)1252 638 y Fm(j)1290 624 y Fp(\()p Fo(k)s Fp(\)\))1445 638 y Fk(1)p Fl(\024)p Fm(j)t Fl(\024)p Fm(p)1663 624 y Fp(\))25 b(the)f(\014elds)f (of)h(rational)i(functions)d(in)h Fo(k)j Fp(and)c(in)h(the)g(v)-5 b(ariables)75 737 y Fo(K)152 751 y Fm(j)189 737 y Fp(\()p Fo(k)s 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Fp(\()p Fo(T)13 b Fp(\))p Fo(:)75 2854 y Fp(Moreo)m(v)m(er,)26 b(the)d(matrices)h Fo(D)s Fp(\()p Fo(k)s Fp(\))f(and)f Fo(N)10 b Fp(\()p Fo(k)s Fp(\))23 b(are)g(comm)m(uting.)39 b(So)22 b(that)h(the)g(matrix) g Fj(K)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))26 b(=)f(exp\()p Fo(\034)10 b(A)p Fp(\()p Fo(k)s Fp(\)\))75 2967 y(has)30 b(co)s(e\016cien)m(ts)i(of)f(the)f(form)75 3223 y(\(17\))935 b Fj(K)1241 3238 y Fm(l)q(;m)1348 3223 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b(=)1724 3105 y Fm(p)1676 3137 y Fn(X)1680 3332 y Fm(j)t Fk(=1)1822 3223 y Fp(exp\()p Fo(i\034)10 b(K)2154 3237 y Fm(j)2192 3223 y Fp(\()p Fo(k)s Fp(\)\))p Fo(p)2393 3238 y Fm(l)q(;m;j)2554 3223 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))75 3522 y(where)31 b Fo(p)385 3537 y Fm(l)q(;m;j)545 3522 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))32 b(are)g(p)s(olynomials)f(in)g Fo(\034)10 b Fp(,)32 b(with)f(co)s(e\016cien)m(ts)i(in)e Fj(F)p Fp(,)g(of)g(degree)h(less)g(than)f Fo(n)3368 3536 y Fm(j)3425 3522 y Fi(\000)20 b Fp(1,)33 b(for)e(all)75 3635 y(1)26 b Fi(\024)f Fo(j)31 b Fi(\024)25 b Fo(p)p Fp(,)30 b(1)c Fi(\024)e Fo(l)r(;)15 b(m)26 b Fi(\024)f Fp(2)p Fo(n)p Fp(.)75 3798 y Fg(\(H3\))33 b Fp(W)-8 b(e)34 b(mak)m(e)g(the)g (assumption)e(that)i(the)g(functions)e Fo(p)2166 3812 y Fk(1)p Fm(;)p Fk(1)p Fm(;j)2312 3798 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\),)36 b(1)30 b Fi(\024)g Fo(j)36 b Fi(\024)30 b Fo(p)p Fp(,)j(are)h(p)s(olynomials)g(in)e Fo(\034)75 3910 y Fp(with)37 b(co)s(e\016cien)m(ts)i(in)e Fo(E)936 3929 y Fk(\(0)p Fm(;)p Fl(\000)p Fm(\026)1115 3938 y Fh(1)p Ff(;)p Fh(1)1200 3929 y Fk(\))1269 3910 y Fp(for)g Fo(\026)1470 3924 y Fk(1)p Fm(;)p Fk(1)1601 3910 y Fi(\025)f Fp(0)i(and)f(that)h(the)f(functions)g Fo(p)2787 3924 y Fk(1)p Fm(;)p Fk(2)p Fm(n;j)2976 3910 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\),)40 b(1)e Fi(\024)e Fo(j)43 b Fi(\024)36 b Fo(p)p Fp(,)j(are)75 4023 y(p)s(olynomials)31 b(in)f Fo(\034)40 b Fp(with)30 b(co)s(e\016cien)m(ts)i(in)e Fo(E)1607 4042 y Fk(\(0)p Fm(;)p Fl(\000)12 b Fk(deg)r(\()p Fm(R)p Fk(\))p Fl(\000)p Fm(s)2058 4051 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)2192 4042 y Fk(\))2224 4023 y Fp(,)31 b(where)e Fo(s)2585 4037 y Fk(0)p Fm(;)p Fk(1)p Fm(;)p Fk(2)2759 4023 y Fi(\024)c Fp(min\()p Fi(\000)p Fo(s)3156 4037 y Fk(0)3195 4023 y Fo(;)15 b Fi(\000)p Fo(s)3349 4037 y Fk(1)3388 4023 y Fo(;)g Fi(\000)p Fo(s)3542 4037 y Fk(2)3581 4023 y Fp(\).)75 4154 y Fg(\(H4\))39 b Fp(The)g(initial)i (condition)f(satis\014es)g Fo(w)1615 4168 y Fk(0)1655 4154 y Fp(\()p Fo(k)s Fp(\))h Fi(2)g Fo(E)1985 4185 y Fk(\()2022 4167 y(~)2012 4185 y Fm(\014)s(;\026)p Fk(+)p Fm(\026)2214 4194 y Fh(1)p Ff(;)p Fh(1)2299 4185 y Fk(\))2330 4154 y Fp(,)h(for)e(giv)m(en)2805 4130 y(~)2792 4154 y Fo(\014)46 b(>)41 b Fp(0)f(and)f Fo(\026)h(>)h Fp(2)p Fo(n)26 b Fp(+)g Fo(\033)k Fp(+)75 4280 y(max\(1)21 b(+)f Fo(s)479 4294 y Fk(0)518 4280 y Fo(;)15 b Fp(1)21 b(+)f Fo(s)758 4294 y Fk(1)797 4280 y Fo(;)15 b Fp(1)21 b(+)f Fo(s)1037 4294 y Fk(2)1076 4280 y Fp(\).)75 4393 y Fg(\(H5\))35 b Fp(Let)g Fo(\014)k Fp(=)700 4369 y(~)687 4393 y Fo(\014)29 b Fi(\000)23 b Fo(M)949 4407 y Fk(0)989 4393 y Fo(\032)g Fi(\000)g Fo(')p Fp(\(0\))37 b(and)e(let)h Fo(\027)j(>)33 b Fp(0.)56 b(W)-8 b(e)37 b(assume)e(that)g(the)h(co)s(e\016cien)m(t)h Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))37 b(b)s(elongs)75 4506 y(to)45 b Fo(E)267 4524 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))531 4506 y Fp(and)f(that)h Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))50 b(=)f Fo(a)p Fp(\()p Fi(\000)p Fo(\034)5 b(;)15 b(k)s Fp(\))46 b(for)e(all)i(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))50 b Fi(2)f Fj(R)2451 4473 y Fk(2)2490 4506 y Fp(.)83 b(Moreo)m(v)m(er,)51 b(w)m(e)45 b(assume)f(that)h(the)75 4619 y(function)37 b Fo(\034)47 b Fi(7!)36 b Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))39 b(is)e(2)p Fo(n)p Fp(-times)h (di\013eren)m(tiable)h(with)d(resp)s(ect)i(to)g Fo(\034)46 b Fi(2)37 b Fj(R)p Fp(,)h(for)f(all)h Fo(k)i Fi(2)c Fj(R)h Fp(and)g(that)75 4732 y Fo(@)128 4699 y Fm(l)123 4754 y(\034)166 4732 y Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b Fi(2)e Fo(E)604 4699 y Fc(R)599 4763 y Fk(\()p Fm(\027;\014)s(;\026)p Fl(\000)p Fm(l)q Fk(\))895 4732 y Fp(,)30 b(for)g(all)h(0)26 b Fi(\024)f Fo(l)i Fi(\024)e Fp(2)p Fo(n)p Fp(.)75 4906 y Fg(Remark:)47 b Fp(Notice)35 b(that)f(if)g(the)f(p)s(olynomials)h Fo(P)1814 4920 y Fm(j)1851 4906 y Fp(\()p Fo(T)13 b Fp(\),)35 b(1)30 b Fi(\024)g Fo(j)36 b Fi(\024)30 b Fo(p)p Fp(,)k(are)g(w)m(ell)g (de\014ned)e(for)i(all)g Fo(k)f Fi(2)d Fj(R)p Fp(,)k(the)75 5019 y(assumption)f Fg(\(H2\))h Fp(implies)g(that)g(the)h(assumption)e Fg(\(H3\))g Fp(holds)h(for)f(some)i Fo(\026)2886 5033 y Fk(1)p Fm(;)p Fk(1)3011 5019 y Fi(\025)c Fp(0)j(and)g(deg)q(\()p Fo(R)q Fp(\))e Fi(\025)f Fp(0.)75 5132 y(F)-8 b(or)40 b(instance,)j(if)d(w)m(e)g(mak)m(e)h(the)e(assumption)h(that)g(the)g (ro)s(ots)f Fo(K)2465 5146 y Fm(j)2502 5132 y Fp(\()p Fo(k)s Fp(\))i(are)f Fd(holomorphic)k(functions)k Fp(on)75 5245 y(some)41 b(domain)g(\012)g(in)f Fj(C)h Fp(con)m(taining)h Fj(R)p Fp(,)i(then)d(the)g(results)f(of)i(the)f(c)m(hapter)g(2)h(of)f ([17)q(])g(sho)m(ws)g(that)g(the)75 5357 y(Dunford)32 b(decomp)s(osition)i(\(16\))g(holds)f(on)g(\012)g(with)g(holomorphic)g (pro)5 b(jectors)33 b Fo(P)2924 5371 y Fm(j)2961 5357 y Fp(\()p Fo(A)p Fp(\()p Fo(k)s Fp(\)\),)j(1)31 b Fi(\024)e Fo(j)36 b Fi(\024)29 b Fo(p)k Fp(and)75 5470 y(holomorphic)i(matrix)g Fo(N)10 b Fp(\()p Fo(k)s Fp(\).)55 b(In)35 b(that)g(case,)j(w)m(e)d (deduce)g(that)g(the)g(assumption)g Fg(\(H3\))f Fp(holds)g(for)h(some) 75 5583 y Fo(\026)130 5597 y Fk(1)p Fm(;)p Fk(1)224 5583 y Fo(;)15 b Fp(deg)r(\()p Fo(R)q Fp(\))26 b Fi(\025)f Fp(0.)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 3764 100 a Fp(11)75 399 y Fg(Example:)40 b Fp(Let)31 b Fo(P)13 b Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))26 b(=)f Fo(T)33 b Fi(\000)20 b Fo(X)1409 366 y Fk(2)1468 399 y Fp(+)g(1)31 b(and)e Fo(R)q Fp(\()p Fo(X)7 b Fp(\))26 b(=)f(1.)41 b(W)-8 b(e)32 b(ha)m(v)m(e)f Fo(Q)p Fp(\()p Fo(T)8 b(;)15 b(k)s Fp(\))26 b(=)f Fo(T)3113 366 y Fk(2)3173 399 y Fp(+)19 b Fo(k)3313 366 y Fk(2)3373 399 y Fp(+)g(1.)41 b(In)30 b(that)75 511 y(case,)i(w)m(e)f(get)g(that) 1423 673 y Fo(A)p Fp(\()p Fo(k)s Fp(\))26 b(=)1733 545 y Fn(\022)2013 616 y Fp(0)254 b(1)1842 729 y Fi(\000)p Fp(\()p Fo(k)1998 696 y Fk(2)2058 729 y Fp(+)19 b(1\))84 b(0)2399 545 y Fn(\023)2481 673 y Fo(;)75 900 y Fp(whic)m(h)30 b(satis\014es)h Fo(A)p Fp(\()p Fo(k)s Fp(\))26 b(=)f Fo(P)13 b Fp(\()p Fo(k)s Fp(\))p Fo(S)5 b Fp(\()p Fo(k)s Fp(\))p Fo(P)1423 867 y Fl(\000)p Fk(1)1519 900 y Fp(\()p Fo(k)s Fp(\),)32 b(where)267 1159 y Fo(P)13 b Fp(\()p Fo(k)s Fp(\))26 b(=)580 1031 y Fn(\022)688 1104 y Fi(\000)p Fo(i)p Fp(\(1)21 b(+)f Fo(k)1032 1071 y Fk(2)1072 1104 y Fp(\))1107 1071 y Fl(\000)p Fk(1)p Fm(=)p Fk(2)1355 1104 y Fo(i)p Fp(\(1)h(+)f Fo(k)1628 1071 y Fk(2)1668 1104 y Fp(\))1703 1071 y Fl(\000)p Fk(1)p Fm(=)p Fk(2)957 1217 y Fp(1)587 b(1)1910 1031 y Fn(\023)1992 1159 y Fo(;)15 b(S)5 b Fp(\()p Fo(k)s Fp(\))26 b(=)2335 1031 y Fn(\022)2444 1103 y Fo(i)p Fp(\()p Fo(k)2560 1070 y Fk(2)2620 1103 y Fp(+)20 b(1\))2791 1070 y Fk(1)p Fm(=)p Fk(2)3227 1103 y Fp(0)2650 1219 y(0)290 b Fi(\000)p Fo(i)p Fp(\()p Fo(k)3172 1186 y Fk(2)3232 1219 y Fp(+)20 b(1\))3403 1186 y Fk(1)p Fm(=)p Fk(2)3555 1031 y Fn(\023)3638 1159 y Fo(;)75 1413 y Fp(for)30 b(all)h Fo(k)e Fi(2)c Fj(R)p Fp(.)40 b(So)30 b(that)323 1635 y Fj(K)394 1649 y Fk(1)p Fm(;)p Fk(1)488 1635 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))26 b(=)825 1574 y(1)p 825 1614 46 4 v 825 1698 a(2)880 1635 y(\()p Fo( )s Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))23 b(+)d Fo( )s Fp(\()p Fi(\000)p Fo(\034)5 b(;)15 b(k)s Fp(\)\))62 b Fo(;)76 b Fj(K)1903 1649 y Fk(1)p Fm(;)p Fk(2)1997 1635 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b(=)2555 1574 y Fo(i)p 2335 1614 473 4 v 2335 1702 a Fp(2\()p Fo(k)2465 1675 y Fk(2)2525 1702 y Fp(+)20 b(1\))2696 1675 y Fk(1)p Fm(=)p Fk(2)2817 1635 y Fp(\()p Fo( )s Fp(\()p Fi(\000)p Fo(\034)5 b(;)15 b(k)s Fp(\))22 b Fi(\000)e Fo( )s Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\)\))75 1904 y(where)29 b Fo( )s Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b(=)e(exp\()p Fo(i\034)10 b Fp(\()p Fo(k)1067 1871 y Fk(2)1126 1904 y Fp(+)18 b(1\))1295 1871 y Fk(1)p Fm(=)p Fk(2)1405 1904 y Fp(\).)41 b(All)30 b(the)g(assumptions)e Fg(\(H0\),\(H1\),\(H2\),\(H3\))f Fp(are)j(ful\014lled)f(in)75 2017 y(this)h(situation.)75 2227 y Fg(Lemma)35 b(6)45 b Fd(L)-5 b(et)31 b Fo(w)763 2241 y Fk(0)828 2227 y Fp(:)25 b Fj(R)g Fi(!)g Fj(R)30 b Fd(and)i Fo(F)38 b Fp(:)26 b Fj(R)1569 2194 y Fk(2)1633 2227 y Fi(!)f Fj(C)30 b Fd(b)-5 b(e)30 b(c)-5 b(ontinuous)32 b(functions.)41 b(Then,)31 b(the)g(unique)e(solution)75 2340 y(of)k(the)g(line)-5 b(ar)34 b(pr)-5 b(oblem)75 2542 y Fp(\(18\))1104 b Fo(R)q Fp(\()p Fo(ik)s Fp(\))p Fo(P)13 b Fp(\()p Fo(@)1719 2504 y Fk(2)1714 2564 y Fm(\034)1760 2542 y Fo(;)i(ik)s Fp(\))p Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))29 b(=)c Fo(F)13 b Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))75 2743 y Fd(for)33 b(initial)g(c)-5 b(onditions)75 2945 y Fp(\(19\))1020 b Fo(w)r Fp(\(0)p Fo(;)15 b(k)s Fp(\))27 b(=)e Fo(w)1715 2959 y Fk(0)1755 2945 y Fp(\()p Fo(k)s Fp(\))66 b Fo(;)80 b Fp(\()p Fo(@)2134 2907 y Fm(j)2129 2967 y(\034)2173 2945 y Fo(w)r Fp(\)\(0)p Fo(;)15 b(k)s Fp(\))28 b Fi(\021)d Fp(0)p Fo(;)75 3147 y Fd(for)33 b(al)5 b(l)33 b Fp(1)26 b Fi(\024)f Fo(j)31 b Fi(\024)25 b Fp(2)p Fo(n)20 b Fi(\000)g Fp(1)p Fd(,)33 b(is)815 3391 y Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b(=)e Fj(K)1281 3405 y Fk(11)1355 3391 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fo(w)1625 3405 y Fk(0)1666 3391 y Fp(\()p Fo(k)s Fp(\))21 b(+)1898 3268 y Fn(Z)1989 3294 y Fm(\034)1949 3474 y Fk(0)2057 3329 y Fj(K)2128 3343 y Fk(1)p Fm(;)p Fk(2)p Fm(n)2265 3329 y Fp(\()p Fo(\034)31 b Fi(\000)20 b Fo(\034)2502 3343 y Fk(1)2541 3329 y Fo(;)15 b(k)s Fp(\))p 2057 3371 610 4 v 2251 3454 a Fo(R)q Fp(\()p Fo(ik)s Fp(\))2677 3391 y Fo(F)e Fp(\()p Fo(\034)2823 3405 y Fk(1)2862 3391 y Fo(;)i(k)s Fp(\))p Fo(d\034)3074 3405 y Fk(1)75 3637 y Fd(for)33 b(al)5 b(l)33 b Fo(\034)j Fi(2)24 b Fj(R)33 b Fd(and)g Fo(k)c Fi(2)24 b Fj(R)p Fd(.)75 3896 y Fg(Pro)s(of)31 b Fp(If)f(w)m(e)h(put)342 4282 y Fo(Y)20 b Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b(=)743 4072 y Fn(0)743 4232 y(B)743 4290 y(@)978 4144 y Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))1102 4237 y(.)1102 4270 y(.)1102 4304 y(.)864 4420 y Fo(w)931 4387 y Fk(\(2)p Fm(n)p Fl(\000)p Fk(1\))1159 4420 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))1407 4072 y Fn(1)1407 4232 y(C)1407 4290 y(A)1501 4282 y Fo(;)g(Y)1594 4296 y Fk(0)1634 4282 y Fp(\()p Fo(k)s Fp(\))26 b(=)1876 4018 y Fn(0)1876 4178 y(B)1876 4232 y(B)1876 4287 y(B)1876 4345 y(@)1997 4089 y Fo(w)2062 4103 y Fk(0)2102 4089 y Fp(\()p Fo(k)s Fp(\))2087 4202 y(0)2097 4296 y(.)2097 4329 y(.)2097 4362 y(.)2087 4475 y(0)2264 4018 y Fn(1)2264 4178 y(C)2264 4232 y(C)2264 4287 y(C)2264 4345 y(A)2359 4282 y Fo(;)15 b(B)5 b Fp(\()p Fo(\034)g(;)15 b(k)s Fp(\))26 b(=)2800 4018 y Fn(0)2800 4178 y(B)2800 4232 y(B)2800 4287 y(B)2800 4345 y(@)3171 4089 y Fp(0)3181 4183 y(.)3181 4216 y(.)3181 4249 y(.)3171 4362 y(0)2921 4475 y Fo(F)13 b Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fo(=R)q Fp(\()p Fo(ik)s Fp(\))3507 4018 y Fn(1)3507 4178 y(C)3507 4232 y(C)3507 4287 y(C)3507 4345 y(A)75 4674 y Fp(Then,)30 b(the)g(problem)g(\(18\),)i(\(19\))g(can)f(b)s(e)e(written)i(in)m(to)g (the)g(di\013eren)m(tial)h(system)1311 4876 y Fo(Y)1384 4838 y Fl(0)1407 4876 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))26 b(=)f Fo(A)p Fp(\()p Fo(k)s Fp(\))p Fo(Y)c Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))22 b(+)e Fo(B)5 b Fp(\()p Fo(\034)g(;)15 b(k)s Fp(\))p Fo(;)75 5077 y Fp(with)31 b(initial)i(condition)g Fo(Y)20 b Fp(\(0)p Fo(;)15 b(k)s Fp(\))29 b(=)e Fo(Y)1406 5091 y Fk(0)1446 5077 y Fp(\()p Fo(k)s Fp(\).)45 b(A)32 b(classical)i(result)e(sho)m(ws)f(that)i(the)e (unique)g(solution)i(of)f(this)75 5190 y(di\013eren)m(tial)g(system)e (is)716 5431 y Fo(Y)20 b Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))26 b(=)f(exp\()p Fo(\034)10 b(A)p Fp(\()p Fo(k)s Fp(\)\))p Fo(Y)1616 5445 y Fk(0)1657 5431 y Fp(\()p Fo(k)s Fp(\))21 b(+)1889 5307 y Fn(Z)1980 5334 y Fm(\034)1940 5513 y Fk(0)2038 5431 y Fp(exp\(\()p Fo(\034)31 b Fi(\000)20 b Fo(\034)2449 5445 y Fk(1)2488 5431 y Fp(\))p Fo(A)p Fp(\()p Fo(k)s Fp(\)\))p Fo(B)5 b Fp(\()p Fo(\034)2895 5445 y Fk(1)2936 5431 y Fo(;)15 b(k)s Fp(\))p Fo(d\034)3148 5445 y Fk(1)3189 5431 y Fo(;)75 5674 y Fp(for)30 b(all)h Fo(\034)5 b(;)15 b(k)29 b Fi(2)c Fj(R)p Fp(.)40 b(The)30 b(lemma)h(6)g(follo)m(ws.)2190 b Fb(2)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 3764 100 a Fp(12)75 399 y Fg(Lemma)35 b(7)45 b Fd(L)-5 b(et)32 b Fo(')g Fd(b)-5 b(e)32 b(the)g(function)g (de\014ne)-5 b(d)32 b(in)g(se)-5 b(ction)32 b(2.)42 b(We)31 b(put)h Fo(\014)f Fp(=)2755 375 y(~)2742 399 y Fo(\014)23 b Fi(\000)18 b Fo(M)2993 413 y Fk(0)3032 399 y Fo(\032)g Fi(\000)g Fo(')p Fp(\(0\))33 b Fd(and)g(assume)75 511 y(that)45 b Fo(\014)50 b(>)45 b Fp(0)e Fd(\(for)i Fo(\032)e Fd(and)h Fo(')p Fp(\(0\))h Fd(smal)5 b(l)45 b(enough\).)75 b(Then,)46 b(ther)-5 b(e)45 b(exist)e Fo(R)q(;)15 b(\016)49 b(>)c Fp(0)f Fd(\(dep)-5 b(ending)45 b(only)f(on)75 624 y Fo(s)118 638 y Fk(0)157 624 y Fo(;)15 b(s)240 638 y Fk(1)280 624 y Fo(;)g(s)363 638 y Fk(2)402 624 y Fo(;)g(\032;)g(\026;)g (\027)q(;)g(P)e Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))p Fo(;)15 b(R)q Fp(\()p Fo(X)7 b Fp(\))p Fd(\))32 b(such)d(that)i(if)d (the)h(functions)g Fo(w)2438 638 y Fk(0)2503 624 y Fi(2)c Fo(E)2656 655 y Fk(\()2693 638 y(~)2683 655 y Fm(\014)t(;\026)p Fk(+)p Fm(\026)2886 664 y Fh(1)p Ff(;)p Fh(1)2970 655 y Fk(\))3030 624 y Fd(and)30 b Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b Fi(2)e Fo(E)3636 643 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))75 737 y Fd(satis\014es)1081 850 y Fi(jj)p Fo(w)1196 864 y Fk(0)1236 850 y Fp(\()p Fo(k)s Fp(\))p Fi(jj)1406 881 y Fk(\()1443 863 y(~)1433 881 y Fm(\014)5 b(;\026)p Fk(+)p Fm(\026)1637 890 y Fh(1)p Ff(;)p Fh(1)1722 881 y Fk(\))1778 850 y Fi(\024)25 b Fo(\016)69 b(;)81 b Fi(jj)p Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fi(jj)2442 869 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2688 850 y Fi(\024)25 b Fo(\016)n(;)75 1017 y Fd(the)33 b(map)h(de\014ne)-5 b(d)33 b(by)158 1289 y Fi(H)q Fp(\()p Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\)\))27 b(:=)f Fj(K)798 1303 y Fk(11)872 1289 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fo(w)1142 1303 y Fk(0)1183 1289 y Fp(\()p Fo(k)s Fp(\))21 b(+)1415 1166 y Fn(Z)1506 1192 y Fm(\034)1465 1372 y Fk(0)1574 1227 y Fj(K)1645 1241 y Fk(1)p Fm(;)p Fk(2)p Fm(n)1782 1227 y Fp(\()p Fo(\034)30 b Fi(\000)20 b Fo(\034)2018 1241 y Fk(1)2058 1227 y Fo(;)15 b(k)s Fp(\))p 1574 1269 610 4 v 1627 1356 a(\(2)p Fo(\031)s Fp(\))1797 1329 y Fk(1)p Fm(=)p Fk(2)1908 1356 y Fo(R)q Fp(\()p Fo(ik)s Fp(\))1002 1549 y Fi(\002)1093 1425 y Fn(Z)1184 1452 y Fm(\034)1215 1461 y Fh(1)1144 1632 y Fk(0)1269 1425 y Fn(Z)1360 1452 y Fm(\034)1391 1461 y Fh(2)1320 1632 y Fk(0)1445 1425 y Fn(Z)1536 1452 y Fk(+)p Fl(1)1496 1632 y(\0001)1666 1549 y Fp(\()p Fo(ik)1779 1563 y Fk(1)1819 1549 y Fp(\))1854 1512 y Fm(s)1887 1521 y Fh(0)1926 1549 y Fo(a)p Fp(\()p Fo(\034)2049 1563 y Fk(2)2109 1549 y Fi(\000)20 b Fo(\034)2240 1563 y Fk(3)2279 1549 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2528 1563 y Fk(1)2567 1549 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)2744 1563 y Fk(3)2785 1549 y Fo(;)15 b(k)2872 1563 y Fk(1)2912 1549 y Fp(\))1374 1751 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)k Fi(\000)c Fo(k)1775 1765 y Fk(1)1815 1751 y Fp(\)\))1885 1713 y Fm(s)1918 1722 y Fh(1)1957 1751 y Fp(\()p Fo(ik)2070 1765 y Fk(1)2110 1751 y Fp(\))2145 1713 y Fm(s)2178 1722 y Fh(2)2217 1751 y Fo(w)r Fp(\()p Fo(\034)2359 1765 y Fk(2)2419 1751 y Fi(\000)g Fo(\034)2550 1765 y Fk(3)2590 1751 y Fo(;)15 b(k)23 b Fi(\000)d Fo(k)2838 1765 y Fk(1)2878 1751 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)3055 1765 y Fk(3)3095 1751 y Fo(;)15 b(k)3182 1765 y Fk(1)3223 1751 y Fp(\))p Fo(dk)3352 1765 y Fk(1)3392 1751 y Fo(d\034)3479 1765 y Fk(3)3519 1751 y Fo(d\034)3606 1765 y Fk(2)3645 1751 y Fo(d\034)3732 1765 y Fk(1)75 1955 y Fd(satis\014es)34 b(the)f(fol)5 b(lowing)33 b(pr)-5 b(op)g(erties.)75 2068 y Fg(i\))37 b Fi(H)q Fp(\()316 2045 y(\026)295 2068 y Fo(D)s Fp(\(0)p Fo(;)15 b(R)q Fp(\)\))36 b Fi(\032)795 2045 y Fp(\026)774 2068 y Fo(D)s Fp(\(0)p Fo(;)15 b(R)q Fp(\))p Fd(,)39 b(wher)-5 b(e)1427 2045 y Fp(\026)1406 2068 y Fo(D)s Fp(\(0)p Fo(;)15 b(R)q Fp(\))39 b Fd(is)e(the)h(close)-5 b(d)39 b(b)-5 b(al)5 b(l)38 b(c)-5 b(enter)g(e)g(d)39 b(at)f(0)g(with)h(r)-5 b(adius)39 b Fo(R)c(>)f Fp(0)k Fd(in)75 2181 y Fo(E)142 2199 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))361 2181 y Fd(.)75 2294 y Fg(ii\))32 b Fd(Ther)-5 b(e)34 b(exists)f Fp(0)25 b Fo(<)g(M)36 b(<)25 b Fp(1)33 b Fd(such)f(that)1067 2498 y Fi(jjH)q Fp(\()p Fo(w)1295 2512 y Fk(1)1335 2498 y Fp(\))20 b Fi(\000)g(H)q Fp(\()p Fo(w)1659 2512 y Fk(2)1699 2498 y Fp(\))p Fi(jj)1784 2516 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2029 2498 y Fi(\024)25 b Fo(M)10 b Fi(jj)p Fo(w)2338 2512 y Fk(1)2398 2498 y Fi(\000)20 b Fo(w)2554 2512 y Fk(2)2593 2498 y Fi(jj)2643 2516 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))75 2702 y Fd(for)33 b(al)5 b(l)33 b Fo(w)415 2716 y Fk(1)455 2702 y Fo(;)15 b(w)560 2716 y Fk(2)625 2702 y Fi(2)732 2679 y Fp(\026)711 2702 y Fo(D)s Fp(\(0)p Fo(;)g(R)q Fp(\))p Fd(.)75 2965 y Fg(Pro)s(of)40 b Fp(W)-8 b(e)40 b(\014rst)e(c)m(hec)m(k)i(the)f(prop)s(ert)m(y)f Fg(i\))p Fp(.)66 b(Let)40 b Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))41 b Fi(2)e Fo(E)2340 2983 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2597 2965 y Fp(suc)m(h)g(that)g Fi(jj)p Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fi(jj)3388 2983 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))3649 2965 y Fi(\024)40 b Fo(R)q Fp(,)75 3077 y(with)30 b Fo(R)d(>)f Fp(0.)42 b(F)-8 b(rom)31 b(Lemma)g(1)g(and)f(Lemma)h(5,)g(there)g(exists)h(a)f (constan)m(t)h Fo(C)2823 3091 y Fk(1)2893 3077 y Fp(\(dep)s(ending)d (on)i Fo(M)3580 3091 y Fk(1)3620 3077 y Fp(,)g Fo(\027)6 b Fp(,)30 b Fo(\032)p Fp(,)75 3190 y Fo(p)121 3204 y Fk(1)p Fm(;)p Fk(1)p Fm(;j)267 3190 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\),)32 b(for)e(1)c Fi(\024)f Fo(j)31 b Fi(\024)25 b Fo(p)p Fp(\))30 b(suc)m(h)g(that)75 3471 y(\(20\))85 b Fi(jj)p Fj(K)441 3485 y Fk(11)516 3471 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fo(w)786 3485 y Fk(0)826 3471 y Fp(\()p Fo(k)s Fp(\))p Fi(jj)996 3490 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))1242 3471 y Fp(=)25 b Fi(jj)1452 3410 y Fj(K)1523 3424 y Fk(11)1597 3410 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p 1398 3450 459 4 v 1398 3533 a(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1725 3505 y Fm(\026)1767 3514 y Fh(1)p Ff(;)p Fh(1)1867 3471 y Fp(\(1)h(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))2194 3434 y Fm(\026)2236 3443 y Fh(1)p Ff(;)p Fh(1)2325 3471 y Fo(w)2390 3485 y Fk(0)2430 3471 y Fp(\()p Fo(k)s Fp(\))p Fi(jj)2600 3490 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))1435 3668 y Fi(\024)25 b Fo(C)1596 3682 y Fk(1)1635 3668 y Fi(jj)p Fp(\(1)d(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))2013 3631 y Fm(\026)2055 3640 y Fh(1)p Ff(;)p Fh(1)2144 3668 y Fo(w)2209 3682 y Fk(0)2249 3668 y Fp(\()p Fo(k)s Fp(\))p Fi(jj)2419 3699 y Fk(\()2456 3682 y(~)2446 3699 y Fm(\014)5 b(;\026)p Fk(\))2610 3668 y Fi(\024)25 b Fo(C)2771 3682 y Fk(1)2810 3668 y Fi(jj)p Fo(w)2925 3682 y Fk(0)2965 3668 y Fp(\()p Fo(k)s Fp(\))p Fi(jj)3135 3699 y Fk(\()3172 3682 y(~)3162 3699 y Fm(\014)5 b(;\026)p Fk(+)p Fm(\026)3366 3708 y Fh(1)p Ff(;)p Fh(1)3450 3699 y Fk(\))3507 3668 y Fi(\024)25 b Fo(C)3668 3682 y Fk(1)3707 3668 y Fo(\016)n(:)75 3886 y Fp(F)-8 b(rom)31 b(Lemma)f(4,)h(there)g(exists)g Fo(C)1288 3900 y Fk(4)1353 3886 y Fo(>)25 b Fp(0)30 b(\(dep)s(ending)g(on)g Fo(\032)p Fp(,)p Fo(\026)p Fp(,)p Fo(s)2318 3900 y Fk(0)2357 3886 y Fp(,)p Fo(s)2425 3900 y Fk(1)2464 3886 y Fp(,)p Fo(s)2532 3900 y Fk(2)2572 3886 y Fp(\))g(suc)m(h)g(that)75 4171 y(\(21\))85 b Fi(jj)p Fp(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))697 4133 y Fm(s)730 4142 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)883 4047 y Fn(Z)974 4073 y Fm(\034)1005 4082 y Fh(2)934 4253 y Fk(0)1059 4047 y Fn(Z)1150 4073 y Fk(+)p Fl(1)1110 4253 y(\0001)1280 4171 y Fp(\()p Fo(ik)1393 4185 y Fk(1)1433 4171 y Fp(\))1468 4133 y Fm(s)1501 4142 y Fh(0)1540 4171 y Fo(a)p Fp(\()p Fo(\034)1663 4185 y Fk(2)1723 4171 y Fi(\000)g Fo(\034)1854 4185 y Fk(3)1893 4171 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2142 4185 y Fk(1)2181 4171 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)2358 4185 y Fk(3)2399 4171 y Fo(;)15 b(k)2486 4185 y Fk(1)2526 4171 y Fp(\))768 4372 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)k Fi(\000)c Fo(k)1169 4386 y Fk(1)1209 4372 y Fp(\)\))1279 4335 y Fm(s)1312 4344 y Fh(1)1351 4372 y Fp(\()p Fo(ik)1464 4386 y Fk(1)1504 4372 y Fp(\))1539 4335 y Fm(s)1572 4344 y Fh(2)1611 4372 y Fo(w)r Fp(\()p Fo(\034)1753 4386 y Fk(2)1813 4372 y Fi(\000)g Fo(\034)1944 4386 y Fk(3)1984 4372 y Fo(;)15 b(k)23 b Fi(\000)d Fo(k)2232 4386 y Fk(1)2272 4372 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)2449 4386 y Fk(3)2489 4372 y Fo(;)15 b(k)2576 4386 y Fk(1)2617 4372 y Fp(\))p Fo(dk)2746 4386 y Fk(1)2786 4372 y Fo(d\034)2873 4386 y Fk(3)2912 4372 y Fi(jj)2962 4391 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))972 4521 y Fi(\024)25 b Fo(C)1133 4535 y Fk(4)1173 4521 y Fp(\()p Fi(jj)p Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fi(jj)1561 4539 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))1782 4521 y Fi(jj)p Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fi(jj)2154 4539 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2396 4521 y Fp(+)20 b Fi(jj)p Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fi(jj)2859 4483 y Fk(2)2859 4548 y(\()p Fm(\027;\014)s(;\026)p Fk(\))3081 4521 y Fp(\))25 b Fi(\024)g Fo(C)3302 4535 y Fk(4)3342 4521 y Fp(\()p Fo(\016)s(R)d Fp(+)e Fo(R)3672 4483 y Fk(2)3711 4521 y Fp(\))p Fo(:)75 4725 y Fp(F)-8 b(rom)31 b(Lemma)f(2,)h(there)g (exists)g Fo(C)1288 4739 y Fk(2)1353 4725 y Fo(>)25 b Fp(0)30 b(\(dep)s(ending)g(on)g Fo(\032)p Fp(,)g Fo(\027)6 b Fp(\))31 b(suc)m(h)f(that)75 5010 y(\(22\))85 b Fi(jj)p Fp(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))697 4973 y Fm(s)730 4982 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)883 4886 y Fn(Z)974 4913 y Fm(\034)1005 4922 y Fh(1)934 5092 y Fk(0)1059 4886 y Fn(Z)1150 4913 y Fm(\034)1181 4922 y Fh(2)1110 5092 y Fk(0)1236 4886 y Fn(Z)1326 4913 y Fk(+)p Fl(1)1286 5092 y(\0001)1456 5010 y Fp(\()p Fo(ik)1569 5024 y Fk(1)1610 5010 y Fp(\))1645 4973 y Fm(s)1678 4982 y Fh(0)1716 5010 y Fo(a)p Fp(\()p Fo(\034)1839 5024 y Fk(2)1899 5010 y Fi(\000)g Fo(\034)2030 5024 y Fk(3)2069 5010 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2318 5024 y Fk(1)2358 5010 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)2535 5024 y Fk(3)2575 5010 y Fo(;)15 b(k)2662 5024 y Fk(1)2702 5010 y Fp(\))705 5212 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)k Fi(\000)c Fo(k)1106 5226 y Fk(1)1146 5212 y Fp(\)\))1216 5174 y Fm(s)1249 5183 y Fh(1)1288 5212 y Fp(\()p Fo(ik)1401 5226 y Fk(1)1441 5212 y Fp(\))1476 5174 y Fm(s)1509 5183 y Fh(2)1548 5212 y Fo(w)r Fp(\()p Fo(\034)1690 5226 y Fk(2)1750 5212 y Fi(\000)g Fo(\034)1881 5226 y Fk(3)1920 5212 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2169 5226 y Fk(1)2209 5212 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)2386 5226 y Fk(3)2426 5212 y Fo(;)15 b(k)2513 5226 y Fk(1)2553 5212 y Fp(\))p Fo(dk)2682 5226 y Fk(1)2723 5212 y Fo(d\034)2810 5226 y Fk(3)2849 5212 y Fo(d\034)2936 5226 y Fk(2)2976 5212 y Fi(jj)3026 5230 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))756 5418 y Fi(\024)25 b Fo(C)917 5432 y Fk(2)957 5418 y Fi(jj)p Fp(\(1)c(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1334 5380 y Fm(s)1367 5389 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)1520 5294 y Fn(Z)1611 5320 y Fm(\034)1642 5329 y Fh(2)1571 5500 y Fk(0)1697 5294 y Fn(Z)1787 5320 y Fk(+)p Fl(1)1747 5500 y(\0001)1917 5418 y Fp(\()p Fo(ik)2030 5432 y Fk(1)2070 5418 y Fp(\))2105 5380 y Fm(s)2138 5389 y Fh(0)2177 5418 y Fo(a)p Fp(\()p Fo(\034)2300 5432 y Fk(2)2360 5418 y Fi(\000)g Fo(\034)2491 5432 y Fk(3)2530 5418 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2779 5432 y Fk(1)2819 5418 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)2996 5432 y Fk(3)3036 5418 y Fo(;)15 b(k)3123 5432 y Fk(1)3163 5418 y Fp(\))1332 5619 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)25 b Fi(\000)20 b Fo(k)1734 5633 y Fk(1)1773 5619 y Fp(\)\))1843 5582 y Fm(s)1876 5591 y Fh(1)1915 5619 y Fp(\()p Fo(ik)2028 5633 y Fk(1)2069 5619 y Fp(\))2104 5582 y Fm(s)2137 5591 y Fh(2)2176 5619 y Fo(w)r Fp(\()p Fo(\034)2318 5633 y Fk(2)2378 5619 y Fi(\000)g Fo(\034)2509 5633 y Fk(3)2548 5619 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2797 5633 y Fk(1)2836 5619 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)3013 5633 y Fk(3)3054 5619 y Fo(;)15 b(k)3141 5633 y Fk(1)3181 5619 y Fp(\))p Fo(dk)3310 5633 y Fk(1)3350 5619 y Fo(d\034)3437 5633 y Fk(3)3477 5619 y Fi(jj)3527 5638 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))3746 5619 y Fo(:)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 3764 100 a Fp(13)75 399 y(F)-8 b(rom)34 b(Lemma)g(3,)h(there)f(exists)g Fo(C)1305 413 y Fk(3)1375 399 y Fo(>)c Fp(0)k(\(dep)s(ending)f(on)g Fo(s)2203 413 y Fk(0)2242 399 y Fp(,)p Fo(s)2310 413 y Fk(1)2350 399 y Fp(,)p Fo(s)2418 413 y Fk(2)2457 399 y Fp(,)p Fo(M)2570 413 y Fk(1)2610 399 y Fp(,)p Fo(\027)6 b Fp(,)p Fo(\032)p Fp(,)p Fo(R)q Fp(\()p Fo(ik)s Fp(\),)36 b Fo(p)3111 413 y Fk(1)p Fm(;)p Fk(2)p Fm(n;j)3300 399 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\),)35 b(for)f(1)d Fi(\024)75 511 y Fo(j)g Fi(\024)25 b Fo(p)p Fp(\))30 b(suc)m(h)g(that)75 745 y(\(23\))85 b Fi(jj)385 621 y Fn(Z)476 647 y Fm(\034)436 827 y Fk(0)740 683 y Fj(K)811 697 y Fk(1)p Fm(;)p Fk(2)p Fm(n)948 683 y Fp(\()p Fo(\034)31 b Fi(\000)20 b Fo(\034)1185 697 y Fk(1)1224 683 y Fo(;)15 b(k)s Fp(\))p 544 724 1002 4 v 544 811 a(\(2)p Fo(\031)s Fp(\))714 785 y Fk(1)p Fm(=)p Fk(2)826 811 y Fo(R)q Fp(\()p Fo(ik)s Fp(\)\(1)22 b(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))1375 782 y Fm(s)1408 791 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)746 1005 y Fi(\002)g Fp(\(1)h(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1164 967 y Fm(s)1197 976 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)1350 881 y Fn(Z)1441 907 y Fm(\034)1472 916 y Fh(1)1400 1087 y Fk(0)1526 881 y Fn(Z)1617 907 y Fm(\034)1648 916 y Fh(2)1576 1087 y Fk(0)1702 881 y Fn(Z)1793 907 y Fk(+)p Fl(1)1753 1087 y(\0001)1923 1005 y Fp(\()p Fo(ik)2036 1019 y Fk(1)2076 1005 y Fp(\))2111 967 y Fm(s)2144 976 y Fh(0)2183 1005 y Fo(a)p Fp(\()p Fo(\034)2306 1019 y Fk(2)2366 1005 y Fi(\000)f Fo(\034)2496 1019 y Fk(3)2536 1005 y Fo(;)c(k)23 b Fi(\000)d Fo(k)2784 1019 y Fk(1)2824 1005 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)3001 1019 y Fk(3)3041 1005 y Fo(;)15 b(k)3128 1019 y Fk(1)3169 1005 y Fp(\))641 1206 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)25 b Fi(\000)20 b Fo(k)1043 1220 y Fk(1)1082 1206 y Fp(\)\))1152 1169 y Fm(s)1185 1178 y Fh(1)1225 1206 y Fp(\()p Fo(ik)1338 1220 y Fk(1)1378 1206 y Fp(\))1413 1169 y Fm(s)1446 1178 y Fh(2)1485 1206 y Fo(w)r Fp(\()p Fo(\034)1627 1220 y Fk(2)1687 1206 y Fi(\000)g Fo(\034)1818 1220 y Fk(3)1857 1206 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2106 1220 y Fk(1)2145 1206 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)2322 1220 y Fk(3)2363 1206 y Fo(;)15 b(k)2450 1220 y Fk(1)2490 1206 y Fp(\))p Fo(dk)2619 1220 y Fk(1)2660 1206 y Fo(d\034)2747 1220 y Fk(3)2786 1206 y Fo(d\034)2873 1220 y Fk(2)2912 1206 y Fo(d\034)2999 1220 y Fk(1)3039 1206 y Fi(jj)3089 1225 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))668 1412 y Fi(\024)25 b Fo(C)829 1426 y Fk(3)869 1412 y Fi(jj)p Fp(\(1)c(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1246 1374 y Fm(s)1279 1383 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)1432 1288 y Fn(Z)1523 1315 y Fm(\034)1554 1324 y Fh(1)1483 1494 y Fk(0)1608 1288 y Fn(Z)1699 1315 y Fm(\034)1730 1324 y Fh(2)1659 1494 y Fk(0)1785 1288 y Fn(Z)1875 1315 y Fk(+)p Fl(1)1835 1494 y(\0001)2005 1412 y Fp(\()p Fo(ik)2118 1426 y Fk(1)2159 1412 y Fp(\))2194 1374 y Fm(s)2227 1383 y Fh(0)2265 1412 y Fo(a)p Fp(\()p Fo(\034)2388 1426 y Fk(2)2448 1412 y Fi(\000)g Fo(\034)2579 1426 y Fk(3)2618 1412 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2867 1426 y Fk(1)2907 1412 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)3084 1426 y Fk(3)3124 1412 y Fo(;)15 b(k)3211 1426 y Fk(1)3251 1412 y Fp(\))1206 1614 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)k Fi(\000)c Fo(k)1607 1628 y Fk(1)1647 1614 y Fp(\)\))1717 1576 y Fm(s)1750 1585 y Fh(1)1789 1614 y Fp(\()p Fo(ik)1902 1628 y Fk(1)1942 1614 y Fp(\))1977 1576 y Fm(s)2010 1585 y Fh(2)2049 1614 y Fo(w)r Fp(\()p Fo(\034)2191 1628 y Fk(2)2251 1614 y Fi(\000)g Fo(\034)2382 1628 y Fk(3)2421 1614 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2670 1628 y Fk(1)2710 1614 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)2887 1628 y Fk(3)2927 1614 y Fo(;)15 b(k)3014 1628 y Fk(1)3055 1614 y Fp(\))p Fo(dk)3184 1628 y Fk(1)3224 1614 y Fo(d\034)3311 1628 y Fk(3)3350 1614 y Fo(d\034)3437 1628 y Fk(2)3477 1614 y Fi(jj)3527 1632 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))3746 1614 y Fo(:)75 1774 y Fp(F)-8 b(rom)31 b(the)f(inequalities)i(\(20\),)g (\(21\),)h(\(22\),)f(\(23\),)g(w)m(e)f(get)g(that)1045 1934 y Fi(jjH)q Fp(\()p Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\)\))p Fi(jj)1565 1952 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))1812 1934 y Fi(\024)25 b Fo(C)1973 1948 y Fk(1)2012 1934 y Fo(\016)f Fp(+)c Fo(C)2232 1948 y Fk(2)2272 1934 y Fo(C)2337 1948 y Fk(3)2376 1934 y Fo(C)2441 1948 y Fk(4)2480 1934 y Fp(\()p Fo(\016)s(R)j Fp(+)c Fo(R)2810 1896 y Fk(2)2850 1934 y Fp(\))75 2094 y(No)m(w,)39 b(w)m(e)e(c)m(ho)s (ose)g Fo(\016)s Fp(,)i Fo(R)e Fp(small)g(enough,)h(suc)m(h)e(that)h Fo(C)2031 2108 y Fk(1)2070 2094 y Fo(\016)28 b Fp(+)c Fo(C)2298 2108 y Fk(2)2337 2094 y Fo(C)2402 2108 y Fk(3)2442 2094 y Fo(C)2507 2108 y Fk(4)2546 2094 y Fp(\()p Fo(\016)s(R)i Fp(+)e Fo(R)2884 2061 y Fk(2)2923 2094 y Fp(\))36 b Fi(\024)f Fo(R)q Fp(.)58 b(This)36 b(yields)g(the)75 2207 y(prop)s(ert)m(y)30 b Fg(i\))p Fp(.)216 2355 y(W)-8 b(e)27 b(c)m(hec)m(k)h(the)e(prop)s (ert)m(y)g Fg(ii\))p Fp(.)38 b(Let)27 b Fo(w)1515 2369 y Fk(1)1555 2355 y Fp(,)g Fo(w)1672 2369 y Fk(2)1737 2355 y Fp(b)s(e)e(in)h Fo(E)2025 2374 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2270 2355 y Fp(suc)m(h)g(that)g Fi(jj)p Fo(w)2778 2369 y Fm(j)2815 2355 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fi(jj)3070 2374 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))3316 2355 y Fi(\024)25 b Fo(R)q Fp(,)i Fo(j)k Fp(=)25 b(1)p Fo(;)15 b Fp(2.)75 2468 y(W)-8 b(e)32 b(write)158 2679 y Fi(H)q Fp(\()p Fo(w)336 2693 y Fk(1)376 2679 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\)\))21 b Fi(\000)f(H)q Fp(\()p Fo(w)906 2693 y Fk(2)946 2679 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\)\))27 b(=)1309 2555 y Fn(Z)1400 2581 y Fm(\034)1359 2761 y Fk(0)1664 2617 y Fj(K)1735 2631 y Fk(1)p Fm(;)p Fk(2)p Fm(n)1872 2617 y Fp(\()p Fo(\034)j Fi(\000)20 b Fo(\034)2108 2631 y Fk(1)2147 2617 y Fo(;)15 b(k)s Fp(\))p 1468 2658 V 1468 2745 a(\(2)p Fo(\031)s Fp(\))1638 2719 y Fk(1)p Fm(=)p Fk(2)1749 2745 y Fo(R)q Fp(\()p Fo(ik)s Fp(\)\(1)22 b(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))2298 2716 y Fm(s)2331 2725 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)446 2938 y Fi(\002)f Fp(\(1)i(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))863 2901 y Fm(s)896 2910 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)1050 2815 y Fn(Z)1141 2841 y Fm(\034)1172 2850 y Fh(1)1100 3021 y Fk(0)1226 2815 y Fn(Z)1317 2841 y Fm(\034)1348 2850 y Fh(2)1276 3021 y Fk(0)1402 2815 y Fn(Z)1493 2841 y Fk(+)p Fl(1)1452 3021 y(\0001)1622 2938 y Fp(\()p Fo(ik)1735 2952 y Fk(1)1776 2938 y Fp(\))1811 2901 y Fm(s)1844 2910 y Fh(0)1883 2938 y Fo(a)p Fp(\()p Fo(\034)2006 2952 y Fk(2)2065 2938 y Fi(\000)g Fo(\034)2196 2952 y Fk(3)2235 2938 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2484 2952 y Fk(1)2524 2938 y Fp(\)\()p Fo(w)2659 2952 y Fk(1)2699 2938 y Fp(\()p Fo(\034)2774 2952 y Fk(3)2814 2938 y Fo(;)15 b(k)2901 2952 y Fk(1)2941 2938 y Fp(\))20 b Fi(\000)g Fo(w)3152 2952 y Fk(2)3192 2938 y Fp(\()p Fo(\034)3267 2952 y Fk(3)3306 2938 y Fo(;)15 b(k)3393 2952 y Fk(1)3434 2938 y Fp(\)\))1768 3140 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)k Fi(\000)c Fo(k)2169 3154 y Fk(1)2209 3140 y Fp(\)\))2279 3102 y Fm(s)2312 3111 y Fh(1)2351 3140 y Fp(\()p Fo(ik)2464 3154 y Fk(1)2505 3140 y Fp(\))2540 3102 y Fm(s)2573 3111 y Fh(2)2611 3140 y Fo(w)2676 3154 y Fk(1)p Fm(;)p Fk(2)2771 3140 y Fp(\()p Fo(\034)2846 3154 y Fk(2)2885 3140 y Fo(;)15 b(\034)2965 3154 y Fk(3)3005 3140 y Fo(;)g(k)s(;)g(k)3182 3154 y Fk(1)3223 3140 y Fp(\))p Fo(dk)3352 3154 y Fk(1)3392 3140 y Fo(d\034)3479 3154 y Fk(3)3519 3140 y Fo(d\034)3606 3154 y Fk(2)3645 3140 y Fo(d\034)3732 3154 y Fk(1)75 3300 y Fp(where)158 3485 y Fo(w)223 3499 y Fk(1)p Fm(;)p Fk(2)317 3485 y Fp(\()p Fo(\034)392 3499 y Fk(2)432 3485 y Fo(;)g(\034)512 3499 y Fk(3)552 3485 y Fo(;)g(k)s(;)g(k)729 3499 y Fk(1)769 3485 y Fp(\))26 b(=)f(\()p Fo(w)1026 3499 y Fk(1)1066 3485 y Fp(\()p Fo(\034)1141 3499 y Fk(2)1200 3485 y Fi(\000)20 b Fo(\034)1331 3499 y Fk(3)1371 3485 y Fo(;)15 b(k)23 b Fi(\000)d Fo(k)1619 3499 y Fk(1)1659 3485 y Fp(\))h Fi(\000)e Fo(w)1870 3499 y Fk(2)1910 3485 y Fp(\()p Fo(\034)1985 3499 y Fk(2)2045 3485 y Fi(\000)h Fo(\034)2176 3499 y Fk(3)2215 3485 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2464 3499 y Fk(1)2503 3485 y Fp(\)\))h Fi(\002)f Fo(w)2750 3499 y Fk(1)2790 3485 y Fp(\()p Fo(\034)2865 3499 y Fk(3)2904 3485 y Fo(;)15 b(k)2991 3499 y Fk(1)3031 3485 y Fp(\))1891 3623 y(+)20 b Fo(w)2047 3637 y Fk(2)2087 3623 y Fp(\()p Fo(\034)2162 3637 y Fk(2)2222 3623 y Fi(\000)f Fo(\034)2352 3637 y Fk(3)2392 3623 y Fo(;)c(k)23 b Fi(\000)d Fo(k)2640 3637 y Fk(1)2680 3623 y Fp(\))h Fi(\002)f Fp(\()p Fo(w)2927 3637 y Fk(1)2966 3623 y Fp(\()p Fo(\034)3041 3637 y Fk(3)3081 3623 y Fo(;)15 b(k)3168 3637 y Fk(1)3208 3623 y Fp(\))21 b Fi(\000)f Fo(w)3420 3637 y Fk(2)3459 3623 y Fp(\()p Fo(\034)3534 3637 y Fk(3)3574 3623 y Fo(;)15 b(k)3661 3637 y Fk(1)3701 3623 y Fp(\)\))75 3783 y(Using)31 b(the)f(same)h(estimates)h(as)f(\(21\),)h(\(22\),)g(\(23\),)g(w)m(e)f (get)g(that)364 3944 y Fi(jjH)q Fp(\()p Fo(w)592 3958 y Fk(1)632 3944 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\)\))21 b Fi(\000)f(H)q Fp(\()p Fo(w)1162 3958 y Fk(2)1202 3944 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\)\))p Fi(jj)1492 3962 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))1738 3944 y Fi(\024)25 b Fp(\()p Fo(\016)f Fp(+)c(2)p Fo(R)q Fp(\))p Fo(C)2239 3958 y Fk(2)2279 3944 y Fo(C)2344 3958 y Fk(3)2384 3944 y Fo(C)2449 3958 y Fk(4)2488 3944 y Fi(jj)p Fo(w)2603 3958 y Fk(1)2643 3944 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))22 b Fi(\000)e Fo(w)3026 3958 y Fk(2)3065 3944 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fi(jj)3320 3962 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))3541 3944 y Fo(:)75 4104 y Fp(No)m(w,)27 b(w)m(e)f(c)m(ho)s(ose)g Fo(R)q(;)15 b(\016)29 b Fp(small)d(enough)e(suc)m(h)h(that)h Fo(M)35 b Fp(=)25 b(\()p Fo(\016)13 b Fp(+)d(2)p Fo(R)q Fp(\))p Fo(C)2429 4118 y Fk(2)2470 4104 y Fo(C)2535 4118 y Fk(3)2574 4104 y Fo(C)2639 4118 y Fk(4)2704 4104 y Fo(<)25 b Fp(1.)39 b(This)24 b(yields)i(the)f(prop)s(ert)m(y)75 4217 y Fg(ii\))p Fp(.)3557 b Fb(2)75 4420 y Fg(Prop)s(osition)36 b(1)46 b Fd(L)-5 b(et)38 b Fo(')g Fd(b)-5 b(e)37 b(the)h(function)g(de\014ne) -5 b(d)39 b(in)e(se)-5 b(ction)39 b(2.)57 b(We)37 b(put)h Fo(\014)i Fp(=)3043 4396 y(~)3030 4420 y Fo(\014)29 b Fi(\000)23 b Fo(M)3292 4434 y Fk(0)3332 4420 y Fo(\032)h Fi(\000)f Fo(')p Fp(\(0\))39 b Fd(and)75 4533 y(assume)29 b(that)g Fo(\014)i(>)25 b Fp(0)j Fd(\(for)h Fo(\032)f Fd(and)h Fo(')p Fp(\(0\))h Fd(smal)5 b(l)29 b(enough\).)41 b(Then,)29 b(ther)-5 b(e)29 b(exist)g Fo(R)q(;)15 b(\016)29 b(>)c Fp(0)j Fd(\(dep)-5 b(ending)30 b(only)f(on)75 4646 y Fo(s)118 4660 y Fk(0)157 4646 y Fo(;)15 b(s)240 4660 y Fk(1)280 4646 y Fo(;)g(s)363 4660 y Fk(2)402 4646 y Fo(;)g(\032;)g(\026;)g(\027)q(;)g(P)e Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))p Fo(;)15 b(R)q Fp(\()p Fo(X)7 b Fp(\))p Fd(\))32 b(such)d(that)i(if)d(the)h(functions)g Fo(w)2438 4660 y Fk(0)2503 4646 y Fi(2)c Fo(E)2656 4677 y Fk(\()2693 4659 y(~)2683 4677 y Fm(\014)t(;\026)p Fk(+)p Fm(\026)2886 4686 y Fh(1)p Ff(;)p Fh(1)2970 4677 y Fk(\))3030 4646 y Fd(and)30 b Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b Fi(2)e Fo(E)3636 4665 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))75 4759 y Fd(satis\014es)1081 4872 y Fi(jj)p Fo(w)1196 4886 y Fk(0)1236 4872 y Fp(\()p Fo(k)s Fp(\))p Fi(jj)1406 4903 y Fk(\()1443 4885 y(~)1433 4903 y Fm(\014)5 b(;\026)p Fk(+)p Fm(\026)1637 4912 y Fh(1)p Ff(;)p Fh(1)1722 4903 y Fk(\))1778 4872 y Fi(\024)25 b Fo(\016)69 b(;)81 b Fi(jj)p Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fi(jj)2442 4890 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2688 4872 y Fi(\024)25 b Fo(\016)n(;)75 5017 y Fd(then)33 b(the)g(pr)-5 b(oblem)75 5245 y Fp(\(24\))85 b Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b(=)e Fj(K)786 5259 y Fk(11)860 5245 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fo(w)1130 5259 y Fk(0)1171 5245 y Fp(\()p Fo(k)s Fp(\))21 b(+)1403 5121 y Fn(Z)1494 5148 y Fm(\034)1453 5328 y Fk(0)1562 5183 y Fj(K)1633 5197 y Fk(1)p Fm(;)p Fk(2)p Fm(n)1770 5183 y Fp(\()p Fo(\034)30 b Fi(\000)20 b Fo(\034)2006 5197 y Fk(1)2046 5183 y Fo(;)15 b(k)s Fp(\))p 1562 5224 610 4 v 1615 5311 a(\(2)p Fo(\031)s Fp(\))1785 5285 y Fk(1)p Fm(=)p Fk(2)1896 5311 y Fo(R)q Fp(\()p Fo(ik)s Fp(\))1002 5505 y Fi(\002)1093 5381 y Fn(Z)1184 5407 y Fm(\034)1215 5416 y Fh(1)1144 5587 y Fk(0)1269 5381 y Fn(Z)1360 5407 y Fm(\034)1391 5416 y Fh(2)1320 5587 y Fk(0)1445 5381 y Fn(Z)1536 5407 y Fk(+)p Fl(1)1496 5587 y(\0001)1666 5505 y Fp(\()p Fo(ik)1779 5519 y Fk(1)1819 5505 y Fp(\))1854 5467 y Fm(s)1887 5476 y Fh(0)1926 5505 y Fo(a)p Fp(\()p Fo(\034)2049 5519 y Fk(2)2109 5505 y Fi(\000)20 b Fo(\034)2240 5519 y Fk(3)2279 5505 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2528 5519 y Fk(1)2567 5505 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)2744 5519 y Fk(3)2785 5505 y Fo(;)15 b(k)2872 5519 y Fk(1)2912 5505 y Fp(\))1374 5706 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)k Fi(\000)c Fo(k)1775 5720 y Fk(1)1815 5706 y Fp(\)\))1885 5669 y Fm(s)1918 5678 y Fh(1)1957 5706 y Fp(\()p Fo(ik)2070 5720 y Fk(1)2110 5706 y Fp(\))2145 5669 y Fm(s)2178 5678 y Fh(2)2217 5706 y Fo(w)r Fp(\()p Fo(\034)2359 5720 y Fk(2)2419 5706 y Fi(\000)g Fo(\034)2550 5720 y Fk(3)2590 5706 y Fo(;)15 b(k)23 b Fi(\000)d Fo(k)2838 5720 y Fk(1)2878 5706 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)3055 5720 y Fk(3)3095 5706 y Fo(;)15 b(k)3182 5720 y Fk(1)3223 5706 y Fp(\))p Fo(dk)3352 5720 y Fk(1)3392 5706 y Fo(d\034)3479 5720 y Fk(3)3519 5706 y Fo(d\034)3606 5720 y Fk(2)3645 5706 y Fo(d\034)3732 5720 y Fk(1)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 3764 100 a Fp(14)75 399 y Fd(has)34 b(a)f(unique)e(solution)j Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))35 b Fd(in)d(the)h(sp)-5 b(ac)g(e)34 b Fo(E)1826 417 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2077 399 y Fd(such)f(that)h Fi(jj)p Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fi(jj)2842 417 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))3089 399 y Fi(\024)25 b Fo(R)q Fd(.)75 646 y Fg(Pro)s(of)40 b Fp(The)e(ball)769 623 y(\026)748 646 y Fo(D)s Fp(\(0)p Fo(;)15 b(R)q Fp(\))40 b(is)e(a)i(complete)g(metric)f(space)g(for)g (the)g(norm)f Fi(jj)p Fo(:)p Fi(jj)2893 664 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))3113 646 y Fp(.)66 b(F)-8 b(rom)39 b(Lemma)g(7,)75 759 y(there)33 b(exists)g Fo(R)d(>)f Fp(0)k(suc)m(h)f(that)h Fi(H)h Fp(is)e(a)h(shrinking)f(map)g(from)2371 736 y(\026)2350 759 y Fo(D)r Fp(\(0)p Fo(;)15 b(R)q Fp(\))34 b(in)m(to)g(itself.)48 b(F)-8 b(rom)33 b(the)g(classical)75 872 y(shrinking)c(map)h(theorem)h(on)f(complete)i(metric)f(spaces,)g (the)f(map)g Fi(H)h Fp(has)f(a)g(unique)g(\014xed)f(p)s(oin)m(t)i Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))75 985 y(in)202 962 y(\026)181 985 y Fo(D)s Fp(\(0)p Fo(;)g(R)q Fp(\))31 b(\(i.e.)42 b Fi(H)q Fp(\()p Fo(w)r Fp(\))27 b(=)d Fo(w)r Fp(\).)42 b(This)30 b(yields)g(the)h(prop)s(osition.)1450 b Fb(2)75 1232 y Fg(Prop)s(osition)36 b(2)46 b Fd(L)-5 b(et)34 b Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))29 b Fi(2)d Fo(E)1351 1250 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))1603 1232 y Fd(b)-5 b(e)34 b(the)g(solution)h(of)e(the)h(pr)-5 b(oblem)36 b(\(24\).)45 b(Then,)34 b(for)g(al)5 b(l)34 b Fo(k)c Fi(2)d Fj(R)p Fd(,)75 1345 y(the)33 b(function)g Fo(\034)i Fi(7!)25 b Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))35 b Fd(is)d(the)h(unique)f(solution)i(of)f(the)g(pr)-5 b(oblem)34 b(\(14\),)g(\(15\),)f(for)g(al)5 b(l)34 b Fo(\034)5 b(;)15 b(k)28 b Fi(2)d Fj(R)p Fd(.)75 1592 y Fg(Pro)s(of)31 b Fp(W)-8 b(e)32 b(put)158 1861 y Fo(F)13 b Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b(=)567 1799 y(\(1)21 b(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))893 1766 y Fm(s)926 1775 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)p 567 1840 499 4 v 675 1927 a Fp(\(2)p Fo(\031)s Fp(\))845 1900 y Fk(1)p Fm(=)p Fk(2)1090 1737 y Fn(Z)1181 1763 y Fm(\034)1140 1943 y Fk(0)1239 1737 y Fn(Z)1330 1763 y Fm(\034)1361 1772 y Fh(1)1289 1943 y Fk(0)1415 1737 y Fn(Z)1506 1763 y Fk(+)p Fl(1)1466 1943 y(\0001)1636 1861 y Fp(\()p Fo(ik)1749 1875 y Fk(1)1789 1861 y Fp(\))1824 1823 y Fm(s)1857 1832 y Fh(0)1896 1861 y Fo(a)p Fp(\()p Fo(\034)2019 1875 y Fk(1)2079 1861 y Fi(\000)g Fo(\034)2209 1875 y Fk(2)2249 1861 y Fo(;)c(k)23 b Fi(\000)d Fo(k)2497 1875 y Fk(1)2537 1861 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)2714 1875 y Fk(2)2754 1861 y Fo(;)15 b(k)2841 1875 y Fk(1)2882 1861 y Fp(\))1500 2062 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)25 b Fi(\000)20 b Fo(k)1902 2076 y Fk(1)1941 2062 y Fp(\)\))2011 2025 y Fm(s)2044 2034 y Fh(1)2084 2062 y Fp(\()p Fo(ik)2197 2076 y Fk(1)2237 2062 y Fp(\))2272 2025 y Fm(s)2305 2034 y Fh(2)2344 2062 y Fo(w)r Fp(\()p Fo(\034)2486 2076 y Fk(1)2546 2062 y Fi(\000)g Fo(\034)2677 2076 y Fk(2)2716 2062 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2965 2076 y Fk(1)3005 2062 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)3182 2076 y Fk(2)3222 2062 y Fo(;)15 b(k)3309 2076 y Fk(1)3349 2062 y Fp(\))p Fo(dk)3478 2076 y Fk(1)3519 2062 y Fo(d\034)3606 2076 y Fk(2)3645 2062 y Fo(d\034)3732 2076 y Fk(1)75 2255 y Fp(for)37 b(all)h Fo(\034)5 b(;)15 b(k)40 b Fi(2)c Fj(R)p Fp(.)61 b(F)-8 b(rom)37 b(the)h(fact)g(that)f Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))39 b Fi(2)d Fo(E)2048 2274 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2267 2255 y Fp(,)j(w)m(e)e(get)h(from)f(\(21\),)k(\(22\))d(that)g Fo(F)13 b Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))38 b(is)75 2368 y(con)m(tin)m(uous)31 b(with)f(resp)s(ect)g(to)i Fo(\034)5 b(;)15 b(k)28 b Fi(2)d Fj(R)p Fp(.)41 b(F)-8 b(rom)30 b(Lemma)h(6,)g(the)g(unique)e(solution)i(of)f(the)h(problem) 1052 2561 y Fo(R)q Fp(\()p Fo(ik)s Fp(\))p Fo(P)13 b Fp(\()p Fo(@)1432 2524 y Fk(2)1427 2584 y Fm(\034)1473 2561 y Fo(;)i(ik)s Fp(\))p Fo(W)e Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))28 b(=)c Fo(F)13 b Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fo(=)p Fp(\(1)23 b(+)d Fi(j)p Fo(k)s Fi(j)p Fp(\))2706 2524 y Fm(s)2739 2533 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)75 2754 y Fp(for)30 b(initial)i(conditions)1228 2867 y Fo(W)13 b Fp(\(0)p Fo(;)i(k)s Fp(\))27 b(=)e Fo(w)1720 2881 y Fk(0)1759 2867 y Fp(\()p Fo(k)s Fp(\))62 b Fo(;)76 b Fp(\()p Fo(@)2130 2829 y Fm(j)2125 2889 y(\034)2169 2867 y Fo(W)13 b Fp(\)\(0)p Fo(;)i(k)s Fp(\))26 b Fi(\021)f Fp(0)p Fo(;)75 3028 y Fp(for)30 b(all)h(1)26 b Fi(\024)f Fo(j)31 b Fi(\024)25 b Fp(2)p Fo(n)20 b Fi(\000)g Fp(1,)31 b(is)744 3264 y Fo(W)13 b Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))26 b(=)f Fj(K)1241 3278 y Fk(11)1315 3264 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fo(w)1585 3278 y Fk(0)1626 3264 y Fp(\()p Fo(k)s Fp(\))21 b(+)1858 3141 y Fn(Z)1949 3167 y Fm(\034)1909 3347 y Fk(0)2073 3202 y Fj(K)2144 3216 y Fk(1)p Fm(;)p Fk(2)p Fm(n)2281 3202 y Fp(\()p Fo(\034)30 b Fi(\000)20 b Fo(\034)2517 3216 y Fk(1)2556 3202 y Fo(;)15 b(k)s Fp(\))p 2017 3243 720 4 v 2017 3327 a Fo(R)q Fp(\()p Fo(ik)s Fp(\)\(1)22 b(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))2566 3298 y Fm(s)2599 3307 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)2747 3264 y Fo(F)13 b Fp(\()p Fo(\034)2893 3278 y Fk(1)2933 3264 y Fo(;)i(k)s Fp(\))p Fo(d\034)3145 3278 y Fk(1)75 3506 y Fp(for)32 b(all)h Fo(\034)38 b Fi(2)27 b Fj(R)32 b Fp(and)f Fo(k)h Fi(2)27 b Fj(R)p Fp(.)46 b(But)32 b(the)g(equation)h(\(24\))g(implies)g(that)f Fo(W)13 b Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))29 b(=)f Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))34 b(for)d(all)i Fo(\034)5 b(;)15 b(k)32 b Fi(2)c Fj(R)p Fp(.)75 3619 y(So)i(that)h Fo(\034)36 b Fi(7!)25 b Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))32 b(is)e(the)h(unique)e(solution)i(of)g (\(14\),)h(\(15\),)g(for)e(all)h Fo(\034)5 b(;)15 b(k)29 b Fi(2)c Fj(R)p Fp(.)834 b Fb(2)75 3866 y Fg(Lemma)35 b(8)45 b Fd(L)-5 b(et)47 b Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))52 b Fi(2)e Fo(E)1216 3885 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))1482 3866 y Fd(b)-5 b(e)46 b(the)h(solution)g(of)g(the)g(pr)-5 b(oblem)48 b(\(24\).)85 b(Then,)50 b(we)c(have)h(that)75 3989 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b Fi(7!)e Fo(@)476 3956 y Fm(l)471 4011 y(\034)514 3989 y Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b Fi(2)e Fo(E)971 3956 y Fc(R)966 4020 y Fk(\()p Fm(\027;\014)s(;\026)p Fl(\000)p Fm(l)q Fk(\))1294 3989 y Fd(for)33 b(al)5 b(l)34 b Fp(0)25 b Fi(\024)g Fo(l)i Fi(\024)e Fp(2)p Fo(n)p Fd(.)75 4248 y Fg(Pro)s(of)31 b Fp(F)-8 b(rom)31 b(the)g(expressions)f (\(17\))i(and)d(the)i(assumption)f Fg(\(H2\))p Fp(,)g(w)m(e)h(get)g (that)75 4515 y(\(25\))605 b(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1167 4477 y Fl(\000)p Fm(l)q Fl(\000)p Fm(\026)1341 4486 y Fh(1)p Ff(;)p Fh(1)1430 4515 y Fo(@)1483 4477 y Fm(l)1478 4537 y(\034)1521 4515 y Fj(K)1592 4529 y Fk(11)1667 4515 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))26 b(=)2042 4396 y Fm(p)1994 4429 y Fn(X)1999 4624 y Fm(j)t Fk(=1)2141 4515 y Fp(exp\()p Fo(i\034)10 b(K)2473 4529 y Fm(j)2510 4515 y Fp(\()p Fo(k)s Fp(\)\))p Fo(p)2711 4477 y Fm(l)2711 4537 y Fk(1)p Fm(;)p Fk(1)p Fm(;j)2858 4515 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fo(;)75 4813 y Fp(where)30 b Fo(p)384 4780 y Fm(l)384 4839 y Fk(1)p Fm(;)p Fk(1)p Fm(;j)530 4813 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))31 b(are)g(p)s(olynomials)g(in)f Fo(\034)40 b Fp(and)30 b(b)s(ounded)e(con)m(tin)m(uous)j(with)f(resp)s(ect)h(to)g Fo(k)i Fp(on)e Fj(R)p Fp(,)f(and)75 5096 y(\(26\))526 b(\(1)21 b(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))1087 5059 y Fl(\000)p Fm(l)1287 5034 y Fo(@)1340 5001 y Fm(l)1335 5057 y(\034)1378 5034 y Fj(K)1449 5048 y Fk(1)p Fm(;)p Fk(2)p Fm(n)1586 5034 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p 1179 5075 V 1179 5159 a Fo(R)q Fp(\()p Fo(ik)s Fp(\)\(1)22 b(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))1728 5130 y Fm(s)1761 5139 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)1935 5096 y Fp(=)2078 4978 y Fm(p)2031 5010 y Fn(X)2035 5205 y Fm(j)t Fk(=1)2177 5096 y Fp(exp\()p Fo(i\034)10 b(K)2509 5110 y Fm(j)2546 5096 y Fp(\()p Fo(k)s Fp(\)\))p Fo(p)2747 5059 y Fm(l)2747 5119 y Fk(1)p Fm(;)p Fk(2)p Fm(n;j)2937 5096 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fo(;)75 5395 y Fp(where)30 b Fo(p)384 5362 y Fm(l)384 5420 y Fk(1)p Fm(;)p Fk(2)p Fm(n;j)573 5395 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))32 b(are)f(p)s(olynomials)g(in)f Fo(\034)41 b Fp(and)30 b(b)s(ounded)f(con)m(tin)m(uous)i(with)g(resp)s(ect)f(to)i Fo(k)i Fp(on)c Fj(R)p Fp(,)h(for)g(all)75 5520 y Fo(l)c Fi(\025)e Fp(0.)41 b(By)31 b(construction,)g(w)m(e)g(ha)m(v)m(e)h(that) f Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b Fi(2)d Fo(E)2022 5487 y Fc(R)2017 5552 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2236 5520 y Fp(.)41 b(Let)31 b(us)f(assume)g(in)g(the)g(follo)m (wing)i(that)75 5740 y(\(27\))1298 b Fo(@)1586 5703 y Fm(l)1581 5763 y(\034)1624 5740 y Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b Fi(2)e Fo(E)2081 5703 y Fc(R)2076 5767 y Fk(\()p Fm(\027;\014)s(;\026)p Fl(\000)p Fm(l)q Fk(\))2372 5740 y Fo(;)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 3764 100 a Fp(15)75 399 y(for)30 b(0)c Fi(\024)f Fo(l)i Fi(\024)e Fo(q)s Fp(,)30 b(with)g Fo(q)f(<)24 b Fp(2)p Fo(n)p Fp(.)41 b(Then,)30 b(w)m(e)h(ha)m(v)m(e)g (to)g(sho)m(w)g(that)g Fo(@)2361 354 y Fm(q)r Fk(+1)2356 410 y Fm(\034)2489 399 y Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b Fi(2)e Fo(E)2946 366 y Fc(R)2941 430 y Fk(\()p Fm(\027;\014)s(;\026)p Fl(\000)p Fk(\()p Fm(q)r Fk(+1\)\))3394 399 y Fp(.)40 b(W)-8 b(e)32 b(put)158 669 y Fo(I)7 b Fp(\()p Fo(\034)280 683 y Fk(1)320 669 y Fo(;)15 b(k)s Fp(\))26 b(=)f(\(1)c(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))894 631 y Fm(s)927 640 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)1080 545 y Fn(Z)1171 571 y Fm(\034)1202 580 y Fh(1)1131 751 y Fk(0)1256 545 y Fn(Z)1347 571 y Fm(\034)1378 580 y Fh(2)1307 751 y Fk(0)1432 545 y Fn(Z)1523 571 y Fk(+)p Fl(1)1483 751 y(\0001)1653 669 y Fp(\()p Fo(ik)1766 683 y Fk(1)1806 669 y Fp(\))1841 631 y Fm(s)1874 640 y Fh(0)1913 669 y Fo(a)p Fp(\()p Fo(\034)2036 683 y Fk(2)2096 669 y Fi(\000)g Fo(\034)2227 683 y Fk(3)2266 669 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2515 683 y Fk(1)2554 669 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)2731 683 y Fk(3)2772 669 y Fo(;)15 b(k)2859 683 y Fk(1)2899 669 y Fp(\))1500 870 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)25 b Fi(\000)20 b Fo(k)1902 884 y Fk(1)1941 870 y Fp(\)\))2011 833 y Fm(s)2044 842 y Fh(1)2084 870 y Fp(\()p Fo(ik)2197 884 y Fk(1)2237 870 y Fp(\))2272 833 y Fm(s)2305 842 y Fh(2)2344 870 y Fo(w)r Fp(\()p Fo(\034)2486 884 y Fk(2)2546 870 y Fi(\000)g Fo(\034)2677 884 y Fk(3)2716 870 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2965 884 y Fk(1)3005 870 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)3182 884 y Fk(3)3222 870 y Fo(;)15 b(k)3309 884 y Fk(1)3349 870 y Fp(\))p Fo(dk)3478 884 y Fk(1)3519 870 y Fo(d\034)3606 884 y Fk(3)3645 870 y Fo(d\034)3732 884 y Fk(2)75 1034 y Fp(for)30 b(all)h Fo(\034)380 1048 y Fk(1)420 1034 y Fo(;)15 b(k)28 b Fi(2)d Fj(R)p Fp(.)41 b(F)-8 b(rom)30 b(the)h(h)m(yp)s(othesis)f(\(27\))i(and)e(the)g (assumption)g Fg(\(H5\))g Fp(w)m(e)h(deduce)f(that)158 1279 y Fo(@)206 1293 y Fm(\034)237 1302 y Fh(1)276 1279 y Fo(I)7 b Fp(\()p Fo(\034)398 1293 y Fk(1)438 1279 y Fo(;)15 b(k)s Fp(\))26 b(=)f(\(1)c(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1012 1242 y Fm(s)1045 1251 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)1198 1155 y Fn(Z)1289 1182 y Fk(+)p Fl(1)1249 1361 y(\0001)1434 1155 y Fn(Z)1525 1182 y Fm(\034)1556 1191 y Fh(1)1485 1361 y Fk(0)1595 1279 y Fp(\()p Fo(ik)1708 1293 y Fk(1)1748 1279 y Fp(\))1783 1242 y Fm(s)1816 1251 y Fh(0)1855 1279 y Fo(a)p Fp(\()p Fo(\034)1978 1293 y Fk(1)2038 1279 y Fi(\000)g Fo(\034)2169 1293 y Fk(3)2208 1279 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2457 1293 y Fk(1)2496 1279 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)2673 1293 y Fk(3)2714 1279 y Fo(;)15 b(k)2801 1293 y Fk(1)2841 1279 y Fp(\))1602 1481 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)k Fi(\000)c Fo(k)2003 1495 y Fk(1)2043 1481 y Fp(\)\))2113 1443 y Fm(s)2146 1452 y Fh(1)2185 1481 y Fp(\()p Fo(ik)2298 1495 y Fk(1)2338 1481 y Fp(\))2373 1443 y Fm(s)2406 1452 y Fh(2)2445 1481 y Fo(w)r Fp(\()p Fo(\034)2587 1495 y Fk(1)2647 1481 y Fi(\000)g Fo(\034)2778 1495 y Fk(3)2817 1481 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)3066 1495 y Fk(1)3106 1481 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)3283 1495 y Fk(3)3323 1481 y Fo(;)15 b(k)3410 1495 y Fk(1)3450 1481 y Fp(\))p Fo(dk)3579 1495 y Fk(1)3620 1481 y Fo(d\034)3707 1495 y Fk(3)3746 1481 y Fo(;)75 1645 y Fp(for)30 b(all)h Fo(\034)380 1659 y Fk(1)420 1645 y Fo(;)15 b(k)28 b Fi(2)d Fj(R)30 b Fp(and)g(for)g(all)h Fo(j)g Fi(\025)25 b Fp(0,)158 1885 y Fo(@)211 1847 y Fm(j)t Fk(+2)206 1907 y Fm(\034)237 1916 y Fh(1)338 1885 y Fo(I)7 b Fp(\()p Fo(\034)460 1899 y Fk(1)500 1885 y Fo(;)15 b(k)s Fp(\))26 b(=)f(\(1)c(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1074 1847 y Fm(s)1107 1856 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)1260 1761 y Fn(Z)1351 1787 y Fk(+)p Fl(1)1311 1967 y(\0001)1481 1885 y Fp(\()1577 1798 y Fn(X)1516 1996 y Fm(l)1537 2005 y Fh(1)1572 1996 y Fk(+)p Fm(l)1648 2005 y Fh(2)1682 1996 y Fk(=)p Fm(j)1769 1885 y Fp(\()p Fo(ik)1882 1899 y Fk(1)1923 1885 y Fp(\))1958 1847 y Fm(s)1991 1856 y Fh(0)2029 1885 y Fp(\()p Fo(@)2117 1847 y Fm(l)2138 1856 y Fh(1)2112 1907 y Fm(\034)2143 1916 y Fh(1)2183 1885 y Fo(a)p Fp(\)\(0)p Fo(;)15 b(k)25 b Fi(\000)20 b Fo(k)2596 1899 y Fk(1)2636 1885 y Fp(\)\()p Fo(@)2759 1847 y Fm(l)2780 1856 y Fh(2)2754 1907 y Fm(\034)2785 1916 y Fh(1)2825 1885 y Fo(w)r Fp(\)\()p Fo(\034)3002 1899 y Fk(1)3042 1885 y Fo(;)15 b(k)3129 1899 y Fk(1)3169 1885 y Fp(\))871 2135 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)k Fi(\000)c Fo(k)1272 2149 y Fk(1)1312 2135 y Fp(\)\))1382 2097 y Fm(s)1415 2106 y Fh(1)1454 2135 y Fp(\()p Fo(ik)1567 2149 y Fk(1)1608 2135 y Fp(\))1643 2097 y Fm(s)1676 2106 y Fh(2)1714 2135 y Fp(\()p Fo(@)1802 2097 y Fm(l)1823 2106 y Fh(1)1797 2157 y Fm(\034)1828 2166 y Fh(1)1868 2135 y Fo(w)r Fp(\)\(0)p Fo(;)15 b(k)26 b Fi(\000)19 b Fo(k)2300 2149 y Fk(1)2340 2135 y Fp(\)\()p Fo(@)2463 2097 y Fm(l)2484 2106 y Fh(2)2458 2157 y Fm(\034)2489 2166 y Fh(1)2529 2135 y Fo(w)r Fp(\)\()p Fo(\034)2706 2149 y Fk(1)2747 2135 y Fo(;)c(k)2834 2149 y Fk(1)2874 2135 y Fp(\)\))p Fo(dk)3038 2149 y Fk(1)708 2339 y Fp(+)20 b(\(1)h(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1126 2302 y Fm(s)1159 2311 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)1313 2216 y Fn(Z)1403 2242 y Fk(+)p Fl(1)1363 2422 y(\0001)1548 2216 y Fn(Z)1639 2242 y Fm(\034)1670 2251 y Fh(1)1599 2422 y Fk(0)1709 2339 y Fp(\()p Fo(ik)1822 2353 y Fk(1)1863 2339 y Fp(\))1898 2302 y Fm(s)1931 2311 y Fh(0)1969 2339 y Fp(\()p Fo(@)2057 2302 y Fm(j)t Fk(+1)2052 2362 y Fm(\034)2083 2371 y Fh(1)2185 2339 y Fo(a)p Fp(\)\()p Fo(\034)2343 2353 y Fk(1)2403 2339 y Fi(\000)g Fo(\034)2534 2353 y Fk(3)2573 2339 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2822 2353 y Fk(1)2861 2339 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)3038 2353 y Fk(3)3079 2339 y Fo(;)15 b(k)3166 2353 y Fk(1)3206 2339 y Fp(\))1376 2543 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)25 b Fi(\000)19 b Fo(k)1777 2557 y Fk(1)1817 2543 y Fp(\)\))1887 2506 y Fm(s)1920 2515 y Fh(1)1959 2543 y Fp(\()p Fo(ik)2072 2557 y Fk(1)2113 2543 y Fp(\))2148 2506 y Fm(s)2181 2515 y Fh(2)2219 2543 y Fp(\()p Fo(@)2307 2506 y Fm(j)t Fk(+1)2302 2566 y Fm(\034)2333 2575 y Fh(1)2435 2543 y Fo(w)r Fp(\)\()p Fo(\034)2612 2557 y Fk(1)2672 2543 y Fi(\000)h Fo(\034)2803 2557 y Fk(3)2843 2543 y Fo(;)15 b(k)23 b Fi(\000)d Fo(k)3091 2557 y Fk(1)3131 2543 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)3308 2557 y Fk(3)3348 2543 y Fo(;)15 b(k)3435 2557 y Fk(1)3476 2543 y Fp(\))p Fo(d\034)3598 2557 y Fk(3)3637 2543 y Fo(dk)3731 2557 y Fk(1)75 2708 y Fp(for)30 b(all)h Fo(\034)380 2722 y Fk(1)420 2708 y Fo(;)15 b(k)28 b Fi(2)d Fj(R)p Fp(,)30 b(all)i(0)25 b Fi(\024)g Fo(j)31 b Fi(\024)25 b Fo(q)e Fi(\000)d Fp(1.)41 b(F)-8 b(rom)31 b(Lemma)g(4)f(and)g(the)h (con)m(tin)m(uous)g(inclusion)1439 2872 y Fo(E)1506 2890 y Fk(\()p Fm(\027;\014)s(;\026)p Fl(\000)p Fm(m)1810 2899 y Fh(1)1845 2890 y Fk(\))1902 2872 y Fo(,)-15 b Fi(!)25 b Fo(E)2095 2890 y Fk(\()p Fm(\027;\014)s(;\026)p Fl(\000)p Fm(m)2399 2899 y Fh(2)2434 2890 y Fk(\))2466 2872 y Fo(;)75 3036 y Fp(if)30 b(0)c Fi(\024)f Fo(m)405 3050 y Fk(1)470 3036 y Fi(\024)f Fo(m)645 3050 y Fk(2)710 3036 y Fo(<)h(\026)p Fp(,)30 b(w)m(e)h(get)h(that)75 3200 y(\(28\))1277 b Fo(@)1565 3162 y Fm(l)1560 3223 y(\034)1591 3232 y Fh(1)1631 3200 y Fo(I)7 b Fp(\()p Fo(\034)1753 3214 y Fk(1)1792 3200 y Fo(;)15 b(k)s Fp(\))26 b Fi(2)f Fo(E)2101 3162 y Fc(R)2096 3227 y Fk(\()p Fm(\027;\014)s (;\026)p Fl(\000)p Fm(l)q Fk(\))2392 3200 y Fo(;)75 3364 y Fp(for)30 b(all)h(0)26 b Fi(\024)f Fo(l)i Fi(\024)e Fo(q)s Fp(.)41 b(F)-8 b(rom)31 b(\(25\),)h(\(26\),)g(\(28\))g(and)e (Lemma)g(3)h(and)f(5,)h(w)m(e)f(get)i(that)158 3553 y(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))485 3516 y Fl(\000)p Fk(\()p Fm(q)r Fk(+1\))724 3553 y Fo(@)777 3516 y Fm(q)r Fk(+1)772 3576 y Fm(\034)906 3553 y Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b(=)e(\(1)c(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))1627 3516 y Fl(\000)p Fk(\()p Fm(q)r Fk(+1\))1866 3553 y Fo(@)1919 3516 y Fm(q)r Fk(+1)1914 3576 y Fm(\034)2048 3553 y Fj(K)2119 3567 y Fk(1)p Fm(;)p Fk(1)2213 3553 y Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fo(w)2483 3567 y Fk(0)2524 3553 y Fp(\()p Fo(k)s Fp(\))841 3761 y(+)994 3675 y Fn(X)932 3873 y Fm(l)953 3882 y Fh(1)988 3873 y Fk(+)p Fm(l)1064 3882 y Fh(2)1098 3873 y Fk(=)p Fm(q)1607 3699 y Fp(\()p Fo(@)1695 3666 y Fm(l)1716 3675 y Fh(1)1690 3722 y Fm(\034)1756 3699 y Fj(K)1827 3713 y Fk(1)p Fm(;)p Fk(2)p Fm(n)1964 3699 y Fp(\)\(0)p Fo(;)g(k)s Fp(\))p 1212 3741 1390 4 v 1212 3828 a(\(2)p Fo(\031)s Fp(\))1382 3801 y Fk(1)p Fm(=)p Fk(2)1493 3828 y Fo(R)q Fp(\()p Fo(ik)s Fp(\)\(1)22 b(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))2042 3799 y Fm(s)2075 3808 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)2213 3828 y Fp(\(1)h(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))2540 3801 y Fm(l)2561 3810 y Fh(1)2677 3700 y Fo(@)2730 3667 y Fm(l)2751 3676 y Fh(2)2725 3722 y Fm(\034)2790 3700 y Fo(I)7 b Fp(\()p Fo(\034)e(;)15 b(k)s Fp(\))p 2621 3741 478 4 v 2621 3824 a(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))2948 3798 y Fm(l)2969 3807 y Fh(2)3005 3798 y Fk(+1)1640 4077 y Fp(+)1731 3953 y Fn(Z)1822 3980 y Fm(\034)1781 4159 y Fk(0)2223 4015 y Fo(@)2276 3971 y Fm(q)r Fk(+1)2271 4027 y Fm(\034)2405 4015 y Fj(K)2476 4029 y Fk(1)p Fm(;)p Fk(2)p Fm(n)2613 4015 y Fp(\()p Fo(\034)30 b Fi(\000)20 b Fo(\034)2849 4029 y Fk(1)2888 4015 y Fo(;)15 b(k)s Fp(\))p 1890 4056 1458 4 v 1890 4143 a(\(2)p Fo(\031)s Fp(\))2060 4117 y Fk(1)p Fm(=)p Fk(2)2171 4143 y Fo(R)q Fp(\()p Fo(ik)s Fp(\)\(1)22 b(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))2720 4115 y Fm(s)2753 4124 y Fh(0)p Ff(;)p Fh(1)p Ff(;)p Fh(2)2891 4143 y Fp(\(1)h(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))3218 4117 y Fm(q)r Fk(+1)3357 4077 y Fo(I)7 b Fp(\()p Fo(\034)3479 4091 y Fk(1)3519 4077 y Fo(;)15 b(k)s Fp(\))p Fo(d\034)3731 4091 y Fk(1)75 4313 y Fp(for)32 b(all)h Fo(\034)5 b(;)15 b(k)32 b Fi(2)c Fj(R)p Fp(.)47 b(W)-8 b(e)33 b(also)g(deduce)f(that)h(\(1)23 b(+)e Fi(j)p Fo(k)s Fi(j)p Fp(\))1911 4280 y Fl(\000)p Fk(\()p Fm(q)r Fk(+1\))2150 4313 y Fo(@)2203 4268 y Fm(q)r Fk(+1)2198 4324 y Fm(\034)2331 4313 y Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))31 b Fi(2)d Fo(E)2795 4280 y Fc(R)2790 4344 y Fk(\()p Fm(\027;\014)s (;\026)p Fk(\))3009 4313 y Fp(.)46 b(F)-8 b(rom)33 b(Lemma)g(5,)g(w)m (e)75 4456 y(\014nally)d(get)i(that)f Fo(@)752 4412 y Fm(q)r Fk(+1)747 4468 y Fm(\034)880 4456 y Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b Fi(2)e Fo(E)1337 4423 y Fc(R)1332 4487 y Fk(\()p Fm(\027;\014)s(;\026)p Fl(\000)p Fk(\()p Fm(q)r Fk(+1\)\))1785 4456 y Fp(.)1946 b Fb(2)75 4676 y Fg(Lemma)35 b(9)45 b Fd(L)-5 b(et)31 b Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b Fi(2)e Fo(E)1150 4695 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))1400 4676 y Fd(b)-5 b(e)30 b(the)h(unique)f(solution)h(of)g(the)g(pr)-5 b(oblem)32 b(\(14\),)h(\(15\).)42 b(Then,)31 b(we)g(get)75 4789 y(that)j Fo(w)r Fp(\()p Fi(\000)p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b(=)e Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))34 b Fd(for)f(al)5 b(l)33 b Fo(\034)5 b(;)15 b(k)29 b Fi(2)c Fj(R)p Fd(.)75 4998 y Fg(Pro)s(of)k Fp(W)-8 b(e)30 b(put)46 b(~)-63 b Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b(:=)e Fo(w)r Fp(\()p Fi(\000)p Fo(\034)5 b(;)15 b(k)s Fp(\))30 b(for)e(all)h Fo(\034)5 b(;)15 b(k)29 b Fi(2)c Fj(R)p Fp(.)40 b(F)-8 b(rom)29 b(the)f(equation)i (\(14\),)g(transforming)e Fo(\034)38 b Fp(in)m(to)75 5111 y Fi(\000)p Fo(\034)i Fp(yields)158 5300 y Fo(R)q Fp(\()p Fo(ik)s Fp(\)\()p Fo(P)13 b Fp(\()p Fo(@)573 5263 y Fk(2)568 5323 y Fm(\034)614 5300 y Fo(;)i(ik)s Fp(\))p Fo(w)r Fp(\)\()p Fi(\000)p Fo(\034)5 b(;)15 b(k)s Fp(\))830 5505 y(=)1054 5443 y(1)p 936 5484 282 4 v 936 5571 a(\(2)p Fo(\031)s Fp(\))1106 5545 y Fk(1)p Fm(=)p Fk(2)1242 5381 y Fn(Z)1333 5407 y Fl(\000)p Fm(\034)1293 5587 y Fk(0)1447 5381 y Fn(Z)1537 5407 y Fm(\034)1568 5416 y Fh(1)1497 5587 y Fk(0)1623 5381 y Fn(Z)1714 5407 y Fk(+)p Fl(1)1673 5587 y(\0001)1843 5505 y Fp(\()p Fo(ik)1956 5519 y Fk(1)1997 5505 y Fp(\))2032 5467 y Fm(s)2065 5476 y Fh(0)2103 5505 y Fo(a)p Fp(\()p Fo(\034)2226 5519 y Fk(1)2286 5505 y Fi(\000)20 b Fo(\034)2417 5519 y Fk(2)2456 5505 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2705 5519 y Fk(1)2745 5505 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)2922 5519 y Fk(2)2962 5505 y Fo(;)15 b(k)3049 5519 y Fk(1)3089 5505 y Fp(\))1500 5706 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)25 b Fi(\000)20 b Fo(k)1902 5720 y Fk(1)1941 5706 y Fp(\)\))2011 5669 y Fm(s)2044 5678 y Fh(1)2084 5706 y Fp(\()p Fo(ik)2197 5720 y Fk(1)2237 5706 y Fp(\))2272 5669 y Fm(s)2305 5678 y Fh(2)2344 5706 y Fo(w)r Fp(\()p Fo(\034)2486 5720 y Fk(1)2546 5706 y Fi(\000)g Fo(\034)2677 5720 y Fk(2)2716 5706 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2965 5720 y Fk(1)3005 5706 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)3182 5720 y Fk(2)3222 5706 y Fo(;)15 b(k)3309 5720 y Fk(1)3349 5706 y Fp(\))p Fo(dk)3478 5720 y Fk(1)3519 5706 y Fo(d\034)3606 5720 y Fk(2)3645 5706 y Fo(d\034)3732 5720 y Fk(1)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 3764 100 a Fp(16)75 399 y(for)30 b(all)h Fo(\034)5 b(;)15 b(k)29 b Fi(2)c Fj(R)p Fp(.)40 b(But)31 b(w)m(e)g(ha)m(v)m(e)g(that)1168 603 y(\()p Fo(P)13 b Fp(\()p Fo(@)1362 565 y Fk(2)1357 625 y Fm(\034)1402 603 y Fo(;)i(ik)s Fp(\))p Fo(w)r Fp(\)\()p Fi(\000)p Fo(\034)5 b(;)15 b(k)s Fp(\))29 b(=)c Fo(P)13 b Fp(\()p Fo(@)2220 565 y Fk(2)2215 625 y Fm(\034)2260 603 y Fo(;)i(ik)s Fp(\)\()20 b(~)-65 b Fo(w)t Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\)\))75 807 y(for)37 b(all)h Fo(\034)5 b(;)15 b(k)41 b Fi(2)36 b Fj(R)p Fp(.)62 b(F)-8 b(rom)38 b(the)f(assumption)g Fg(\(H5\))g Fp(and)f(using)h(the)h(c)m(hange)g(of)g(v)-5 b(ariable)38 b Fo(\034)3325 821 y Fk(2)3401 807 y Fp(=)f Fi(\000)p Fo(u)3632 821 y Fk(2)3708 807 y Fp(and)75 920 y Fo(\034)115 934 y Fk(1)179 920 y Fp(=)25 b Fi(\000)p Fo(u)398 934 y Fk(1)438 920 y Fp(,)30 b(w)m(e)h(get)g(that)173 1076 y Fn(Z)264 1102 y Fl(\000)p Fm(\034)224 1282 y Fk(0)377 1076 y Fn(Z)468 1102 y Fm(\034)499 1111 y Fh(1)428 1282 y Fk(0)553 1076 y Fn(Z)644 1102 y Fk(+)p Fl(1)604 1282 y(\0001)774 1200 y Fp(\()p Fo(ik)887 1214 y Fk(1)927 1200 y Fp(\))962 1162 y Fm(s)995 1171 y Fh(0)1034 1200 y Fo(a)p Fp(\()p Fo(\034)1157 1214 y Fk(1)1217 1200 y Fi(\000)20 b Fo(\034)1348 1214 y Fk(2)1387 1200 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)1636 1214 y Fk(1)1675 1200 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)1852 1214 y Fk(2)1893 1200 y Fo(;)15 b(k)1980 1214 y Fk(1)2020 1200 y Fp(\))839 1401 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)25 b Fi(\000)20 b Fo(k)1241 1415 y Fk(1)1280 1401 y Fp(\)\))1350 1364 y Fm(s)1383 1373 y Fh(1)1422 1401 y Fp(\()p Fo(ik)1535 1415 y Fk(1)1576 1401 y Fp(\))1611 1364 y Fm(s)1644 1373 y Fh(2)1683 1401 y Fo(w)r Fp(\()p Fo(\034)1825 1415 y Fk(1)1885 1401 y Fi(\000)g Fo(\034)2016 1415 y Fk(2)2055 1401 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2304 1415 y Fk(1)2343 1401 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)2520 1415 y Fk(2)2561 1401 y Fo(;)15 b(k)2648 1415 y Fk(1)2688 1401 y Fp(\))p Fo(dk)2817 1415 y Fk(1)2857 1401 y Fo(d\034)2944 1415 y Fk(2)2984 1401 y Fo(d\034)3071 1415 y Fk(1)993 1606 y Fp(=)1089 1482 y Fn(Z)1179 1508 y Fm(\034)1139 1688 y Fk(0)1238 1482 y Fn(Z)1329 1508 y Fm(u)1370 1517 y Fh(1)1288 1688 y Fk(0)1423 1482 y Fn(Z)1514 1508 y Fk(+)p Fl(1)1474 1688 y(\0001)1644 1606 y Fp(\()p Fo(ik)1757 1620 y Fk(1)1797 1606 y Fp(\))1832 1568 y Fm(s)1865 1577 y Fh(0)1904 1606 y Fo(a)p Fp(\()p Fo(u)2039 1620 y Fk(1)2099 1606 y Fi(\000)20 b Fo(u)2242 1620 y Fk(2)2282 1606 y Fo(;)15 b(k)23 b Fi(\000)d Fo(k)2530 1620 y Fk(1)2570 1606 y Fp(\))f(~)-64 b Fo(w)s Fp(\()p Fo(u)2760 1620 y Fk(2)2800 1606 y Fo(;)15 b(k)2887 1620 y Fk(1)2927 1606 y Fp(\))1414 1808 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)k Fi(\000)c Fo(k)1815 1822 y Fk(1)1855 1808 y Fp(\)\))1925 1770 y Fm(s)1958 1779 y Fh(1)1997 1808 y Fp(\()p Fo(ik)2110 1822 y Fk(1)2150 1808 y Fp(\))2185 1770 y Fm(s)2218 1779 y Fh(2)2276 1808 y Fp(~)-64 b Fo(w)r Fp(\()p Fo(u)2411 1822 y Fk(1)2471 1808 y Fi(\000)20 b Fo(u)2614 1822 y Fk(2)2654 1808 y Fo(;)15 b(k)24 b Fi(\000)c Fo(k)2903 1822 y Fk(1)2942 1808 y Fp(\))f(~)-64 b Fo(w)s Fp(\()p Fo(u)3132 1822 y Fk(2)3172 1808 y Fo(;)15 b(k)3259 1822 y Fk(1)3299 1808 y Fp(\))p Fo(dk)3428 1822 y Fk(1)3469 1808 y Fo(du)3568 1822 y Fk(2)3607 1808 y Fo(du)3706 1822 y Fk(1)3746 1808 y Fo(:)75 2012 y Fp(for)31 b(all)h Fo(\034)5 b(;)15 b(k)31 b Fi(2)26 b Fj(R)p Fp(.)43 b(W)-8 b(e)32 b(deduce)f(that)51 b(~)-64 b Fo(w)s Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))32 b(satis\014es)g(the)f(equation)i(\(14\).)45 b(Moreo)m(v)m(er,)33 b(b)m(y)f(construction,)75 2125 y(w)m(e)f(also)g(ha)m(v)m(e)1278 2238 y(~)-64 b Fo(w)s Fp(\(0)p Fo(;)15 b(k)s Fp(\))27 b(=)e Fo(w)1720 2252 y Fk(0)1759 2238 y Fp(\()p Fo(k)s Fp(\))62 b Fo(;)76 b Fp(\()p Fo(@)2130 2200 y Fm(j)2125 2260 y(\034)2187 2238 y Fp(~)-63 b Fo(w)r Fp(\)\(0)p Fo(;)15 b(k)s Fp(\))28 b Fi(\021)c Fp(0)p Fo(;)75 2404 y Fp(for)31 b(all)h(1)27 b Fi(\024)g Fo(j)32 b Fi(\024)26 b Fp(2)p Fo(n)21 b Fi(\000)g Fp(1.)43 b(Finally)-8 b(,)34 b(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b Fi(7!)46 b Fp(~)-64 b Fo(w)s Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))32 b(satis\014es)g(the)f(problem)g (\(14\),)i(\(15\))g(for)e(all)h Fo(\034)5 b(;)15 b(k)30 b Fi(2)d Fj(R)p Fp(.)75 2517 y(F)-8 b(rom)31 b(the)f(unicit)m(y)h(of)g (the)g(solution)g(of)f(\(14\),)i(\(15\),)g(w)m(e)f(get)h(that)49 b(~)-63 b Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b(=)e Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))32 b(for)e(all)h Fo(\034)5 b(;)15 b(k)29 b Fi(2)c Fj(R)p Fp(.)56 b Fb(2)75 2854 y Fq(4)135 b(Cauc)l(h)l(y)57 b(problems)g(with)g (a)g(real)h(parameter)g(and)f(Laplace)g(trans-)277 3003 y(forms)75 3206 y Fp(Let)33 b Fo(P)13 b Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))p Fo(;)15 b(R)q Fp(\()p Fo(X)7 b Fp(\))32 b Fi(2)d Fj(C)p Fp([)p Fo(T)8 b(;)15 b(X)7 b Fp(])33 b(b)s(e)f(p)s(olynomials)h(with)g(deg)2261 3228 y Fm(T)2332 3206 y Fo(P)13 b Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))30 b(=)f Fo(n)g Fi(\025)g Fp(1)k(and)f Fo(s)3270 3220 y Fk(0)3309 3206 y Fo(;)15 b(s)3392 3220 y Fk(1)3431 3206 y Fo(;)g(s)3514 3220 y Fk(2)3583 3206 y Fi(\025)29 b Fp(0)k(b)s(e)75 3319 y(non)d(negativ)m(e)i(in)m(tegers.)42 b(In)30 b(this)g(section,)i(w)m(e)f(will)g(consider)f(the)g(follo)m (wing)i(problem)75 3599 y(\(29\))85 b Fo(R)q Fp(\()p Fo(ik)s Fp(\))p Fo(P)13 b Fp(\()p Fo(@)695 3613 y Fm(t)726 3599 y Fo(;)i(ik)s Fp(\))p Fo(X)7 b Fp(\()p Fo(t;)15 b(k)s Fp(\))28 b(=)1409 3538 y(1)p 1291 3578 282 4 v 1291 3666 a(\(2)p Fo(\031)s Fp(\))1461 3639 y Fk(1)p Fm(=)p Fk(2)1597 3476 y Fn(Z)1688 3502 y Fm(t)1648 3682 y Fk(0)1733 3476 y Fn(Z)1824 3502 y Fk(+)p Fl(1)1783 3682 y(\0001)1953 3599 y Fp(\()p Fo(ik)2066 3613 y Fk(1)2107 3599 y Fp(\))2142 3562 y Fm(s)2175 3571 y Fh(0)2214 3599 y Fo(A)p Fp(\()p Fo(t)20 b Fi(\000)g Fo(s;)15 b(k)23 b Fi(\000)d Fo(k)2752 3613 y Fk(1)2792 3599 y Fp(\))p Fo(X)7 b Fp(\()p Fo(s;)15 b(k)3074 3613 y Fk(1)3115 3599 y Fp(\))1753 3801 y(+)20 b(\()p Fo(i)p Fp(\()p Fo(k)k Fi(\000)c Fo(k)2154 3815 y Fk(1)2194 3801 y Fp(\)\))2264 3763 y Fm(s)2297 3772 y Fh(1)2336 3801 y Fp(\()p Fo(ik)2449 3815 y Fk(1)2490 3801 y Fp(\))2525 3763 y Fm(s)2558 3772 y Fh(2)2596 3801 y Fo(X)7 b Fp(\()p Fo(t)21 b Fi(\000)f Fo(s;)15 b(k)24 b Fi(\000)19 b Fo(k)3149 3815 y Fk(1)3189 3801 y Fp(\))p Fo(X)7 b Fp(\()p Fo(s;)15 b(k)3471 3815 y Fk(1)3512 3801 y Fp(\))p Fo(dk)3641 3815 y Fk(1)3682 3801 y Fo(ds)75 4005 y Fp(with)30 b(initial)i(conditions)75 4209 y(\(30\))1010 b Fo(X)7 b Fp(\(0)p Fo(;)15 b(k)s Fp(\))28 b(=)d Fo(w)1721 4223 y Fk(0)1760 4209 y Fp(\()p Fo(k)s Fp(\))62 b Fo(;)76 b Fp(\()p Fo(@)2131 4165 y Fm(j)2126 4234 y(t)2168 4209 y Fo(X)7 b Fp(\)\(0)p Fo(;)15 b(k)s Fp(\))28 b Fi(\021)d Fp(0)p Fo(;)75 4414 y Fp(for)31 b(all)h(1)27 b Fi(\024)f Fo(j)33 b Fi(\024)26 b Fo(n)20 b Fi(\000)h Fp(1)31 b(and)g(where)f Fo(A)p Fp(\()p Fo(t;)15 b(k)s Fp(\))28 b(=)e Fo(X)1823 4428 y Fm(a)1865 4414 y Fp(\()p Fo(t;)15 b(k)s Fp(\))33 b(is)e(the)g(mo)s(di\014ed)f(Laplace) j(transform)d(of)h(order)g(1)75 4527 y(\(de\014ned)e(in)i(Prop)s (osition)f(3\))h(of)g(a)f(function)g Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))32 b(whic)m(h)e(satis\014es)h(the)g(assumption)e Fg(\(H5\))p Fp(.)216 4689 y(In)35 b(this)g(section,)j(w)m(e)e(mak)m(e)h (the)f(assumptions)e(that)i(the)g(p)s(olynomials)g Fo(P)13 b Fp(,)37 b Fo(R)q Fp(,)f(the)g(initial)h(condition)75 4802 y Fo(w)140 4816 y Fk(0)180 4802 y Fp(\()p Fo(k)s Fp(\))31 b(satisfy)g(the)f(conditions)h Fg(\(Hj\))p Fp(,)f(for)g(0)25 b Fi(\024)g Fo(j)31 b Fi(\024)25 b Fp(4.)41 b(Moreo)m(v)m(er)33 b(w)m(e)e(assume)f(that)1146 5006 y Fi(jj)p Fo(w)1261 5020 y Fk(0)1301 5006 y Fi(jj)1351 5037 y Fk(\()1388 5020 y(~)1378 5037 y Fm(\014)t(;\026)p Fk(+)p Fm(\026)1581 5046 y Fh(1)p Ff(;)p Fh(1)1666 5037 y Fk(\))1722 5006 y Fi(\024)25 b Fo(\016)65 b(;)76 b Fi(jj)p Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fi(jj)2377 5025 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2623 5006 y Fi(\024)25 b Fo(\016)n(;)75 5211 y Fp(where)30 b Fo(\016)f(>)c Fp(0)31 b(is)f(de\014ned)f(in)h(the)h(prop)s(osition)f(1.)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 3764 100 a Fp(17)75 399 y Fa(4.1)112 b(De\014nition)38 b(and)h(prop)s(erties)f(of)f(the)h(Laplace)h (transform)75 570 y Fp(In)22 b(this)i(subsection,)h(w)m(e)e(de\014ne)g (the)g(Laplace)i(transform)d(of)h(functions)g(that)h(b)s(elong)f(to)h (the)g(Banac)m(h)g(spaces)75 683 y(in)m(tro)s(duced)36 b(in)f(the)i(section)g(2)g(and)e(w)m(e)i(giv)m(e)g(some)g(relations)g (that)g(will)f(b)s(e)g(useful)f(in)h(the)h(sequel.)58 b(The)75 796 y(next)28 b(prop)s(osition)f(is)g(a)h(consequence)g(of)g (Theorem)f(58)h(in)f([3)q(])h(applied)f(to)h(the)g(acceleration)i(op)s (erator)e Fi(A)3761 810 y Fk(2)p Fm(;)p Fk(1)75 909 y Fp(in)m(tro)s(duced)i(b)m(y)g(J.)767 886 y(\023)758 909 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3645 y Fk(0)2400 3562 y Fo(a)p Fp(\()p Fo(\034)2523 3576 y Fk(1)2583 3562 y Fi(\000)20 b Fo(\034)2714 3576 y Fk(2)2753 3562 y Fo(;)15 b(k)24 b Fi(\000)19 b Fo(k)3001 3576 y Fk(1)3041 3562 y Fp(\))p Fo(w)r Fp(\()p Fo(\034)3218 3576 y Fk(2)3259 3562 y Fo(;)c(k)3346 3576 y Fk(1)3386 3562 y Fp(\))p Fo(d\034)3508 3576 y Fk(2)3548 3562 y Fo(d\034)3635 3576 y Fk(1)3674 3562 y Fo(d\034)75 3795 y Fd(for)33 b(al)5 b(l)33 b Fo(t)25 b Fi(2)g Fo(S)550 3813 y Fk(0)p Fm(;\031)r(=)p Fk(2)722 3795 y Fd(,)33 b Fo(k)s(;)15 b(k)920 3809 y Fk(1)986 3795 y Fi(2)24 b Fj(R)p Fd(.)75 3994 y Fp(In)31 b(the)g(follo)m(wing)i(prop)s (osition,)e(w)m(e)h(analyse)g(the)g(asymptotic)g(expansion)f(of)h(the)f (Laplace)i(transform)e(on)75 4107 y(a)g(sector)g(near)f(the)h(origin.) 41 b(This)30 b(prop)s(osition)g(follo)m(ws)h(from)f(Theorem)g(2.4)i(in) e([21)q(].)75 4284 y Fg(Prop)s(osition)36 b(4)46 b Fd(L)-5 b(et)46 b Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))49 b Fi(2)f Fo(E)1405 4303 y Fk(\()p Fm(\027;\014)s(;\026)p 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b(the)f(Cauchy)g(formula\))h(and)f(which)g(is)75 5047 y(c)-5 b(onver)g(gent)27 b(for)f(al)5 b(l)26 b Fi(j)p Fo(\034)10 b Fi(j)26 b Fo(<)f(\032)p Fd(.)40 b(Then,)27 b(the)f(L)-5 b(aplac)g(e)28 b(tr)-5 b(ansform)28 b Fo(X)2325 5061 y Fm(w)2382 5047 y Fp(\()p Fo(t;)15 b(k)s Fp(\))27 b Fd(satis\014es)g(the)f(fol)5 b(lowing)27 b(estimates.)75 5160 y(F)-7 b(or)34 b(al)5 b(l)33 b Fp(0)26 b Fo(<)e(\022)k(<)d(\031)s (=)p Fp(2)p Fd(,)33 b(ther)-5 b(e)34 b(exist)f Fo(C)q(;)15 b(K)q(;)g(R)27 b(>)e Fp(0)33 b Fd(such)g(that)75 5437 y Fp(\(35\))509 b(sup)745 5516 y Fm(k)r Fl(2)p Fc(R)880 5437 y Fp(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1207 5400 y Fm(\026)1270 5437 y Fp(exp\()p Fo(\014)5 b Fi(j)p Fo(k)s Fi(j)p Fp(\))p Fi(j)p Fo(X)1735 5451 y Fm(w)1794 5437 y Fp(\()p Fo(t;)15 b(k)s Fp(\))21 b Fi(\000)2099 5324 y Fm(N)7 b Fl(\000)p Fk(1)2110 5351 y Fn(X)2120 5548 y Fm(l)q Fk(=0)2268 5437 y Fo(w)2333 5452 y Fm(l)2359 5437 y Fp(\()p Fo(k)s Fp(\))2489 5376 y Fo(t)2522 5343 y Fm(l)p 2489 5416 59 4 v 2491 5500 a Fo(l)r Fp(!)2558 5437 y Fi(j)26 b(\024)f Fo(C)7 b(K)2861 5400 y Fm(N)2927 5437 y Fi(j)p Fo(t)p Fi(j)3010 5400 y Fm(N)3078 5437 y Fo(N)j Fp(!)75 5713 y Fd(for)33 b(al)5 b(l)33 b Fo(t)25 b Fi(2)g Fo(S)550 5728 y Fk(0)p Fm(;\022)r(;R)742 5713 y Fp(=)g Fi(f)p Fo(t)h Fi(2)f Fj(C)1094 5680 y Fl(\003)1133 5713 y Fo(=)p Fi(j)p Fp(arg)r(\()p Fo(t)p Fp(\))p Fi(j)h(\024)f Fo(\022)s(;)15 b Fi(j)p Fo(t)p Fi(j)25 b(\024)g Fo(R)q Fi(g)p Fd(,)33 b(al)5 b(l)33 b Fo(N)i Fi(\025)25 b Fp(1)p Fd(.)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 3764 100 a Fp(18)75 399 y Fa(4.2)112 b(F)-9 b(ormal)31 b(p)s(o)m(w)m(er)e(series)h(solutions)h(of)e(the)h (Cauc)m(h)m(y)g(problem)g(and)h(Gevrey)f(asymp-)330 515 y(totic)37 b(expansion)75 686 y Fp(In)25 b(the)h(next)g(prop)s (osition,)g(w)m(e)g(construct)g(an)g(actual)h(sectorial)h(holomorphic)d (solution)i(with)e(v)-5 b(alues)26 b(in)f(the)75 799 y(Banac)m(h)j(space)g Fo(E)703 818 y Fk(\()p Fm(\014)s(;\026)p Fk(\))895 799 y Fp(of)f(the)h(Cauc)m(h)m(y)g(problem)e(\(29\),)k (\(30\))f(and)e(w)m(e)h(analyse)g(it's)g(asymptotic)h(expansion)75 912 y(near)h(the)h(origin)g(in)f Fj(C)p Fp(.)216 1025 y(F)-8 b(rom)36 b(the)f(assumption)g Fg(\(H5\))p Fp(,)h(w)m(e)f(ha)m(v) m(e)i(the)e(follo)m(wing)i(expansion)e Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))34 b(=)3105 957 y Fn(P)3201 1052 y Fm(l)q Fl(\025)p Fk(0)3332 1025 y Fo(a)3380 1040 y Fm(l)3407 1025 y Fp(\()p Fo(k)s Fp(\))p Fo(\034)3577 992 y Fk(2)p Fm(l)3639 1025 y Fo(=)p Fp(\(2)p Fo(l)r Fp(\)!)75 1158 y(where)47 b(the)h(co)s(e\016cien)m(ts)i Fo(a)1055 1173 y Fm(l)1081 1158 y Fp(\()p Fo(k)s Fp(\),)j Fo(l)j Fi(\025)e Fp(0,)f(b)s(elong)48 b(to)h Fo(E)2114 1176 y Fk(\()p Fm(\014)s(;\026)p Fk(\))2278 1158 y Fp(.)93 b(In)47 b(the)h(follo)m(wing,)54 b(w)m(e)48 b(put)3491 1135 y(^)3467 1158 y Fo(A)p Fp(\()p Fo(t;)15 b(k)s Fp(\))56 b(=)75 1212 y Fn(P)171 1307 y Fm(l)q Fl(\025)p Fk(0)302 1280 y Fo(a)350 1295 y Fm(l)376 1280 y Fp(\()p Fo(k)s Fp(\))p Fo(t)529 1247 y Fm(l)556 1280 y Fo(=l)r Fp(!.)75 1470 y Fg(Prop)s(osition)36 b(5)46 b Fd(Ther)-5 b(e)33 b(exists)g(a)g(unique)f(formal)i(p)-5 b(ower)35 b(series)75 1706 y Fp(\(36\))1573 1683 y(^)1547 1706 y Fo(X)8 b Fp(\()p Fo(t;)15 b(k)s Fp(\))26 b(=)1945 1619 y Fn(X)1955 1817 y Fm(l)q Fl(\025)p Fk(0)2092 1706 y Fo(w)2157 1721 y Fm(l)2183 1706 y Fp(\()p Fo(k)s Fp(\))2313 1644 y Fo(t)2346 1611 y Fm(l)p 2313 1685 59 4 v 2315 1768 a Fo(l)r Fp(!)75 1978 y Fd(with)33 b(c)-5 b(o)g(e\016cients)34 b Fo(w)801 1993 y Fm(l)852 1978 y Fi(2)25 b Fo(E)1005 1996 y Fk(\()p Fm(\014)s(;\026)p Fk(\))1169 1978 y Fd(,)33 b(for)g(al)5 b(l)33 b Fo(l)27 b Fi(\025)e Fp(0)p Fd(,)33 b(which)h(is)e(a)h (solution)h(of)f(the)g(pr)-5 b(oblem)75 2253 y Fp(\(37\))85 b Fo(R)q Fp(\()p Fo(ik)s Fp(\))p Fo(P)13 b Fp(\()p Fo(@)695 2267 y Fm(t)726 2253 y Fo(;)i(ik)s Fp(\))909 2230 y(^)882 2253 y Fo(X)9 b Fp(\()p Fo(t;)15 b(k)s Fp(\))26 b(=)1409 2192 y(1)p 1291 2232 282 4 v 1291 2319 a(\(2)p Fo(\031)s Fp(\))1461 2293 y Fk(1)p Fm(=)p 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Fp(\(38\))1006 b Fo(X)7 b Fp(\(0)p Fo(;)15 b(k)s Fp(\))27 b(=)e Fo(w)1716 2840 y Fk(0)1756 2826 y Fp(\()p Fo(k)s Fp(\))66 b Fo(;)80 b Fp(\()p Fo(@)2135 2781 y Fm(j)2130 2850 y(t)2172 2826 y Fo(X)7 b Fp(\)\(0)p Fo(;)15 b(k)s Fp(\))28 b Fi(\021)d Fp(0)p Fo(:)75 3007 y Fd(Mor)-5 b(e)g(over,)37 b(ther)-5 b(e)37 b(exists)f(an)g(actual)h (solution)f Fo(t)31 b Fi(7!)g Fo(X)7 b Fp(\()p Fo(t;)15 b(k)s Fp(\))37 b Fd(of)f(\(29\),)h(\(30\),)g(which)g(is)f(holomorphic)j (with)75 3120 y(r)-5 b(esp)g(e)g(ct)38 b(to)e Fo(t)g Fd(on)g(the)h(se)-5 b(ctor)37 b Fo(S)1167 3139 y Fk(0)p Fm(;\031)r(=)p Fk(2)1375 3120 y Fd(with)g(values)f(in)g Fo(E)2033 3139 y Fk(\()p Fm(\014)s(;\026)p Fk(\))2233 3120 y Fd(satisfying)h(the)f(fol)5 b(lowing)37 b(pr)-5 b(op)g(erty:)52 b(F)-7 b(or)37 b(al)5 b(l)75 3233 y Fp(0)26 b Fo(<)f(\022)i(<)e(\031)s(=)p Fp(2)p Fd(,)33 b(ther)-5 b(e)34 b(exist)f Fo(R)q(;)15 b(C)q(;)g(K)33 b(>)25 b Fp(0)33 b Fd(such)f(that)75 3501 y Fp(\(39\))533 b(sup)770 3580 y Fm(k)r Fl(2)p Fc(R)905 3501 y Fp(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))1232 3464 y Fm(\026)1295 3501 y Fp(exp\()p Fo(\014)5 b Fi(j)p Fo(k)s Fi(j)p Fp(\))p Fi(j)p Fo(X)i Fp(\()p Fo(t;)15 b(k)s Fp(\))23 b Fi(\000)2074 3388 y Fm(N)7 b Fl(\000)p Fk(1)2085 3415 y Fn(X)2095 3613 y Fm(l)q Fk(=0)2243 3501 y Fo(w)2308 3516 y Fm(l)2334 3501 y Fp(\()p Fo(k)s Fp(\))2464 3440 y Fo(t)2497 3407 y Fm(l)p 2464 3480 59 4 v 2466 3564 a Fo(l)r Fp(!)2534 3501 y Fi(j)25 b(\024)g Fo(C)7 b(K)2836 3464 y Fm(N)2902 3501 y Fi(j)p Fo(t)p Fi(j)2985 3464 y Fm(N)3053 3501 y Fo(N)j Fp(!)75 3762 y Fd(for)33 b(al)5 b(l)33 b Fo(t)25 b Fi(2)g Fo(S)550 3777 y Fk(0)p Fm(;\022)r(;R)717 3762 y Fd(,)32 b(al)5 b(l)34 b Fo(N)h Fi(\025)25 b Fp(1)p Fd(.)75 3994 y Fg(Pro)s(of)35 b Fp(F)-8 b(rom)34 b(Prop)s(osition)g(2,) h(let)g Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))35 b(b)s(e)e(the)h(unique)f(solution)h(of)g(the)g(problem)f(\(14\),)k (\(15\))e(in)e(the)75 4107 y(space)e Fo(E)385 4126 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))604 4107 y Fp(.)42 b(F)-8 b(rom)32 b(Lemma)f(9,)g(the)g(function)g Fo(w)i Fp(satis\014es)e Fo(w)r Fp(\()p Fi(\000)p Fo(\034)5 b(;)15 b(k)s Fp(\))28 b(=)d Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\),)33 b(for)e(all)h Fo(\034)5 b(;)15 b(k)29 b Fi(2)d Fj(R)p Fp(.)41 b(W)-8 b(e)75 4220 y(deduce)30 b(that)h Fo(w)i Fp(has)d(a)h(con)m(v)m(ergen)m(t)i(T)-8 b(a)m(ylor)31 b(expansion)f(of)h(the)f(form)1493 4456 y Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b(=)1888 4369 y Fn(X)1898 4567 y Fm(l)q Fl(\025)p Fk(0)2034 4456 y Fo(w)2099 4471 y Fm(l)2125 4456 y Fp(\()p Fo(k)s Fp(\))2286 4394 y Fo(\034)2336 4361 y Fk(2)p Fm(l)p 2255 4435 171 4 v 2255 4518 a Fp(\(2)p Fo(l)r Fp(\)!)75 4733 y(where)35 b(the)h(co)s(e\016cien)m(ts)h Fo(w)1035 4748 y Fm(l)1096 4733 y Fi(2)c Fo(E)1257 4751 y Fk(\()p Fm(\014)s(;\026)p Fk(\))1421 4733 y Fp(,)38 b(for)d Fo(l)h Fi(\025)e Fp(0,)k(for)d(all)i Fo(\034)44 b Fi(2)33 b Fo(D)s Fp(\(0)p Fo(;)15 b(\032)p Fp(\).)58 b(W)-8 b(e)37 b(consider)f(no)m(w)f(the)h(formal)75 4862 y(series)31 b(\(36\))h(constructed)e(with)g(the)h(co)s(e\016cien)m (ts)h Fo(w)1890 4877 y Fm(l)1916 4862 y Fp(,)f Fo(l)c Fi(\025)e Fp(0.)41 b(A)31 b(direct)g(computation)g(sho)m(ws)f(that)3604 4839 y(^)3578 4862 y Fo(X)7 b Fp(\()p Fo(t;)15 b(k)s Fp(\))75 4975 y(is)39 b(the)g(unique)f(formal)h(solution)h(of)f (\(37\),)k(\(38\).)67 b(Let)40 b Fo(X)7 b Fp(\()p Fo(t;)15 b(k)s Fp(\))40 b(b)s(e)e(de\014ned)g(b)m(y)h(the)g(Laplace)h(transform) 75 5088 y Fo(X)150 5102 y Fm(w)207 5088 y Fp(\()p Fo(t;)15 b(k)s Fp(\).)41 b(F)-8 b(rom)29 b(Lemma)f(8,)i(w)m(e)f(kno)m(w)f(that)h Fo(@)1740 5055 y Fm(l)1735 5111 y(\034)1779 5088 y Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b Fi(2)d Fo(E)2235 5055 y Fc(R)2230 5120 y Fk(\()p Fm(\027;\014)s(;\026)p Fl(\000)p Fm(l)q Fk(\))2526 5088 y Fp(,)29 b(for)f(all)i(0)25 b Fi(\024)g Fo(l)i Fi(\024)e Fp(2)p Fo(n)p Fp(.)40 b(F)-8 b(rom)29 b(Lemma)75 5224 y(9,)i(w)m(e)f(get)i(that)f(\()p Fo(@)746 5191 y Fk(2)p Fm(l)q Fk(+1)741 5246 y Fm(\034)898 5224 y Fo(w)r Fp(\)\(0)p Fo(;)15 b(k)s Fp(\))28 b Fi(\021)d Fp(0,)30 b(for)g(0)c Fi(\024)f Fo(l)i Fi(\024)e Fo(n)19 b Fi(\000)h Fp(1.)41 b(W)-8 b(e)31 b(deduce)f(from)g(the)g(form)m(ula)h (\(32\))g(and)f(\(33\))75 5336 y(that)37 b(if)f Fo(w)r Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))38 b(satis\014es)f(the)g(equation) g(\(14\),)i(then)d Fo(X)2070 5350 y Fm(w)2127 5336 y Fp(\()p Fo(t;)15 b(k)s Fp(\))38 b(satis\014es)f(the)f(equation)h (\(29\).)60 b(F)-8 b(rom)37 b(the)75 5449 y(prop)s(osition)30 b(4,)g(w)m(e)h(get)g(that)f(the)g(estimates)i(\(39\))f(hold.)40 b(As)30 b(a)h(consequence,)g(w)m(e)f(get)h(also)g(that)g Fo(X)3604 5463 y Fm(w)3660 5449 y Fp(\()p Fo(t;)15 b(k)s Fp(\))75 5562 y(satis\014es)31 b(the)g(initial)h(conditions)g(\(30\).) 43 b(Finally)-8 b(,)33 b Fo(X)7 b Fp(\()p Fo(t;)15 b(k)s Fp(\))32 b(is)f(an)g(actual)h(solution)g(of)f(\(29\),)h(\(30\))h (satisfying)75 5675 y(\(39\).)3496 b Fb(2)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 3764 100 a Fp(19)75 399 y Fq(5)135 b(Cauc)l(h)l(y)45 b(problems)g(in)g(holomorphic)g(functions)g(spaces)75 601 y Fp(In)30 b(the)g(follo)m(wing)i(lemma,)f(w)m(e)g(recall)h(some)f (prop)s(erties)e(of)i(the)f(in)m(v)m(erse)i(F)-8 b(ourier)31 b(transform)75 789 y Fg(Lemma)k(10)46 b Fd(L)-5 b(et)30 b Fo(f)35 b Fi(2)25 b Fo(E)983 808 y Fk(\()p Fm(\014)s(;\026)p Fk(\))1177 789 y Fd(with)32 b Fo(\014)e(>)25 b Fp(0)p Fd(,)31 b Fo(\026)25 b(>)g Fp(1)p Fd(.)42 b(The)31 b(inverse)f(F)-7 b(ourier)31 b(tr)-5 b(ansform)33 b(of)e Fo(f)40 b Fd(is)30 b(de\014ne)-5 b(d)32 b(by)1105 1061 y Fi(F)1179 1024 y Fl(\000)p Fk(1)1273 1061 y Fp(\()p Fo(f)10 b Fp(\)\()p Fo(x)p Fp(\))26 b(=)1770 1000 y(1)p 1652 1040 282 4 v 1652 1128 a(\(2)p Fo(\031)s Fp(\))1822 1101 y Fk(1)p Fm(=)p Fk(2)1958 938 y Fn(Z)2049 964 y Fk(+)p Fl(1)2009 1144 y(\0001)2194 1061 y Fo(f)10 b Fp(\()p Fo(k)s Fp(\))15 b(exp\()p Fo(ixk)s Fp(\))p Fo(dk)75 1326 y Fd(for)33 b(al)5 b(l)33 b Fo(x)26 b Fi(2)e Fj(R)p Fd(.)42 b(The)33 b(function)f Fi(F)1266 1293 y Fl(\000)p Fk(1)1361 1326 y Fp(\()p Fo(f)10 b Fp(\))33 b Fd(extends)g(to)g(an)g(analytic)h (function)f(on)g(the)g(strip)1424 1530 y Fo(H)1500 1545 y Fm(\014)1572 1530 y Fp(=)25 b Fi(f)p Fo(z)30 b Fi(2)24 b Fj(C)p Fo(=)p Fi(j)p Fp(Im\()p Fo(z)t Fp(\))p Fi(j)j Fo(<)e(\014)5 b Fi(g)p Fo(:)75 1735 y Fd(L)-5 b(et)33 b Fo(\036)p Fp(\()p Fo(k)s Fp(\))26 b(=)f Fo(ik)s(f)10 b Fp(\()p Fo(k)s Fp(\))26 b Fi(2)f Fo(E)963 1753 y Fk(\()p Fm(\014)s(;\026)p Fl(\000)p Fk(1\))1217 1735 y Fd(.)42 b(Then,)33 b(we)g(have)75 1950 y Fp(\(40\))1215 b Fo(@)1498 1964 y Fm(z)1538 1950 y Fi(F)1612 1912 y Fl(\000)p Fk(1)1706 1950 y Fp(\()p Fo(f)10 b Fp(\)\()p Fo(z)t Fp(\))26 b(=)f Fi(F)2143 1912 y Fl(\000)p Fk(1)2238 1950 y Fp(\()p Fo(\036)p Fp(\)\()p Fo(z)t Fp(\))75 2154 y Fd(for)k(al)5 b(l)30 b Fo(z)f Fi(2)c Fo(H)576 2169 y Fm(\014)623 2154 y Fd(.)40 b(L)-5 b(et)29 b Fo(g)g Fi(2)c Fo(E)1069 2172 y Fk(\()p Fm(\014)s(;\026)p Fk(\))1262 2154 y Fd(and)k(let)g Fo( )s Fp(\()p Fo(k)s Fp(\))e(=)1970 2118 y Fk(1)p 1873 2133 230 4 v 1873 2194 a(\(2)p Fm(\031)r Fk(\))2005 2174 y Fh(1)p Ff(=)p Fh(2)2128 2081 y Fn(R)2188 2107 y Fk(+)p Fl(1)2171 2186 y(\0001)2333 2154 y Fo(f)10 b Fp(\()p Fo(k)15 b Fi(\000)d Fo(k)2615 2168 y Fk(1)2654 2154 y Fp(\))p Fo(g)s Fp(\()p Fo(k)2817 2168 y Fk(1)2858 2154 y Fp(\))p Fo(dk)2987 2168 y Fk(1)3027 2154 y Fd(,)30 b(for)f(al)5 b(l)29 b Fo(k)g Fi(2)c Fj(R)p Fd(.)40 b(F)-7 b(r)i(om)75 2282 y(L)g(emma)34 b(4,)f(we)f(know)i(that)g Fo( )28 b Fi(2)d Fo(E)1297 2300 y Fk(\()p Fm(\014)s(;\026)p Fk(\))1461 2282 y Fd(.)42 b(Then,)33 b(we)g(have)75 2497 y Fp(\(41\))1053 b Fi(F)1362 2459 y Fl(\000)p Fk(1)1457 2497 y Fp(\()p Fo(f)10 b Fp(\)\()p Fo(z)t Fp(\))p Fi(F)1772 2459 y Fl(\000)p Fk(1)1867 2497 y Fp(\()p Fo(g)s Fp(\)\()p Fo(z)t Fp(\))28 b(=)d Fi(F)2297 2459 y Fl(\000)p Fk(1)2391 2497 y Fp(\()p Fo( )s Fp(\)\()p Fo(z)t Fp(\))75 2701 y Fd(for)33 b(al)5 b(l)33 b Fo(z)d Fi(2)25 b Fo(H)584 2716 y Fm(\014)630 2701 y Fd(.)75 2938 y Fg(Pro)s(of)34 b Fp(The)f(fact)i(that)f Fi(F)1009 2905 y Fl(\000)p Fk(1)1103 2938 y Fp(\()p Fo(f)10 b Fp(\))34 b(extends)f(to)h(an)g(analytic)h (function)e(on)g(the)h(strip)f Fo(H)3129 2953 y Fm(\014)3209 2938 y Fp(follo)m(ws)h(from)f(the)75 3051 y(next)e(inequalit)m(y)-8 b(.)42 b(There)30 b(exists)h Fo(C)h(>)25 b Fp(0)30 b(suc)m(h)g(that) 1071 3292 y Fi(j)p Fo(f)10 b Fp(\()p Fo(k)s Fp(\))p Fi(jj)15 b Fp(exp)q(\()p Fo(iz)t(k)s Fp(\))p Fi(j)27 b(\024)1983 3231 y Fo(C)p 1831 3271 375 4 v 1831 3355 a Fp(\(1)21 b(+)f Fi(j)p Fo(k)s Fi(j)p Fp(\))2158 3328 y Fm(\026)2231 3292 y Fp(exp\(\()p Fo(\014)2496 3255 y Fl(0)2540 3292 y Fi(\000)g Fo(\014)5 b Fp(\))p Fi(j)p Fo(k)s Fi(j)p Fp(\))75 3547 y(for)32 b(all)h Fo(k)e Fi(2)c Fj(R)p Fp(,)32 b Fo(z)h Fi(2)27 b Fo(H)872 3562 y Fm(\014)915 3543 y Fe(0)941 3547 y Fp(,)33 b(with)f Fo(\014)1264 3514 y Fl(0)1315 3547 y Fo(<)c(\014)5 b Fp(.)45 b(The)32 b(relations)h (\(40\),)h(\(41\))g(are)e(classical)i(and)d(can)i(b)s(e)e(found)g(for) 75 3660 y(instance)g(in)f([22)q(].)3055 b Fb(2)75 3822 y Fp(W)-8 b(e)32 b(can)e(no)m(w)h(state)g(the)g(main)f(result)g(of)h (the)g(pap)s(er.)75 4035 y Fg(Theorem)k(1)46 b Fd(L)-5 b(et)44 b Fo(P)13 b Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))p Fo(;)15 b(R)q Fp(\()p Fo(X)7 b Fp(\))49 b Fi(2)c Fj(C)p Fp([)p Fo(T)8 b(;)15 b(X)7 b Fp(])45 b Fd(b)-5 b(e)44 b(p)-5 b(olynomials)47 b(with)e Fp(deg)2846 4057 y Fm(T)2916 4035 y Fo(P)13 b Fp(\()p Fo(T)8 b(;)15 b(X)7 b Fp(\))47 b(=)f Fo(n)f Fi(\025)h Fp(1)e Fd(and)75 4148 y Fo(s)118 4162 y Fk(0)157 4148 y Fo(;)15 b(s)240 4162 y Fk(1)280 4148 y Fo(;)g(s)363 4162 y Fk(2)427 4148 y Fi(\025)25 b Fp(0)33 b Fd(b)-5 b(e)33 b(non)g(ne)-5 b(gative)33 b(inte)-5 b(gers.)42 b(We)32 b(c)-5 b(onsider)34 b(the)f(fol)5 b(lowing)34 b(Cauchy)f(pr)-5 b(oblem)75 4404 y Fp(\(42\))231 b Fo(R)q Fp(\()p Fo(@)619 4418 y Fm(z)659 4404 y Fp(\))p Fo(P)13 b Fp(\()p Fo(@)848 4418 y Fm(t)879 4404 y Fo(;)i(@)967 4418 y Fm(z)1007 4404 y Fp(\))p Fo(u)p Fp(\()p Fo(t;)g(z)t Fp(\))27 b(=)1406 4280 y Fn(Z)1497 4306 y Fm(t)1456 4486 y Fk(0)1542 4404 y Fo(b)p Fp(\()p Fo(t)20 b Fi(\000)g Fo(s;)15 b(z)t Fp(\))p Fo(@)1977 4366 y Fm(s)2010 4375 y Fh(0)1972 4426 y Fm(z)2049 4404 y Fo(u)p Fp(\()p Fo(s;)g(z)t Fp(\))p Fo(ds)21 b Fp(+)2502 4280 y Fn(Z)2593 4306 y Fm(t)2553 4486 y Fk(0)2638 4404 y Fo(@)2691 4366 y Fm(s)2724 4375 y Fh(1)2686 4426 y Fm(z)2763 4404 y Fo(u)p Fp(\()p Fo(t)f Fi(\000)g Fo(s;)15 b(z)t Fp(\))p Fo(@)3211 4366 y Fm(s)3244 4375 y Fh(2)3206 4426 y Fm(z)3283 4404 y Fo(u)p Fp(\()p Fo(s;)g(z)t Fp(\))p Fo(ds)75 4649 y Fd(with)33 b(initial)h(c)-5 b(onditions)75 4853 y Fp(\(43\))1049 b Fo(u)p Fp(\(0)p Fo(;)15 b(z)t Fp(\))27 b(=)e Fo(u)1712 4867 y Fk(0)1751 4853 y Fp(\()p Fo(z)t Fp(\))66 b Fo(;)81 b Fp(\()p Fo(@)2127 4809 y Fm(j)2122 4877 y(t)2164 4853 y Fo(u)p Fp(\)\(0)p Fo(;)15 b(z)t Fp(\))27 b Fi(\021)e Fp(0)p Fo(;)75 5057 y Fd(for)33 b(al)5 b(l)33 b Fp(1)26 b Fi(\024)f Fo(j)31 b Fi(\024)25 b Fo(n)20 b Fi(\000)f Fp(1)p Fd(.)75 5170 y Fg(1\))32 b Fd(Assume)h(that)h(the)f(c)-5 b(onditions)34 b Fg(\(Hj\))p Fd(,)e Fp(0)25 b Fi(\024)g Fo(j)31 b Fi(\024)25 b Fp(3)p Fd(,)33 b(hold)h(for)f(the)g(p)-5 b(olynomials)36 b Fo(P)13 b Fd(,)33 b Fo(R)q Fd(.)75 5283 y Fg(2\))44 b Fd(L)-5 b(et)44 b Fo(w)445 5297 y Fk(0)531 5283 y Fi(2)i Fo(E)705 5314 y Fk(\()742 5296 y(~)732 5314 y Fm(\014)s(;\026)p Fk(+)p Fm(\026)934 5323 y Fh(1)p Ff(;)p Fh(1)1019 5314 y Fk(\))1050 5283 y Fd(,)h(wher)-5 b(e)1406 5259 y Fp(~)1393 5283 y Fo(\014)52 b(>)46 b Fp(0)p Fd(,)h Fo(\026)f(>)g Fp(2)p Fo(n)28 b Fp(+)g Fo(\033)k Fp(+)c(max)q(\(1)h(+)f Fo(s)2782 5297 y Fk(0)2821 5283 y Fo(;)15 b Fp(1)30 b(+)e Fo(s)3078 5297 y Fk(1)3117 5283 y Fo(;)15 b Fp(1)29 b(+)g Fo(s)3374 5297 y Fk(2)3413 5283 y Fp(\))p Fd(.)76 b(We)44 b(put)75 5417 y Fo(u)127 5431 y Fk(0)166 5417 y Fp(\()p Fo(z)t Fp(\))27 b(=)e Fi(F)479 5384 y Fl(\000)p Fk(1)573 5417 y Fp(\()p Fo(w)673 5431 y Fk(0)713 5417 y Fp(\)\()p Fo(z)t Fp(\))p Fd(,)34 b(which)g(is)e(a)h(holomorphic)j(function)d(on)g(the)g(strip)g Fo(H)2812 5430 y Fk(~)2802 5448 y Fm(\014)2849 5417 y Fd(.)75 5530 y Fg(3\))f Fd(L)-5 b(et)33 b Fo(M)445 5544 y Fk(0)509 5530 y Fo(>)25 b Fp(0)33 b Fd(b)-5 b(e)32 b(a)h(c)-5 b(onstant)34 b(and)f Fo(')f Fd(b)-5 b(e)32 b(the)h(function)f(de\014ne)-5 b(d)33 b(in)g(Se)-5 b(ction)32 b(2.)42 b(We)32 b(cho)-5 b(ose)34 b Fo(\032;)15 b(')p Fp(\(0\))27 b Fo(>)e Fp(0)75 5643 y Fd(in)30 b(such)f(a)h(way)h(that)g Fo(\014)f Fp(=)1017 5619 y(~)1004 5643 y Fo(\014)19 b Fi(\000)14 b Fo(M)1247 5657 y Fk(0)1286 5643 y Fo(\032)g Fi(\000)g Fo(')p Fp(\(0\))26 b Fo(>)f Fp(0)p Fd(.)41 b(L)-5 b(et)30 b Fo(\027)h(>)25 b Fp(0)30 b Fd(and)g(c)-5 b(onsider)31 b(a)f(function)g Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))27 b Fi(2)e Fo(E)3636 5661 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))p eop end %%Page: 20 20 TeXDict begin 20 19 bop 3764 100 a Fp(20)75 399 y Fd(which)43 b(satis\014es)g(the)g(assumption)h Fg(\(H5\))p Fd(.)69 b(In)42 b(p)-5 b(articular,)47 b(we)42 b(have)h(the)f(T)-7 b(aylor)44 b(exp)-5 b(ansion)44 b Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))44 b(=)75 443 y Fn(P)171 538 y Fm(l)q Fl(\025)p Fk(0)302 511 y Fo(a)350 526 y Fm(l)376 511 y Fp(\()p Fo(k)s Fp(\))p Fo(\034)546 478 y Fk(2)p Fm(l)609 511 y Fo(=)p Fp(\(2)p Fo(l)r Fp(\)!)34 b Fd(wher)-5 b(e)34 b(the)f(c)-5 b(o)g(e\016cients)33 b Fo(a)1776 526 y Fm(l)1802 511 y Fp(\()p Fo(k)s Fp(\))p Fd(,)g Fo(l)28 b Fi(\025)d Fp(0)p Fd(,)32 b(b)-5 b(elong)34 b(to)f Fo(E)2690 530 y Fk(\()p Fm(\014)s(;\026)p Fk(\))2854 511 y Fd(.)42 b(We)32 b(put)1570 719 y Fp(^)1573 743 y Fo(b)p Fp(\()p Fo(t;)15 b(z)t Fp(\))27 b(=)1924 656 y Fn(X)1933 854 y Fm(l)q Fl(\025)p Fk(0)2070 743 y Fo(b)2109 758 y Fm(l)2135 743 y Fp(\()p Fo(z)t Fp(\))2261 681 y Fo(t)2294 648 y Fm(l)p 2261 722 59 4 v 2263 805 a Fo(l)r Fp(!)2331 743 y Fo(;)75 1003 y Fd(wher)-5 b(e)35 b Fo(b)372 1018 y Fm(l)398 1003 y Fp(\()p Fo(z)t Fp(\))30 b(=)e Fi(F)717 970 y Fl(\000)p Fk(1)811 1003 y Fp(\()p Fo(a)894 1018 y Fm(l)921 1003 y Fp(\)\()p Fo(z)t Fp(\))p Fd(,)36 b Fo(l)30 b Fi(\025)e Fp(0)p Fd(,)35 b(ar)-5 b(e)35 b(holomorphic)j (functions)c(on)h(the)g(strip)g Fo(H)3046 1018 y Fm(\014)3093 1003 y Fd(.)47 b(L)-5 b(et)34 b Fo(A)p Fp(\()p Fo(t;)15 b(k)s Fp(\))36 b Fd(b)-5 b(e)34 b(the)75 1116 y(mo)-5 b(di\014e)g(d)35 b(L)-5 b(aplac)g(e)34 b(tr)-5 b(ansform)35 b(of)e(or)-5 b(der)35 b(1)e(of)f Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fd(.)44 b(We)32 b(put)h Fo(b)p Fp(\()p Fo(t;)15 b(z)t Fp(\))26 b(=)f Fi(F)2781 1083 y Fl(\000)p Fk(1)2876 1116 y Fp(\()p Fo(k)k Fi(7!)c Fo(A)p Fp(\()p Fo(t;)15 b(k)s Fp(\)\)\()p Fo(z)t Fp(\))p Fd(.)75 1229 y Fg(4\))32 b Fd(Assume)h(that)1142 1341 y Fi(jj)p Fo(w)1257 1355 y Fk(0)1297 1341 y Fi(jj)1347 1372 y Fk(\()1384 1355 y(~)1374 1372 y Fm(\014)t(;\026)p Fk(+)p Fm(\026)1577 1381 y Fh(1)p Ff(;)p Fh(1)1661 1372 y Fk(\))1718 1341 y Fi(\024)25 b Fo(\016)69 b(;)80 b Fi(jj)p Fo(a)p Fp(\()p Fo(\034)5 b(;)15 b(k)s Fp(\))p Fi(jj)2381 1360 y Fk(\()p Fm(\027;\014)s(;\026)p Fk(\))2628 1341 y Fi(\024)25 b Fo(\016)n(;)75 1485 y Fd(wher)-5 b(e)34 b Fo(\016)29 b(>)c Fp(0)33 b Fd(is)f(de\014ne)-5 b(d)34 b(in)f(the)g(pr)-5 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4281 y Fm(p)1393 4314 y Fd(-r)-5 b(e)g(gularity)31 b(for)g(hyp)-5 b(erb)g(olic)32 b(inte)-5 b(gr)g(o)g(di\013er)g(ential)32 b(e)-5 b(quations.)29 b Fp(V)-8 b(olterra)262 4427 y(equations)23 b(and)f(applications)i(\(Arlington,)h(TX,)d(1996\),)27 b(67{82,)f(Stabilit)m(y)e(Con)m(trol)g(Theory)e(Metho)s(ds)262 4540 y(Appl.,)30 b(10,)i(Gordon)e(and)g(Breac)m(h,)h(Amsterdam,)g (2000.)75 4727 y([14])47 b(H.)d(Engler,)j Fd(Glob)-5 b(al)47 b(smo)-5 b(oth)47 b(solutions)f(for)g(a)f(class)h(of)f(p)-5 b(ar)g(ab)g(olic)48 b(inte)-5 b(gr)g(o)g(di\013er)g(ential)48 b(e)-5 b(quations.)262 4840 y Fp(T)d(rans.)30 b(Amer.)g(Math.)i(So)s (c.)e(348)i(\(1996\),)h(no.)d(1,)h(267{290.)75 5027 y([15])47 b(Z.)35 b(Gruji)m(\023)-43 b(c,)38 b(I.)e(Kuk)-5 b(a)m(vica,)38 b Fd(Sp)-5 b(ac)g(e)39 b(analyticity)f(for)g(the)g(Navier-Stokes)g(and) g(r)-5 b(elate)g(d)39 b(e)-5 b(quations)39 b(with)262 5140 y(initial)33 b(data)h(in)f Fo(L)905 5107 y Fm(p)944 5140 y Fd(.)d Fp(J.)h(F)-8 b(unct.)31 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