Content-Type: multipart/mixed; boundary="-------------0909020500295" This is a multi-part message in MIME format. ---------------0909020500295 Content-Type: text/plain; name="09-151.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="09-151.keywords" Cauchy problem, partial integro-differential equation, Gevrey estimates, singular perturbation ---------------0909020500295 Content-Type: application/postscript; name="irrspconv2009.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="irrspconv2009.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.96.1 Copyright 2007 Radical Eye Software %%Title: irrspconv2009.dvi %%CreationDate: Wed Sep 2 11:44:48 2009 %%Pages: 22 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR17 CMR12 CMBX10 CMR10 CMSY10 CMBX12 CMMI10 CMR8 %%+ CMMI8 CMEX10 CMR6 CMSY8 MSBM10 CMTI10 LASY10 CMSY6 CMMI6 MSBM7 %%+ CMMI12 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -D600 -o irrspconv2009.ps irrspconv2009.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2009.09.02:1143 %%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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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 14 /delta put dup 26 /rho put dup 59 /comma put dup 75 /K put dup 100 /d put dup 102 /f put dup 105 /i put dup 114 /r put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 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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: 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: 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 54 /six put dup 55 /seven put dup 58 /colon 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 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 83 /S put dup 84 /T put dup 85 /U 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 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 readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 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 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 72 /H put dup 75 /K 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 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 117 /u put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict 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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. 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 15 /bullet put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity 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 69 /E put dup 76 /L put dup 83 /S put dup 84 /T put dup 91 /union put dup 92 /intersection 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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0000000000000000000000000000000000000000000000000000000000000000 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 2 /multiply put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 69 /E put dup 83 /S put dup 84 /T put dup 91 /union put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-20 -250 1193 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 0 /parenleftbig put dup 1 /parenrightbig put dup 12 /vextendsingle put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 80 /summationtext 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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5341FAA302A6A01926D2109B29974548DD348D0F3BD795C67B26847B9F6A0025 882726BC684888FD785284D2D02FF7374970CDD16EF314AC51623033FBF79A11 CD4096AFFE9EF7117A8D3452ABF158A8348CB904DF23A07552C7FE142353D75B D2B3F189407A372A4D37A2F99A48E9FFA805E96F25EA4378106E6483C1D7A5E2 F4F0E6B6536AA1BCB48C1C475FE7E3CFB6175BAAC0933111E388E1BD71AF8446 AA12A8A98DF130FE1804087C322D97A80054640D092BCCB7E3EA5EED96BD6FB2 B8C98EA7D8AD3227466AA9645C18289A909727892CAEED5901625DDA45A741F0 31F9EBEA99B4BA6C1BFBDEFA44B5816DCB41C2604F9889935CE7F814C87DF795 2479338D1BF0D91113CE823CE3D63D0EEB04094F6ED50F656477B8F5B1A1C0A3 AA14703ABDE6D99BFD26CAD307331EFD6103FE9F2D7CFBB9564039DE85851D1C 81ADE44F7B99715E0C41B5E491AFBC4A5191616FCC5158356B3744109A2ABF3B F6A5B0F0F0D83572D49C26B7FA6988E16338E58DAA0617EE6A59844349537233 DBBDEF7FF4528094D3A257097D4145D6B75811FD28ED610DFA65F371DB268E99 CCFDEF5ADE4501B99A7C18BEEF24580C0BC45AA0CF7031385CC5460E74084F4F 1C461284ECF6208E1EC25C5947F276274896FB20536ABD8CA5F6A7FDC67E9247 1B305AEC8C778C71791EB1F85A8EBC7AFA7F9A211BD0C23A0F62604D66CA180C 9CA5A658B4E745CD59690E8DD0B9126892D78924C0869037F2511B989841C3F5 2AE5480DA1AD55945BAC3F1F96183284F7929B4B2A1474A6196F947684F4852A 71C22C651C72D3021B8893BE0F33EA13623966C839E1E64B2056A9BAA6814071 24C4C67745EC3F6D2CAC040C2D2C7AC3DE1DA5918F641FCFE472308F1BB5DFBB B9ADAA52C373CFC9D321A5BFF13DAFE00C8452022C656BE866A26F8F6F632A00 0E29559306970078EAC2142A4C270F2C8C82A068B6AC438A44E31A1560D04F38 1380EA7CE96A100F6AA617566C391638EAEC2392421A8619700F8D2525BB0AB9 72FE9BC749392850B574C0E3FCC196D56FCD312C901EDBDDF13A58A7FD341963 DDC08A873ECDC593C6B79488201720E6B7CF63AC2B78CB959D793D4F7A1C0780 39E4985CD41921B21334CCE6BDAC38B9C16A9FC4BAF40909EECFC1BF4C764023 C00892020EB7C18174289104600576859F4A73D792F922B21EE7BC436B341DE2 5AC6B398F7EB5702D4D82F8267E0463DFFF9AD29E5A0547D5101E9D6ECB59F37 6860952FB748D15B681070BA9D08B9436B65B8042C6A375D86F448769DB96E5E 3AA4782A9EF626951ADD2B4FA632E9236696C5BEF549027DF050978E354FCB84 0C5F4ACF86206AD2E7046CB392102A83 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 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 59 /comma put dup 61 /slash put dup 68 /D put dup 71 /G put dup 75 /K put dup 78 /N put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 88 /X put dup 100 /d 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 112 /p put dup 113 /q put dup 114 /r put dup 116 /t put dup 120 /x put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def 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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 61 /equal put readonly def /FontBBox{-36 -250 1070 750}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 13 /gamma put dup 14 /delta put dup 15 /epsilon1 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 28 /tau put dup 31 /chi put dup 39 /phi1 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 70 /F put dup 71 /G put dup 73 /I put dup 75 /K put dup 77 /M put dup 78 /N 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 97 /a put dup 98 /b put dup 99 /c put dup 100 /d 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 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 readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 990CD249827D9648574955F61BAAA11263A91B6C3D47A5190165B0C25ABF6D3E 6EC187E4B05182126BB0D0323D943170B795255260F9FD25F2248D04F45DFBFB 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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 16 /dotlessi put dup 18 /grave 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 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 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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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. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 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 19 /acute put dup 39 /quoteright 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 57 /nine put dup 64 /at put dup 65 /A put dup 67 /C put dup 70 /F put dup 76 /L put dup 77 /M put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V 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 104 /h put dup 105 /i 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 118 /v put dup 120 /x put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec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%!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. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 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 45 /hyphen put dup 79 /O put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l 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 119 /w put dup 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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Fq(0)p Fp(;)g(\017)p Fq(\),)35 b(0)27 b Fj(\024)g Fp(j)33 b Fj(\024)27 b Fp(S)f Fj(\000)21 b Fq(1)32 b(\(Theorem)g(2\).)45 b(These)75 2220 y(1)p Fj(\000)p Fq(sums)32 b Fp(Y)475 2234 y Fn(i)503 2220 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\),)36 b Fp(i)31 b Fj(2)f Fp(I)40 b Fq(where)33 b Fp(I)41 b Fq(is)33 b(some)h(\014nite)f (subset)g(of)h Fi(N)p Fq(,)g(are)g(solutions)f(of)h(\(25\))h(and)e(are) h(holo-)75 2333 y(morphic)28 b(and)f(b)s(ounded)f(functions)i(on)g(pro) s(ducts)e Fp(T)1911 2347 y Fn(i)1955 2333 y Fj(\002)16 b Fp(D)s Fq(\(0)p Fp(;)f(\032)p Fq(\))i Fj(\002)e(E)2473 2347 y Fn(i)2501 2333 y Fq(,)29 b(where)f Fj(E)2864 2347 y Fn(i)2920 2333 y Fq(are)g(sectors)h(whose)f(union)75 2446 y(form)k(a)g(go)s(o)s(d)g(co)m(v)m(ering)i(of)e(a)h(neigh)m(b)s (orho)s(o)s(d)d(of)i(the)h(origin)f(in)g Fi(C)g Fq(and)f Fp(T)2666 2460 y Fn(i)2726 2446 y Fq(are)i(w)m(ell)g(c)m(hosen)f (sectors.)47 b(The)75 2559 y(main)35 b(to)s(ol)h(in)f(the)g(pro)s(of)f 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Fp(i)e Fj(2)f Fp(I)7 b Fq(,)28 b(satisfy)g(a)g(cohomological)75 3123 y(criterion)37 b(for)f(1)p Fj(\000)p Fq(summabilit)m(y)h(in)f(the) h(p)s(erturbation)e(parameter)i Fp(\017)f Fq(of)g(formal)h(series)g (with)f(co)s(e\016cien)m(ts)75 3236 y(in)43 b(a)g(Banac)m(h)i(space)f (\(Theorem)f(CC\).)g(This)f(\014nally)h(yields)h(the)f(existence)i(and) d(unicit)m(y)i(of)g(a)f(formal)75 3349 y(series)342 3326 y(^)316 3349 y Fp(X)7 b Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))30 b(with)d(resp)s(ect)g(to)i Fp(\017)p Fq(,)f(with)g(co)s (e\016cien)m(ts)h(b)s(eing)e(holomorphic)h(and)f(b)s(ounded)e (functions)j(on)75 3462 y Fj(T)38 b(\002)16 b Fp(D)s Fq(\(0)p Fp(;)f(\032)p Fq(\))29 b(whic)m(h)f(is)h(solution)f(of)h (\(1\).)41 b(Moreo)m(v)m(er,)31 b(the)d(1)p Fj(\000)p Fq(sums)f(of)2578 3439 y(^)2552 3462 y Fp(X)35 b Fq(with)28 b(resp)s(ect)g(to)h Fp(\017)f Fq(on)g Fj(E)3522 3476 y Fn(i)3579 3462 y Fq(are)g(the)75 3575 y(functions)i Fp(\017)25 b Fj(7!)g Fp(X)720 3589 y Fn(i)749 3575 y Fq(\()p Fp(\017;)15 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Fq(\(2\))184 b Fp(@)427 1414 y Fn(S)422 1474 y(x)478 1452 y Fp(u)p Fq(\()p Fp(t;)15 b(x)p Fq(\))26 b(=)937 1366 y Fm(X)847 1567 y Fo(\()p Fn(k)911 1576 y Fl(0)946 1567 y Fn(;k)1003 1576 y Fl(1)1037 1567 y Fo(\))p Fk(2S)1174 1452 y Fp(a)1222 1470 y Fo(\()p Fn(k)1286 1479 y Fl(0)1321 1470 y Fn(;k)1378 1479 y Fl(1)1412 1470 y Fo(\))1444 1452 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(@)1692 1411 y Fn(k)1729 1420 y Fl(0)1687 1476 y Fn(t)1768 1452 y Fp(@)1821 1414 y Fn(k)1858 1423 y Fl(1)1816 1474 y Fn(x)1897 1452 y Fp(u)p Fq(\()p Fp(t;)g(x)p Fq(\))21 b(+)f Fp(a)p Fq(\()p Fp(t;)15 b(x)p Fq(\)\()p Fp(u)p Fq(\()p Fp(t;)g(x)p Fq(\))2781 1414 y Fo(2)2843 1452 y Fq(+)20 b Fp(u)p Fq(\(0)p Fp(;)15 b(x)p Fq(\))p Fp(@)3241 1466 y Fn(t)3272 1452 y Fp(u)p Fq(\()p Fp(t;)g(x)p Fq(\)\))75 1745 y Fg(wher)-5 b(e)41 b Fj(S)47 b Fg(is)40 b(a)h(\014nite)f(subset)g(of)g Fi(N)26 b Fj(\002)f Fi(N)p Fg(.)64 b(The)41 b(c)-5 b(o)g(e\016cients)41 b Fp(a)2325 1763 y Fo(\()p Fn(k)2389 1772 y Fl(0)2423 1763 y Fn(;k)2480 1772 y Fl(1)2514 1763 y Fo(\))2546 1745 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(;)g(a)p Fq(\()p Fp(t;)g(x)p Fq(\))42 b Fg(b)-5 b(elong)41 b(to)g Fp(G)p Fq(\()p Fp(T)3624 1759 y Fo(0)3664 1745 y Fp(;)15 b(X)3779 1759 y Fo(0)3819 1745 y Fq(\))75 1858 y Fg(wher)-5 b(e)34 b Fp(T)385 1872 y Fo(0)424 1858 y Fp(;)15 b(X)539 1872 y Fo(0)605 1858 y Fp(>)25 b Fq(0)p Fg(.)42 b(We)32 b(assume)h(that)h (the)f(fol)5 b(lowing)34 b(assumption)g(hold:)75 2015 y Fh(\(H1\))c Fg(F)-7 b(or)33 b(al)5 b(l)34 b Fq(\()p Fp(k)703 2029 y Fo(0)743 2015 y Fp(;)15 b(k)830 2029 y Fo(1)870 2015 y Fq(\))25 b Fj(2)g(S)7 b Fg(,)33 b(we)f(have)h Fp(S)e Fj(\025)25 b Fp(k)1712 2029 y Fo(0)1772 2015 y Fq(+)19 b Fp(k)1909 2029 y Fo(1)1982 2015 y Fg(and)33 b Fp(S)e(>)24 b(k)2387 2029 y Fo(1)2427 2015 y Fg(.)75 2173 y(We)32 b(imp)-5 b(ose)34 b(the)f(initial)g(c)-5 b(onditions)35 b(:)41 b(F)-7 b(or)34 b(al)5 b(l)33 b Fq(0)26 b Fj(\024)f Fp(k)j Fj(\024)d Fp(S)h Fj(\000)19 b Fq(1)p Fg(,)75 2362 y Fq(\(3\))1395 b(\()p Fp(@)1673 2324 y 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Fl(0)2599 2871 y Fn(;X)2677 2880 y Fl(0)2711 2871 y Fo(\))2768 2852 y Fj(\024)25 b Fp(\016)n(;)75 3041 y Fg(for)33 b(al)5 b(l)33 b Fq(\()p Fp(k)432 3055 y Fo(0)472 3041 y Fp(;)15 b(k)559 3055 y Fo(1)600 3041 y Fq(\))25 b Fj(2)g(S)7 b Fg(,)32 b(then)h(the)g(pr)-5 b(oblem)35 b(\(2\),)e(\(3\))g(has)h(a)f(unique)f(solution)i Fp(u)p Fq(\()p Fp(t;)15 b(x)p Fq(\))33 b Fg(in)f Fp(G)p Fq(\()p Fp(T)8 b(;)15 b(X)7 b Fq(\))p Fg(.)75 3199 y Fq(W)-8 b(e)32 b(start)e(with)h(some)f(preliminaries.)75 3396 y Fh(Lemma)35 b(1)45 b Fg(L)-5 b(et)40 b Fp(U)769 3410 y Fo(1)809 3396 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fg(,)42 b Fp(U)1136 3410 y Fo(2)1176 3396 y Fq(\()p Fp(t;)15 b(x)p Fq(\))40 b Fg(b)-5 b(e)40 b(in)f Fp(G)p Fq(\()p Fp(T)8 b(;)15 b(X)7 b Fq(\))p Fg(.)65 b(Then,)41 b(the)f(pr)-5 b(o)g(duct)42 b Fp(U)2887 3410 y Fo(1)2927 3396 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(U)3184 3410 y Fo(2)3224 3396 y Fq(\()p Fp(t;)g(x)p Fq(\))41 b Fg(b)-5 b(elongs)40 b(to)75 3509 y Fp(G)p Fq(\()p Fp(T)8 b(;)15 b(X)7 b Fq(\))p Fg(.)43 b(Mor)-5 b(e)g(over,)75 3698 y Fq(\(4\))672 b Fj(jj)p Fp(U)974 3712 y Fo(1)1014 3698 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(U)1271 3712 y Fo(2)1311 3698 y Fq(\()p Fp(t;)g(x)p Fq(\))p Fj(jj)1556 3716 y Fo(\()p Fn(T)6 b(;X)f Fo(\))1772 3698 y Fj(\024)25 b(jj)p Fp(U)1980 3712 y Fo(1)2020 3698 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)2265 3716 y Fo(\()p Fn(T)6 b(;X)f Fo(\))2455 3698 y Fj(jj)p Fp(U)2567 3712 y Fo(2)2607 3698 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)2852 3716 y Fo(\()p Fn(T)6 b(;X)f Fo(\))3042 3698 y Fp(:)75 3886 y Fg(In)33 b(other)g(wor)-5 b(ds,)35 b(the)e(norme)-5 b(d)35 b(ve)-5 b(ctor)33 b(sp)-5 b(ac)g(e)34 b Fq(\()p Fp(G)p Fq(\()p Fp(T)8 b(;)15 b(X)7 b Fq(\))p Fp(;)15 b Fj(jj)p Fp(:)p Fj(jj)2221 3905 y Fo(\()p Fn(T)6 b(;X)f Fo(\))2413 3886 y Fq(\))33 b Fg(is)g(in)f(fact)h (a)g(Banach)h(algebr)-5 b(a.)75 4128 y Fh(Pro)s(of)31 b Fq(F)-8 b(or)31 b Fp(k)e Fq(=)c(1)p Fp(;)15 b Fq(2,)32 b(let)1477 4295 y Fp(U)1539 4310 y Fn(k)1582 4295 y Fq(\()p Fp(t;)15 b(x)p Fq(\))26 b(=)1921 4208 y Fm(X)1899 4406 y Fn(\014)s(;l)q Fk(\025)p Fo(0)2089 4295 y Fp(u)2141 4257 y Fn(k)2141 4318 y(\014)s(;l)2239 4233 y Fp(x)2291 4200 y Fn(\014)p 2239 4274 100 4 v 2248 4357 a Fp(\014)5 b Fq(!)2358 4233 y Fp(t)2391 4200 y Fn(l)p 2358 4274 59 4 v 2360 4357 a Fp(l)r Fq(!)2427 4295 y Fp(:)75 4546 y Fq(W)-8 b(e)32 b(ha)m(v)m(e)1334 4695 y Fp(U)1396 4709 y Fo(1)1436 4695 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(U)1693 4709 y Fo(2)1733 4695 y Fq(\()p Fp(t;)g(x)p Fq(\))26 b(=)2072 4608 y Fm(X)2050 4806 y Fn(\014)s(;l)q Fk(\025)p Fo(0)2240 4695 y Fp(v)2284 4710 y Fn(\014)s(;l)2382 4633 y Fp(x)2434 4600 y Fn(\014)p 2382 4674 100 4 v 2391 4757 a Fp(\014)5 b Fq(!)2501 4633 y Fp(t)2534 4600 y Fn(l)p 2501 4674 59 4 v 2503 4757 a Fp(l)r Fq(!)2570 4695 y Fp(;)75 4946 y Fq(where)1215 5106 y Fp(v)1259 5121 y Fn(\014)s(;l)1372 5106 y Fq(=)1553 5019 y Fm(X)1468 5217 y Fn(\014)1508 5226 y Fl(1)1543 5217 y Fo(+)p Fn(\014)1638 5226 y Fl(2)1672 5217 y Fo(=)p Fn(\014)1841 5019 y Fm(X)1785 5217 y Fn(l)1806 5226 y Fl(1)1840 5217 y Fo(+)p Fn(l)1916 5226 y Fl(2)1951 5217 y Fo(=)p Fn(l)2028 5106 y Fq(\()2074 5030 y Fp(u)2126 4997 y Fo(1)2126 5057 y Fn(\014)2166 5066 y Fl(1)2200 5057 y Fn(;l)2241 5066 y Fl(1)p 2073 5085 208 4 v 2073 5168 a Fp(\014)2124 5182 y Fo(1)2164 5168 y Fq(!)p Fp(l)2216 5182 y Fo(1)2256 5168 y Fq(!)2302 5030 y Fp(u)2354 4997 y Fo(2)2354 5057 y Fn(\014)2394 5066 y Fl(2)2428 5057 y Fn(;l)2469 5066 y Fl(2)p 2301 5085 V 2301 5168 a Fp(\014)2352 5182 y Fo(2)2392 5168 y Fq(!)p Fp(l)2444 5182 y Fo(2)2484 5168 y Fq(!)2519 5106 y Fp(\014)g Fq(!)p Fp(l)r Fq(!\))p Fp(:)75 5357 y Fq(By)31 b(de\014nition,)f(w)m(e)h(ha)m(v)m(e)350 5645 y Fj(jj)p Fp(U)462 5659 y Fo(1)502 5645 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)747 5664 y Fo(\()p Fn(T)6 b(;X)f Fo(\))937 5645 y Fj(jj)p Fp(U)1049 5659 y Fo(2)1089 5645 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)1334 5664 y Fo(\()p Fn(T)6 b(;X)f Fo(\))1550 5645 y Fq(=)1667 5559 y Fm(X)1646 5756 y Fn(\014)s(;l)q Fk(\025)p Fo(0)1835 5462 y Fm(0)1835 5626 y(@)2000 5559 y(X)1915 5756 y Fn(\014)1955 5765 y Fl(1)1989 5756 y Fo(+)p Fn(\014)2084 5765 y Fl(2)2118 5756 y Fo(=)p Fn(\014)2287 5559 y Fm(X)2231 5756 y Fn(l)2252 5765 y Fl(1)2287 5756 y Fo(+)p Fn(l)2363 5765 y Fl(2)2397 5756 y Fo(=)p Fn(l)2608 5569 y Fj(j)p Fp(u)2685 5536 y Fo(1)2685 5597 y Fn(\014)2725 5606 y Fl(1)2759 5597 y Fn(;l)2800 5606 y Fl(1)2839 5569 y Fj(jj)p Fp(u)2941 5536 y Fo(2)2941 5597 y Fn(\014)2981 5606 y Fl(2)3015 5597 y Fn(;l)3056 5606 y Fl(2)3095 5569 y Fj(j)p 2499 5624 729 4 v 2499 5707 a Fq(\()p Fp(l)2561 5721 y Fo(1)2621 5707 y Fq(+)20 b Fp(\014)2763 5721 y Fo(1)2803 5707 y Fq(\)!\()p Fp(l)2925 5721 y Fo(2)2986 5707 y Fq(+)g Fp(\014)3128 5721 y Fo(2)3168 5707 y Fq(\)!)3238 5462 y Fm(1)3238 5626 y(A)3333 5645 y Fp(X)3415 5607 y Fn(\014)3463 5645 y Fp(T)3529 5607 y Fn(l)3554 5645 y Fp(;)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 3809 100 a Fq(4)75 399 y(and)602 593 y Fj(jj)p Fp(U)714 607 y Fo(1)754 593 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(U)1011 607 y Fo(2)1052 593 y Fq(\()p Fp(t;)g(x)p Fq(\))p Fj(jj)1297 612 y Fo(\()p Fn(T)6 b(;X)f Fo(\))1512 593 y Fq(=)1630 507 y Fm(X)1608 704 y Fn(\014)s(;l)q Fk(\025)p Fo(0)1798 407 y Fm(\014)1798 461 y(\014)1798 516 y(\014)1798 570 y(\014)1798 625 y(\014)1798 680 y(\014)1913 507 y(X)1828 704 y Fn(\014)1868 713 y Fl(1)1903 704 y Fo(+)p Fn(\014)1998 713 y Fl(2)2032 704 y Fo(=)p Fn(\014)2200 507 y Fm(X)2145 704 y Fn(l)2166 713 y Fl(1)2200 704 y Fo(+)p Fn(l)2276 713 y Fl(2)2311 704 y Fo(=)p Fn(l)2413 517 y Fp(u)2465 484 y Fo(1)2465 545 y Fn(\014)2505 554 y Fl(1)2539 545 y Fn(;l)2580 554 y Fl(1)2618 517 y Fp(u)2670 484 y Fo(2)2670 545 y Fn(\014)2710 554 y Fl(2)2745 545 y Fn(;l)2786 554 y Fl(2)2824 517 y Fp(l)r Fq(!)p Fp(\014)g Fq(!)p 2413 572 547 4 v 2478 656 a Fp(l)2505 670 y Fo(1)2545 656 y Fq(!)p Fp(\014)2621 670 y Fo(1)2661 656 y Fq(!)p Fp(l)2713 670 y Fo(2)2753 656 y Fq(!)p Fp(\014)2829 670 y Fo(2)2869 656 y Fq(!)2970 407 y Fm(\014)2970 461 y(\014)2970 516 y(\014)2970 570 y(\014)2970 625 y(\014)2970 680 y(\014)3060 532 y Fp(X)3142 499 y Fn(\014)3190 532 y Fp(T)3256 499 y Fn(l)p 3025 572 293 4 v 3025 656 a Fq(\()p Fp(l)22 b Fq(+)e Fp(\014)5 b Fq(\)!)75 850 y(On)30 b(the)g(other)h(hand)e(w)m (e)i(ha)m(v)m(e)g(that)75 1059 y(\(5\))1462 998 y Fp(l)r Fq(!)p Fp(\014)5 b Fq(!)p 1334 1038 391 4 v 1334 1122 a Fp(l)1361 1136 y Fo(1)1401 1122 y Fq(!)p Fp(\014)1477 1136 y Fo(1)1517 1122 y Fq(!)p Fp(l)1569 1136 y Fo(2)1609 1122 y Fq(!)p Fp(\014)1685 1136 y Fo(2)1760 1059 y Fj(\024)2085 998 y Fq(\()p Fp(l)22 b Fq(+)e Fp(\014)5 b Fq(\)!)p 1866 1038 729 4 v 1866 1122 a(\()p Fp(l)1928 1136 y Fo(1)1988 1122 y Fq(+)20 b Fp(\014)2130 1136 y Fo(1)2170 1122 y Fq(\)!\()p Fp(l)2292 1136 y Fo(2)2353 1122 y Fq(+)g Fp(\014)2495 1136 y Fo(2)2535 1122 y Fq(\)!)75 1287 y(for)30 b(all)h Fp(l)367 1301 y Fo(1)407 1287 y Fp(;)15 b(l)474 1301 y Fo(2)539 1287 y Fj(\025)25 b Fq(0)31 b(suc)m(h)f(that)h Fp(l)1140 1301 y Fo(1)1200 1287 y Fq(+)20 b Fp(l)1318 1301 y Fo(2)1382 1287 y Fq(=)25 b Fp(l)33 b Fq(and)c Fp(\014)1765 1301 y Fo(1)1805 1287 y Fp(;)15 b(\014)1896 1301 y Fo(2)1962 1287 y Fj(\025)25 b Fq(0)30 b(suc)m(h)g(that)h Fp(\014)2586 1301 y Fo(1)2647 1287 y Fq(+)19 b Fp(\014)2788 1301 y Fo(2)2854 1287 y Fq(=)24 b Fp(\014)5 b Fq(.)216 1400 y(Finally)-8 b(,)35 b(from)d(this)h(inequalit)m(y)h(\(5\),)g(w)m (e)f(deduce)g(that)g Fj(jj)p Fp(U)2294 1414 y Fo(1)2334 1400 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(U)2591 1414 y Fo(2)2631 1400 y Fq(\()p Fp(t;)g(x)p Fq(\))p Fj(jj)2876 1419 y Fo(\()p Fn(T)6 b(;X)f Fo(\))3099 1400 y Fq(con)m(v)m(erges)35 b(and)d(that)75 1513 y(the)f(estimates)h(\(4\))f(hold.)2781 b Ff(2)75 1749 y Fh(Lemma)35 b(2)45 b Fg(L)-5 b(et)29 b Fp(h)748 1763 y Fo(0)788 1749 y Fp(;)15 b(h)880 1763 y Fo(1)945 1749 y Fj(2)25 b Fi(N)j Fg(such)h(that)h Fp(h)1562 1763 y Fo(0)1627 1749 y Fj(\024)25 b Fp(h)1775 1763 y Fo(1)1814 1749 y Fg(.)41 b(The)28 b(op)-5 b(er)g(ator)32 b Fp(@)2471 1709 y Fn(h)2512 1718 y Fl(0)2466 1774 y Fn(t)2551 1749 y Fp(@)2604 1716 y Fk(\000)p Fn(h)2700 1725 y Fl(1)2599 1772 y Fn(x)2767 1749 y Fg(is)c(a)h(b)-5 b(ounde)g(d)30 b(line)-5 b(ar)29 b(op)-5 b(er)g(ator)75 1862 y(fr)g(om)46 b Fq(\()p Fp(G)p Fq(\()p Fp(T)8 b(;)15 b(X)7 b Fq(\))p Fp(;)15 b Fj(jj)p Fp(:)p Fj(jj)826 1881 y Fo(\()p Fn(T)6 b(;X)f Fo(\))1018 1862 y Fq(\))45 b Fg(into)f(itself.)77 b(Mor)-5 b(e)g(over,)48 b(ther)-5 b(e)46 b(exists)e(a)h(c)-5 b(onstant)46 b Fp(C)53 b(>)46 b Fq(0)e Fg(such)h(that)g(the)75 1975 y(estimates)34 b(hold)75 2160 y Fq(\(6\))767 b Fj(jj)p Fp(@)1060 2119 y Fn(h)1101 2128 y Fl(0)1055 2184 y Fn(t)1140 2160 y Fp(@)1193 2122 y Fk(\000)p Fn(h)1289 2131 y Fl(1)1188 2182 y Fn(x)1328 2160 y Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)1645 2178 y Fo(\()p Fn(T)6 b(;X)f Fo(\))1860 2160 y Fj(\024)25 b Fp(C)7 b(X)2110 2122 y Fn(h)2151 2131 y Fl(1)2190 2160 y Fp(T)2256 2122 y Fk(\000)p Fn(h)2352 2131 y Fl(0)2390 2160 y Fj(jj)p Fp(U)j Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)2757 2178 y Fo(\()p Fn(T)6 b(;X)f Fo(\))2947 2160 y Fp(;)75 2344 y Fg(for)33 b(al)5 b(l)33 b Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))26 b Fj(2)f Fp(G)p Fq(\()p Fp(T)8 b(;)15 b(X)7 b Fq(\))p Fg(.)75 2581 y Fh(Pro)s(of)31 b Fq(Let)g Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))31 b(b)s(e)f(in)g Fp(G)p Fq(\()p Fp(T)8 b(;)15 b(X)7 b Fq(\).)42 b(W)-8 b(e)32 b(ha)m(v)m(e,)1237 2824 y Fp(@)1290 2783 y Fn(h)1331 2792 y Fl(0)1285 2848 y Fn(t)1370 2824 y Fp(@)1423 2786 y Fk(\000)p Fn(h)1519 2795 y Fl(1)1418 2846 y Fn(x)1558 2824 y Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))26 b(=)1947 2737 y Fm(X)1970 2935 y Fn(\014)s(;l)2093 2824 y Fp(u)2145 2839 y Fn(\014)s Fk(\000)p Fn(h)2284 2848 y Fl(1)2319 2839 y Fn(;l)q Fo(+)p Fn(h)2457 2848 y Fl(0)2504 2762 y Fp(x)2556 2729 y Fn(\014)p 2504 2803 100 4 v 2513 2886 a Fp(\014)5 b Fq(!)2623 2762 y Fp(t)2656 2729 y Fn(l)p 2623 2803 59 4 v 2625 2886 a Fp(l)r Fq(!)75 3100 y(By)31 b(de\014nition,)f(w)m(e)h(ha)m(v)m(e)335 3329 y Fp(X)417 3292 y Fn(h)458 3301 y Fl(1)497 3329 y Fp(T)563 3292 y Fk(\000)p Fn(h)659 3301 y Fl(0)697 3329 y Fj(jj)p Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)1064 3348 y Fo(\()p Fn(T)6 b(;X)f Fo(\))1280 3329 y Fq(=)1376 3243 y Fm(X)1400 3441 y Fn(\014)s(;l)1583 3266 y Fj(j)p Fp(u)1660 3281 y Fn(\014)s(;l)1749 3266 y Fj(j)p 1533 3308 293 4 v 1533 3392 a Fq(\()p Fp(l)22 b Fq(+)e Fp(\014)5 b Fq(\)!)1835 3329 y Fp(X)1917 3292 y Fn(\014)s Fo(+)p Fn(h)2056 3301 y Fl(1)2095 3329 y Fp(T)2161 3292 y Fn(l)q Fk(\000)p Fn(h)2279 3301 y Fl(0)2342 3329 y Fq(=)2438 3243 y Fm(X)2461 3441 y Fn(\014)s(;l)2703 3266 y Fj(j)p Fp(u)2780 3281 y Fn(\014)s Fk(\000)p Fn(h)2919 3290 y Fl(1)2953 3281 y Fn(;l)q Fo(+)p Fn(h)3091 3290 y Fl(0)3125 3281 y Fk(\000)p Fn(\013)3229 3266 y Fj(j)p 2594 3308 769 4 v 2594 3392 a Fq(\(\()p Fp(l)23 b Fq(+)d Fp(h)2857 3406 y Fo(0)2897 3392 y Fq(\))g(+)g Fp(\014)26 b Fj(\000)19 b Fp(h)3262 3406 y Fo(1)3302 3392 y Fq(\)!)3373 3329 y Fp(X)3455 3292 y Fn(\014)3502 3329 y Fp(T)3568 3292 y Fn(l)75 3606 y Fq(and)1033 3745 y Fj(jj)p Fp(@)1136 3705 y Fn(h)1177 3714 y Fl(0)1131 3769 y Fn(t)1216 3745 y Fp(@)1269 3708 y Fk(\000)p Fn(h)1365 3717 y Fl(1)1264 3768 y Fn(x)1404 3745 y Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)1721 3764 y Fo(\()p Fn(T)6 b(;X)f Fo(\))1936 3745 y Fq(=)2032 3659 y Fm(X)2056 3857 y Fn(\014)s(;l)2189 3682 y Fj(j)p Fp(u)2266 3697 y Fn(\014)s Fk(\000)p Fn(h)2405 3706 y Fl(1)2439 3697 y Fn(;l)q Fo(+)p Fn(h)2577 3706 y Fl(0)2615 3682 y Fj(j)p 2189 3724 452 4 v 2268 3808 a Fq(\()p Fp(l)23 b Fq(+)d Fp(\014)5 b Fq(\)!)2650 3745 y Fp(X)2732 3708 y Fn(\014)2780 3745 y Fp(T)2846 3708 y Fn(l)2871 3745 y Fp(:)75 4018 y Fq(Finally)-8 b(,)31 b(from)e(the)h(inequalit)m(y)h(\()p Fp(l)21 b Fq(+)d Fp(h)1416 4032 y Fo(0)1474 4018 y Fq(+)h Fp(\014)k Fj(\000)c Fp(h)1780 4032 y Fo(1)1820 4018 y Fq(\)!)25 b Fj(\024)g Fq(\()p Fp(l)c Fq(+)d Fp(\014)5 b Fq(\)!,)31 b(w)m(e)f(deduce)f(that)h Fj(jj)p Fp(@)3079 3977 y Fn(h)3120 3986 y Fl(0)3074 4042 y Fn(t)3160 4018 y Fp(@)3213 3985 y Fk(\000)p Fn(h)3309 3994 y Fl(1)3208 4040 y Fn(z)3347 4018 y Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)3664 4036 y Fo(\()p Fn(T)6 b(;X)f Fo(\))75 4131 y Fq(con)m(v)m(erges)32 b(and)e(that)h(the)g (estimates)h(\(6\))f(hold.)2000 b Ff(2)75 4367 y Fh(Lemma)35 b(3)45 b Fg(L)-5 b(et)40 b Fp(h)759 4381 y Fo(1)799 4367 y Fp(;)15 b(h)891 4381 y Fo(2)968 4367 y Fj(2)36 b Fi(N)j Fg(and)h(let)f Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))40 b Fg(b)-5 b(e)39 b(in)g Fp(G)p Fq(\()p Fp(T)2189 4381 y Fo(0)2229 4367 y Fp(;)15 b(X)2344 4381 y Fo(0)2384 4367 y Fq(\))p Fg(.)62 b(Then,)41 b(ther)-5 b(e)40 b(exist)f Fp(X)r(;)15 b(T)50 b(>)37 b Fq(0)i Fg(smal)5 b(l)75 4480 y(enough)46 b(\(dep)-5 b(ending)47 b(on)f Fp(h)1065 4494 y Fo(1)1105 4480 y Fp(;)15 b(h)1197 4494 y Fo(2)1237 4480 y Fg(\))46 b(such)g(that)h(the)f(formal)h(series)f Fp(@)2528 4439 y Fn(h)2569 4448 y Fl(1)2523 4504 y Fn(t)2608 4480 y Fp(@)2661 4447 y Fn(h)2702 4456 y Fl(2)2656 4502 y Fn(x)2741 4480 y Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))46 b Fg(b)-5 b(elongs)47 b(to)f Fp(G)p Fq(\()p Fp(T)8 b(;)15 b(X)7 b Fq(\))p Fg(.)75 4593 y(Mor)-5 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Fp(h)3154 5683 y Fo(2)3194 5669 y Fq(\)!)3265 5607 y Fp(X)3347 5569 y Fn(\014)s Fo(+)p Fn(h)3486 5578 y Fl(2)3524 5607 y Fp(T)3590 5569 y Fn(l)q Fo(+)p Fn(h)3708 5578 y Fl(1)3746 5607 y Fp(:)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 3809 100 a Fq(5)75 399 y(But)31 b(w)m(e)f(ha)m(v)m(e)1269 475 y(\(\()p Fp(l)23 b Fq(+)d Fp(h)1532 489 y Fo(1)1571 475 y Fq(\))h(+)f Fp(\014)25 b Fq(+)20 b Fp(h)1937 489 y Fo(2)1977 475 y Fq(\)!)p 1269 516 769 4 v 1507 599 a(\()p Fp(l)j Fq(+)d Fp(\014)5 b Fq(\)!)2073 536 y Fj(\030)25 b Fq(\()p Fp(l)d Fq(+)e Fp(\014)5 b Fq(\))2435 499 y Fn(h)2476 508 y Fl(1)2511 499 y Fo(+)p Fn(h)2607 508 y Fl(2)2645 536 y Fp(;)75 735 y Fq(as)31 b Fp(l)r(;)15 b(\014)35 b Fq(tend)c(to)g(in\014nit)m(y) -8 b(.)41 b(So)30 b(that)h(there)f(exist)i Fp(K)q(;)15 b(C)1956 749 y Fo(1)1996 735 y Fp(;)g(C)2101 749 y Fo(2)2166 735 y Fp(>)25 b Fq(0)31 b(suc)m(h)f(that)303 960 y Fj(jj)p Fq(\()p Fp(@)441 919 y Fn(h)482 928 y Fl(1)436 984 y Fn(t)522 960 y Fp(@)575 922 y Fn(h)616 931 y 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Fp(a)1923 4770 y Fo(\()p Fn(k)1987 4779 y Fl(0)2022 4770 y Fn(;k)2079 4779 y Fl(1)2113 4770 y Fo(\))2144 4751 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)2389 4770 y Fo(\()p Fn(T)6 b(;X)f Fo(\))2579 4751 y Fp(X)2661 4714 y Fo(\()p Fn(S)t Fk(\000)p Fn(k)2827 4723 y Fl(1)2862 4714 y Fo(\))2894 4751 y Fp(T)2960 4714 y Fk(\000)p Fn(k)3052 4723 y Fl(0)3090 4751 y Fp(R)3035 4903 y Fj(\024)25 b Fp(C)7 b(\016)s(X)3328 4866 y Fn(S)t Fk(\000)p Fn(k)3467 4875 y Fl(1)3506 4903 y Fp(T)3572 4866 y Fk(\000)p Fn(k)3664 4875 y Fl(0)3702 4903 y Fp(R)75 5084 y Fq(for)31 b(all)i Fp(v)s Fq(\()p Fp(t;)15 b(x)p Fq(\))29 b Fj(2)e Fp(B)771 5098 y Fn(R)860 5084 y Fq(and)k(for)g(all)i (\()p Fp(k)1388 5098 y Fo(0)1428 5084 y Fp(;)15 b(k)1515 5098 y Fo(1)1555 5084 y Fq(\))28 b Fj(2)f(S)7 b Fq(.)44 b(On)31 b(the)h(other)g(hand,)f(due)g(to)h(lemma)g(3,)h(there)f(exists) g(a)75 5196 y(constan)m(t)g Fp(C)f(>)25 b Fq(0)31 b(suc)m(h)f(that)75 5424 y(\(9\))84 b Fj(jj)p Fp(a)372 5442 y Fo(\()p Fn(k)436 5451 y Fl(0)472 5442 y Fn(;k)529 5451 y Fl(1)562 5442 y Fo(\))594 5424 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(@)842 5383 y Fn(k)879 5392 y Fl(0)837 5448 y Fn(t)919 5424 y Fp(')978 5438 y Fn(j)1015 5424 y Fq(\()p Fp(t)p Fq(\))1128 5362 y Fp(x)1180 5329 y Fn(q)p 1128 5403 90 4 v 1138 5486 a Fp(q)s Fq(!)1228 5424 y Fj(jj)1278 5442 y Fo(\()p Fn(T)6 b(;X)f Fo(\))616 5660 y Fj(\024)24 b Fp(C)7 b Fj(jj)p Fp(a)881 5679 y Fo(\()p Fn(k)945 5688 y Fl(0)980 5679 y Fn(;k)1037 5688 y Fl(1)1071 5679 y Fo(\))1103 5660 y Fq(\()p Fp(t;)15 b(x)p Fq(\))1308 5599 y Fp(x)1360 5566 y Fn(q)p 1308 5639 V 1318 5723 a Fp(q)s Fq(!)1409 5660 y Fj(jj)1459 5679 y Fo(\()p Fn(T)6 b(;X)f Fo(\))1648 5660 y Fp(T)1714 5623 y Fk(\000)p Fn(k)1806 5632 y Fl(0)1844 5660 y Fj(jj)p Fp(')1953 5674 y Fn(j)1991 5660 y Fq(\()p Fp(t)p Fq(\))p Fj(jj)2144 5679 y Fo(\()p Fn(T)2212 5688 y Fl(0)2248 5679 y Fn(;X)2326 5688 y Fl(0)2360 5679 y Fo(\))2417 5660 y Fj(\024)25 b Fp(C)7 b(\016)s Fj(jj)2688 5599 y Fp(x)2740 5566 y Fn(q)p 2688 5639 V 2698 5723 a Fp(q)s Fq(!)2789 5660 y Fj(jj)2839 5679 y Fo(\()p Fn(T)f(;X)f Fo(\))3028 5660 y Fp(T)3094 5623 y Fk(\000)p Fn(k)3186 5632 y Fl(0)3224 5660 y Fj(jj)p Fp(')3333 5674 y Fn(j)3371 5660 y Fq(\()p Fp(t)p Fq(\))p Fj(jj)3524 5679 y Fo(\()p Fn(T)3592 5688 y Fl(0)3628 5679 y Fn(;X)3706 5688 y Fl(0)3740 5679 y Fo(\))p eop end %%Page: 6 6 TeXDict begin 6 5 bop 3809 100 a Fq(6)75 399 y(for)29 b(all)h(0)c Fj(\024)f Fp(j;)15 b(q)29 b Fj(\024)c Fp(S)d Fj(\000)c Fq(1)29 b(and)g(all)h(\()p Fp(k)1372 413 y Fo(0)1412 399 y Fp(;)15 b(k)1499 413 y Fo(1)1539 399 y Fq(\))26 b Fj(2)e(S)7 b Fq(,)30 b(for)f Fp(X)r(;)15 b(T)42 b Fq(small)30 b(enough.)40 b(Moreo)m(v)m(er)31 b(there)f(exists)f Fp(C)j(>)25 b Fq(0)75 511 y(suc)m(h)30 b(that)75 720 y(\(10\))85 b Fj(jj)p Fp(a)p Fq(\()p Fp(t;)15 b(x)p Fq(\)\()p Fp(@)701 683 y Fk(\000)p Fn(S)696 743 y(x)808 720 y Fp(v)s Fq(\()p Fp(t;)g(x)p Fq(\))22 b(+)e Fp(w)r Fq(\()p Fp(t;)15 b(x)p Fq(\)\))1460 683 y Fo(2)1501 720 y Fj(jj)1551 739 y Fo(\()p Fn(T)6 b(;X)f Fo(\))1373 871 y Fj(\024)25 b Fp(C)7 b(\016)s Fq(\(\()p Fp(X)1736 834 y Fn(S)1788 871 y Fp(R)q Fq(\))1893 834 y Fo(2)1953 871 y Fq(+)20 b(2)p Fp(X)2171 834 y Fn(S)2222 871 y Fp(R)q Fj(jj)p Fp(w)r Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)2654 890 y Fo(\()p Fn(T)2722 899 y Fl(0)2760 890 y Fn(;X)2838 899 y Fl(0)2872 890 y Fo(\))2924 871 y Fq(+)20 b(\()p Fj(jj)p Fp(w)r Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)3412 890 y Fo(\()p Fn(T)3480 899 y Fl(0)3518 890 y Fn(;X)3596 899 y Fl(0)3630 890 y Fo(\))3661 871 y Fq(\))3696 834 y Fo(2)3736 871 y Fq(\))75 1055 y(and)75 1264 y(\(11\))85 b Fj(jj)p Fp(a)p Fq(\()p Fp(t;)15 b(x)p Fq(\)\()p Fp(@)701 1226 y Fk(\000)p Fn(S)696 1286 y(x)808 1264 y Fp(v)s Fq(\(0)p Fp(;)g(x)p Fq(\))22 b(+)e Fp(w)r Fq(\(0)p Fp(;)15 b(x)p Fq(\)\)\()p Fp(@)1567 1278 y Fn(t)1600 1264 y Fp(@)1653 1226 y Fk(\000)p Fn(S)1648 1286 y(x)1759 1264 y Fp(v)s Fq(\()p Fp(t;)g(x)p Fq(\))21 b(+)f Fp(@)2161 1278 y Fn(t)2191 1264 y Fp(w)r Fq(\()p Fp(t;)15 b(x)p Fq(\)\))p Fj(jj)2538 1282 y Fo(\()p Fn(T)6 b(;X)f Fo(\))860 1415 y Fj(\024)25 b Fp(C)7 b(\016)1086 1341 y Fm(\000)1128 1415 y Fp(X)1210 1377 y Fo(2)p Fn(S)1296 1415 y Fp(T)1362 1377 y Fk(\000)p Fo(1)1457 1415 y Fp(R)1527 1377 y Fo(2)1586 1415 y Fq(+)20 b(2)p Fp(X)1804 1377 y Fn(S)1856 1415 y Fp(R)q(T)1992 1377 y Fk(\000)p Fo(1)2085 1415 y Fj(jj)p Fp(w)r Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)2447 1433 y Fo(\()p Fn(T)2515 1442 y Fl(0)2553 1433 y Fn(;X)2631 1442 y Fl(0)2665 1433 y Fo(\))2717 1415 y Fq(+)20 b Fp(T)2874 1377 y Fk(\000)p Fo(1)2968 1415 y Fq(\()p Fj(jj)p Fp(w)r Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)3365 1433 y Fo(\()p Fn(T)3433 1442 y Fl(0)3471 1433 y Fn(;X)3549 1442 y Fl(0)3583 1433 y Fo(\))3615 1415 y Fq(\))3650 1377 y Fo(2)3689 1341 y Fm(\001)3746 1415 y Fp(:)75 1599 y Fh(Conclusion:)52 b Fq(If)36 b Fp(X)r(;)15 b(T)8 b(;)15 b(\016)n(;)g(R)37 b(>)d Fq(0)i(are)g(small)h(enough,)g(the)f(sum)f(o)m(v)m(er)i(\()p Fp(k)2728 1613 y Fo(0)2768 1599 y Fp(;)15 b(k)2855 1613 y Fo(1)2895 1599 y Fq(\))35 b Fj(2)f(S)7 b Fq(,)37 b(0)e Fj(\024)f Fp(j;)15 b(q)38 b Fj(\024)c Fp(S)29 b Fj(\000)23 b Fq(1)75 1712 y(of)31 b(the)g(righ)m(t)g(hand)f(side)h(of)g(the)g (inequalities)h(\(8\),)g(\(9\),)g(\(10\),)h(\(11\))f(can)f(b)s(e)f(c)m (hosen)i(lo)m(w)m(er)g(than)e Fp(R)q Fq(,)h(for)g(all)75 1825 y Fp(v)s Fq(\()p Fp(t;)15 b(x)p Fq(\))26 b Fj(2)f Fp(G)p Fq(\()p Fp(T)8 b(;)15 b(X)7 b Fq(\))32 b(with)e Fj(jj)p Fp(v)s Fj(jj)1139 1843 y Fo(\()p Fn(T)6 b(;X)f Fo(\))1355 1825 y Fj(\024)25 b Fp(R)q Fq(.)40 b(So)30 b(that,)i(if)e Fp(v)f Fj(2)24 b Fp(B)2245 1839 y Fn(R)2333 1825 y Fq(then)31 b Fp(A)p Fq(\()p Fp(v)s Fq(\))26 b Fj(2)f Fp(B)2907 1839 y Fn(R)2965 1825 y Fq(.)75 1980 y(Pro)s(of)30 b(of)h Fh(ii)p Fq(\).)41 b(Due)30 b(to)h(Lemma)g(1,2)g (there)g(exists)g(a)g(constan)m(t)g Fp(C)h(>)25 b Fq(0)31 b(suc)m(h)f(that)75 2189 y(\(12\))85 b Fj(jj)p Fp(a)418 2208 y Fo(\()p Fn(k)482 2217 y Fl(0)517 2208 y Fn(;k)574 2217 y Fl(1)608 2208 y Fo(\))640 2189 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(@)888 2149 y Fn(k)925 2158 y Fl(0)883 2213 y Fn(t)964 2189 y Fp(@)1017 2152 y Fn(k)1054 2161 y Fl(1)1089 2152 y Fk(\000)p Fn(S)1012 2212 y(x)1195 2189 y Fq(\()p Fp(v)1274 2203 y Fo(2)1313 2189 y Fq(\()p Fp(t;)g(x)p Fq(\))21 b Fj(\000)f Fp(v)1664 2203 y Fo(1)1704 2189 y Fq(\()p Fp(t;)15 b(x)p Fq(\)\))p Fj(jj)1984 2208 y Fo(\()p Fn(T)6 b(;X)f Fo(\))1353 2345 y Fj(\024)25 b Fp(C)7 b Fj(jj)p Fp(a)1619 2363 y Fo(\()p Fn(k)1683 2372 y Fl(0)1718 2363 y Fn(;k)1775 2372 y Fl(1)1809 2363 y Fo(\))1841 2345 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)2086 2363 y Fo(\()p Fn(T)6 b(;X)f Fo(\))2276 2345 y Fp(X)2358 2307 y Fo(\()p Fn(S)t Fk(\000)p Fn(k)2524 2316 y Fl(1)2559 2307 y Fo(\))2590 2345 y Fp(T)2656 2307 y Fk(\000)p Fn(k)2748 2316 y Fl(0)2787 2345 y Fj(jj)p Fp(v)2881 2359 y Fo(2)2921 2345 y Fq(\()p Fp(t;)15 b(x)p Fq(\))21 b Fj(\000)f Fp(v)3272 2359 y Fo(1)3311 2345 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)3556 2363 y Fo(\()p Fn(T)6 b(;X)f Fo(\))3746 2345 y Fp(;)75 2529 y Fq(for)30 b(all)h Fp(v)384 2543 y Fo(1)424 2529 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(;)g(v)703 2543 y Fo(2)744 2529 y Fq(\()p Fp(t;)g(x)p Fq(\))26 b Fj(2)f Fp(B)1120 2543 y Fn(R)1208 2529 y Fq(and)k(for)i(all)g(\()p Fp(k)1732 2543 y Fo(0)1772 2529 y Fp(;)15 b(k)1859 2543 y Fo(1)1899 2529 y Fq(\))26 b Fj(2)f(S)7 b Fq(.)40 b(If)30 b(w)m(e)h(denote)g(b)m(y)158 2737 y Fj(A)231 2751 y Fo(2)p Fn(;)p Fo(1)325 2737 y Fq(\()p Fp(t;)15 b(x)p Fq(\))26 b(=)f(\()p Fp(@)730 2700 y Fk(\000)p Fn(S)725 2760 y(x)836 2737 y Fp(v)880 2751 y Fo(2)920 2737 y Fq(\(0)p Fp(;)15 b(x)p Fq(\))22 b(+)e Fp(w)r Fq(\(0)p Fp(;)15 b(x)p Fq(\)\)\()p Fp(@)1632 2751 y Fn(t)1664 2737 y Fp(@)1717 2700 y Fk(\000)p Fn(S)1712 2760 y(x)1823 2737 y Fp(v)1867 2751 y Fo(2)1907 2737 y Fq(\()p Fp(t;)g(x)p Fq(\))21 b(+)f Fp(w)r Fq(\()p Fp(t;)15 b(x)p Fq(\)\))1810 2887 y Fj(\000)20 b Fq(\()p Fp(@)1989 2850 y Fk(\000)p Fn(S)1984 2910 y(x)2095 2887 y Fp(v)2139 2901 y Fo(1)2179 2887 y Fq(\(0)p Fp(;)15 b(x)p Fq(\))22 b(+)e Fp(w)r Fq(\(0)p Fp(;)15 b(x)p Fq(\)\)\()p Fp(@)2891 2901 y Fn(t)2923 2887 y Fp(@)2976 2850 y Fk(\000)p Fn(S)2971 2910 y(x)3082 2887 y Fp(v)3126 2901 y Fo(1)3166 2887 y Fq(\()p Fp(t;)g(x)p Fq(\))21 b(+)f Fp(w)r Fq(\()p Fp(t;)15 b(x)p Fq(\)\))75 3071 y(w)m(e)31 b(also)g(ha)m(v)m(e)158 3280 y Fj(A)231 3294 y Fo(2)p Fn(;)p Fo(1)325 3280 y Fq(\()p Fp(t;)15 b(x)p Fq(\))26 b(=)f(\()p Fp(@)730 3242 y Fk(\000)p Fn(S)725 3302 y(x)836 3280 y Fq(\()p Fp(v)915 3294 y Fo(2)955 3280 y Fq(\(0)p Fp(;)15 b(x)p Fq(\))22 b Fj(\000)e Fp(v)1319 3294 y Fo(1)1358 3280 y Fq(\(0)p Fp(;)15 b(x)p Fq(\)\)\)\()p Fp(@)1718 3294 y Fn(t)1751 3280 y Fp(@)1804 3242 y Fk(\000)p Fn(S)1799 3302 y(x)1910 3280 y Fp(v)1954 3294 y Fo(2)1993 3280 y Fq(\()p Fp(t;)g(x)p Fq(\))21 b(+)f Fp(w)r Fq(\()p Fp(t;)15 b(x)p Fq(\)\))1698 3430 y(+)20 b(\()p Fp(@)1872 3444 y Fn(t)1902 3430 y Fp(@)1955 3392 y Fk(\000)p Fn(S)1950 3452 y(x)2061 3430 y Fq(\()p Fp(v)2140 3444 y Fo(2)2180 3430 y Fq(\()p Fp(t;)15 b(x)p Fq(\))21 b Fj(\000)f Fp(v)2531 3444 y Fo(1)2571 3430 y Fq(\()p Fp(t;)15 b(x)p Fq(\)\)\)\()p Fp(@)2924 3392 y Fk(\000)p Fn(S)2919 3452 y(x)3032 3430 y Fp(v)3076 3444 y Fo(1)3115 3430 y Fq(\(0)p Fp(;)g(x)p Fq(\))22 b(+)e Fp(w)r Fq(\(0)p Fp(;)15 b(x)p Fq(\)\))p Fp(:)75 3613 y Fq(W)-8 b(e)32 b(deduce)e(that)h(there)f(exists)h(a)g(constan)m (t)h Fp(C)f(>)25 b Fq(0)31 b(suc)m(h)f(that)75 3822 y(\(13\))85 b Fj(jj)p Fp(a)p Fq(\()p Fp(t;)15 b(x)p Fq(\))628 3748 y Fm(\000)671 3822 y Fq(\()p Fp(@)759 3784 y Fk(\000)p Fn(S)754 3844 y(x)865 3822 y Fp(v)909 3836 y Fo(2)949 3822 y Fq(\()p Fp(t;)g(x)p Fq(\))21 b(+)f Fp(w)r Fq(\()p Fp(t;)15 b(x)p Fq(\)\))1553 3784 y Fo(2)1614 3822 y Fj(\000)20 b Fq(\()p Fp(@)1793 3784 y Fk(\000)p Fn(S)1788 3844 y(x)1899 3822 y Fp(v)1943 3836 y Fo(1)1983 3822 y Fq(\()p Fp(t;)15 b(x)p Fq(\))21 b(+)f Fp(w)r Fq(\()p Fp(t;)15 b(x)p Fq(\)\))2587 3784 y Fo(2)2648 3822 y Fq(+)20 b Fj(A)2812 3836 y Fo(2)p Fn(;)p Fo(1)2906 3822 y Fq(\()p Fp(t;)15 b(x)p Fq(\))3101 3748 y Fm(\001)3159 3822 y Fj(jj)3209 3841 y Fo(\()p Fn(T)6 b(;X)f Fo(\))311 3973 y Fj(\024)25 b(jj)p Fp(a)p Fq(\()p 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5570 y Fm(X)1928 5766 y Fn(n)p Fk(\025)p Fo(0)2076 5656 y Fp(C)2141 5670 y Fn(n)2188 5656 y Fq(\()p Fp(t)p Fq(\))p Fp(z)2337 5619 y Fn(n)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 3809 100 a Fq(8)75 399 y(is)30 b(holomorphic)h(on)f(\()p Fp(G)21 b Fj([)f Fp(D)1096 413 y Fn(\032)1137 399 y Fq(\))g Fj(\002)g Fp(D)1358 413 y Fn(r)1427 399 y Fq(and)29 b(w)m(eigh)m(ted)j(with)e(resp)s(ect)h (to)g Fp(t)f Fq(on)g(\()p Fp(G)21 b Fj([)f Fp(D)3084 413 y Fn(\032)3124 399 y Fq(\))h Fj(\002)f Fp(D)3346 413 y Fn(r)3384 399 y Fq(.)216 561 y(W)-8 b(e)32 b(study)d(formal)i (series)g Fp(V)20 b Fq(\()p Fp(t;)15 b(z)t Fq(\))31 b(in)f(the)h(v)-5 b(ariable)31 b Fp(z)k Fq(of)30 b(the)h(form)75 813 y(\(15\))1292 b Fp(V)21 b Fq(\()p Fp(t;)15 b(z)t Fq(\))26 b(=)1913 727 y Fm(X)1912 923 y Fn(n)p Fk(\025)p Fo(0)2070 752 y Fp(V)2123 766 y Fn(n)2170 752 y Fq(\()p Fp(t)p Fq(\))p 2070 792 204 4 v 2132 876 a Fp(n)p Fq(!)2284 813 y Fp(z)2330 776 y Fn(n)2377 813 y Fp(;)75 1111 y Fq(with)k(holomorphic)h(co)s (e\016cien)m(ts)h Fp(V)1316 1125 y Fn(n)1363 1111 y Fq(\()p Fp(t)p Fq(\),)f Fp(n)25 b Fj(\025)f Fq(0,)32 b(on)e Fp(G)20 b Fj([)g Fp(D)2172 1125 y Fn(\032)2213 1111 y Fq(,)30 b(solutions)h(of)g(\(14\).)216 1274 y(In)g(the)i(next)f(lemma)g(w)m(e)h (describ)s(e)e(formal)i(series)f(solutions)g(of)h(\(14\))g(in)f(term)g (of)g(in)m(tegro-di\013eren)m(tial)75 1387 y(relations.)75 1574 y Fh(Lemma)j(5)45 b Fg(The)38 b(formal)h(series)e Fp(V)20 b Fq(\()p Fp(t;)15 b(z)t Fq(\))38 b Fg(is)f(a)h(solution)g(of)g (\(14\))g(if)f(and)h(only)g(if)f(the)g(fol)5 b(lowing)39 b(inte)-5 b(gr)g(o-)75 1687 y(di\013er)g(ential)34 b(r)-5 b(elation)34 b(is)f(satis\014e)-5 b(d,)75 1977 y Fq(\(16\))330 1916 y Fp(V)383 1930 y Fn(m)p Fo(+)p Fn(S)551 1916 y Fq(\()p Fp(t)p Fq(\))p 330 1957 325 4 v 439 2040 a Fp(m)p Fq(!)689 1977 y(=)876 1891 y Fm(X)785 2092 y Fo(\()p Fn(k)849 2101 y Fl(0)884 2092 y Fn(;k)941 2101 y Fl(1)975 2092 y Fo(\))p Fk(2S)1165 1891 y Fm(X)1112 2089 y Fn(l)q Fo(+)p Fn(n)p Fo(=)p Fn(m)1364 1977 y Fp(b)1403 1992 y Fn(k)1440 2001 y Fl(0)1475 1992 y Fn(;k)1532 2001 y Fl(1)1566 1992 y Fn(;l)1611 1977 y Fq(\()p Fp(t)p Fq(\))1724 1916 y Fp(@)1777 1875 y Fn(k)1814 1884 y Fl(0)1772 1940 y Fn(t)1854 1916 y Fp(V)1907 1931 y Fn(n)p Fo(+)p Fn(k)2042 1940 y Fl(1)2080 1916 y Fq(\()p Fp(t)p Fq(\))p 1724 1957 459 4 v 1914 2040 a Fp(n)p Fq(!)2193 1977 y(+)423 2203 y Fm(X)276 2395 y Fn(q)308 2404 y Fl(1)342 2395 y Fo(+)p Fn(q)429 2404 y Fl(2)463 2395 y Fo(+)p Fn(q)550 2404 y Fl(3)584 2395 y Fo(=)p Fn(m)727 2228 y Fp(b)766 2242 y Fo(1)p Fn(;q)853 2251 y Fl(3)891 2228 y Fq(\()p Fp(t)p Fq(\))p 727 2269 268 4 v 755 2352 a Fp(q)796 2366 y Fo(1)835 2352 y Fq(!)p Fp(q)901 2366 y Fo(2)940 2352 y Fq(!)1004 2290 y Fp(@)1052 2304 y Fn(t)1097 2166 y Fm(Z)1188 2192 y Fn(t)1148 2372 y Fo(0)1233 2290 y Fp(V)1286 2304 y Fn(q)1318 2313 y Fl(1)1356 2290 y Fq(\()p Fp(t)21 b Fj(\000)e Fp(\034)10 b Fq(\))p Fp(V)1673 2304 y Fn(q)1705 2313 y Fl(2)1744 2290 y Fq(\()p Fp(\034)g Fq(\))p Fp(d\034)31 b Fq(+)2220 2203 y Fm(X)2073 2395 y Fn(q)2105 2404 y Fl(1)2139 2395 y Fo(+)p Fn(q)2226 2404 y Fl(2)2260 2395 y Fo(+)p Fn(q)2347 2404 y Fl(3)2381 2395 y Fo(=)p Fn(m)2524 2228 y Fp(b)2563 2242 y Fo(2)p Fn(;q)2650 2251 y Fl(3)2688 2228 y Fq(\()p Fp(t)p Fq(\))p 2524 2269 V 2605 2352 a Fp(q)2646 2366 y Fo(1)2685 2352 y Fq(!)2801 2290 y Fp(@)2849 2304 y Fn(t)2895 2166 y Fm(Z)2985 2192 y Fn(t)2945 2372 y Fo(0)3030 2290 y Fp(V)3083 2304 y Fn(q)3115 2313 y Fl(1)3154 2290 y Fq(\()p Fp(t)20 b Fj(\000)g Fp(\034)10 b Fq(\))p Fp(C)3483 2304 y Fn(q)3515 2313 y Fl(2)3553 2290 y Fq(\()p Fp(\034)g Fq(\))p Fp(d\034)75 2580 y Fg(for)33 b(al)5 b(l)33 b Fp(m)25 b Fj(\025)g Fq(0)p Fg(.)75 2767 y Fq(In)k(the)g(next)h(prop)s(osition,)g(w)m(e)g(giv)m(e)h(su\016cien)m (t)f(conditions)g(for)f(the)g(existence)i(and)e(uniqueness)g(of)g (formal)75 2880 y(series)i(\(15\))h(that)f(are)f(solutions)h(of)g (\(14\).)75 3068 y Fh(Prop)s(osition)36 b(2)46 b Fg(We)32 b(assume)h(that)h(the)f(\014nite)g(set)g Fj(S)39 b Fg(satis\014es)1279 3272 y Fj(S)33 b(\032)25 b(f)p Fq(\()p Fp(k)1590 3286 y Fo(0)1630 3272 y Fp(;)15 b(k)1717 3286 y Fo(1)1757 3272 y Fp(;)g(l)r Fq(\))26 b Fj(2)f Fi(N)2039 3235 y Fo(3)2103 3272 y Fq(:)h Fp(k)2201 3286 y Fo(1)2266 3272 y Fj(\024)f Fp(S)g Fj(\000)20 b Fq(1)p Fj(g)p Fp(:)75 3476 y Fg(Mor)-5 b(e)g(over,)38 b(assume)e(that)h(holomorphic)i (functions)d Fp(V)1988 3490 y Fn(\026)2035 3476 y Fq(\()p Fp(t)p Fq(\))p Fg(,)h(for)f Fq(0)31 b Fj(\024)g Fp(\026)g Fj(\024)g Fp(S)c Fj(\000)22 b Fq(1)p Fg(,)37 b(ar)-5 b(e)37 b(given)e(on)h Fp(G)23 b Fj([)f Fp(D)3786 3490 y Fn(\032)3827 3476 y Fg(.)75 3589 y(Then,)45 b(ther)-5 b(e)44 b(exists)e(a)h(unique)f(se)-5 b(quenc)g(e)42 b Fq(\()p Fp(V)1706 3603 y Fn(\026)1753 3589 y Fq(\()p Fp(t)p Fq(\)\))1891 3603 y Fn(\026)p Fk(\025)p Fo(0)2071 3589 y Fg(of)h(holomorphic)i(functions)e(which)h(satis\014es)f(the)75 3702 y(inte)-5 b(gr)g(o-di\013er)g(ential)48 b(r)-5 b(e)g(curr)g(enc)g (e)48 b(\(16\).)83 b(As)46 b(a)h(r)-5 b(esult,)50 b(for)d(given)e (holomorphic)50 b(functions)c Fp(V)3514 3716 y Fn(\026)3561 3702 y Fq(\()p Fp(t)p Fq(\))p Fg(,)k(for)75 3815 y Fq(0)26 b Fj(\024)f Fp(\026)g Fj(\024)g Fp(S)g Fj(\000)20 b Fq(1)33 b Fg(on)g Fp(G)21 b Fj([)e Fp(D)1045 3829 y Fn(\032)1086 3815 y Fg(,)32 b(ther)-5 b(e)34 b(exists)f(a)g(unique)e(formal)k (series)d(\(15\),)i(solution)g(of)f(\(14\).)216 4003 y Fq(The)41 b(next)g(result)g(giv)m(es)i(su\016cien)m(t)e(conditions)h (under)e(whic)m(h)g(there)i(exist)g(holomorphic)f(solutions)75 4116 y Fp(V)20 b Fq(\()p Fp(t;)15 b(z)t Fq(\))32 b(to)f(\(14\))h(whic)m (h)e(are)h(w)m(eigh)m(ted)g(with)g(resp)s(ect)f(to)h Fp(t)p Fq(.)75 4303 y Fh(Theorem)k(1)46 b Fg(Consider)33 b(the)g(p)-5 b(artial)35 b(inte)-5 b(gr)g(o-di\013er)g(ential)35 b(e)-5 b(quation)34 b(\(14\))158 4546 y Fp(@)211 4508 y Fn(S)206 4568 y(z)262 4546 y Fp(V)20 b Fq(\()p Fp(t;)15 b(z)t Fq(\))27 b(=)737 4460 y Fm(X)647 4661 y Fo(\()p Fn(k)711 4670 y Fl(0)746 4661 y Fn(;k)803 4670 y Fl(1)836 4661 y Fo(\))p Fk(2S)974 4546 y Fp(b)1013 4561 y Fn(k)1050 4570 y Fl(0)1084 4561 y Fn(;k)1141 4570 y Fl(1)1179 4546 y Fq(\()p Fp(t;)15 b(z)t Fq(\))p Fp(@)1421 4505 y Fn(k)1458 4514 y Fl(0)1416 4570 y Fn(t)1498 4546 y Fp(@)1551 4508 y Fn(k)1588 4517 y Fl(1)1546 4568 y Fn(z)1627 4546 y Fp(V)20 b Fq(\()p Fp(t;)15 b(z)t Fq(\)+)1082 4858 y Fp(b)1121 4872 y Fo(1)1161 4858 y Fq(\()p Fp(t;)g(z)t Fq(\))p Fp(@)1398 4872 y Fn(t)1444 4734 y Fm(Z)1535 4761 y Fn(t)1495 4941 y Fo(0)1580 4858 y Fp(V)20 b Fq(\()p Fp(t)g Fj(\000)g Fp(\034)5 b(;)15 b(z)t Fq(\))p Fp(V)22 b Fq(\()p Fp(\034)5 b(;)15 b(z)t Fq(\))p Fp(d\034)31 b Fq(+)20 b Fp(b)2522 4872 y Fo(2)2562 4858 y Fq(\()p Fp(t;)15 b(z)t Fq(\))p Fp(@)2799 4872 y Fn(t)2845 4734 y Fm(Z)2936 4761 y Fn(t)2896 4941 y Fo(0)2981 4858 y Fp(V)20 b Fq(\()p Fp(t)g Fj(\000)g Fp(\034)5 b(;)15 b(z)t Fq(\))p Fp(C)7 b Fq(\()p Fp(\034)e(;)15 b(z)t Fq(\))p Fp(d\034)75 5116 y Fg(wher)-5 b(e)34 b Fj(S)39 b Fg(is)33 b(a)g(\014nite)f(subset)h(of)g Fi(N)1280 5083 y Fo(2)1351 5116 y Fg(which)h(satis\014es)f(the)g(hyp)-5 b(othesis)35 b(of)e(Pr)-5 b(op)g(osition)36 b(2.)42 b(The)33 b(c)-5 b(o)g(e\016cients)75 5229 y Fp(b)114 5244 y Fn(k)151 5253 y Fl(0)185 5244 y Fn(;k)242 5253 y Fl(1)280 5229 y Fq(\()p Fp(t;)15 b(z)t Fq(\))p Fp(;)g(b)548 5243 y Fo(1)590 5229 y Fq(\()p Fp(t;)g(z)t Fq(\))p Fp(;)g(b)858 5243 y Fo(2)899 5229 y Fq(\()p Fp(t;)g(z)t Fq(\))33 b Fg(ar)-5 b(e)33 b(holomorphic)j(on)d(a)f(c)-5 b(ommon)34 b(domain)g Fq(\()p Fp(G)21 b Fj([)e Fp(D)2969 5243 y Fn(\032)3009 5229 y Fq(\))h Fj(\002)f Fp(D)3229 5243 y Fn(r)3300 5229 y Fg(wher)-5 b(e)33 b Fp(G)g Fg(is)f(an)75 5342 y(op)-5 b(en)31 b(se)-5 b(ctor)31 b(with)g(bise)-5 b(cting)30 b(dir)-5 b(e)g(ction)31 b Fp(d)26 b Fj(2)f Fi(R)k Fg(with)i(in\014nite)f(r)-5 b(adius)31 b(c)-5 b(enter)g(e)g(d)32 b(at)e(0)g(and)h Fp(D)3292 5356 y Fn(\032)3363 5342 y Fg(an)f(op)-5 b(en)31 b(disc)75 5455 y(c)-5 b(enter)g(e)g(d)31 b(at)g(0)g(with)g(r)-5 b(adius)31 b Fp(\032)p Fg(.)41 b(The)30 b(function)h Fp(C)7 b Fq(\()p Fp(t;)15 b(z)t Fq(\))30 b Fg(is)h(holomorphic)i(on)e Fq(\()p Fp(G)15 b Fj([)g Fp(D)3029 5469 y Fn(\032)3069 5455 y Fq(\))g Fj(\002)g Fp(D)3280 5469 y Fn(r)3348 5455 y Fg(and)31 b(weighte)-5 b(d)75 5568 y(with)33 b(r)-5 b(esp)g(e)g(ct)35 b(to)e Fp(t)f Fg(on)h Fq(\()p Fp(G)21 b Fj([)f Fp(D)1156 5582 y Fn(\032)1197 5568 y Fq(\))g Fj(\002)g Fp(D)1418 5582 y Fn(r)1456 5568 y Fg(.)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 3809 100 a Fq(9)75 399 y Fg(1\))33 b(Assume)g(that)g(the)g(quantity)141 671 y Fq(sup)75 749 y Fn(t)p Fk(2)p Fn(G)p Fk([)p Fn(D)307 757 y Fd(\032)369 608 y Fj(j)p Fp(@)447 567 y Fn(l)468 576 y Fl(0)442 632 y Fn(t)507 608 y Fp(b)546 623 y Fn(k)583 632 y Fl(0)617 623 y Fn(;k)674 632 y Fl(1)708 623 y Fn(;l)754 608 y Fq(\()p Fp(t)p Fq(\))p Fj(j)p 369 650 515 4 v 580 733 a Fp(l)607 747 y Fo(0)646 733 y Fq(!)918 671 y(=)25 b Fj(j)p Fp(b)p Fj(j)1103 686 y Fn(k)1140 695 y Fl(0)1175 686 y Fn(;k)1232 695 y Fl(1)1266 686 y Fn(;l)q(;l)1349 695 y Fl(0)1448 671 y Fp(;)143 b Fq(sup)1550 749 y Fn(t)p Fk(2)p Fn(G)p Fk([)p Fn(D)1782 757 y Fd(\032)1844 608 y Fj(j)p Fp(@)1922 567 y Fn(l)1943 576 y Fl(0)1917 632 y Fn(t)1982 608 y Fp(b)2021 623 y Fo(1)p Fn(;l)2102 608 y Fq(\()p Fp(t)p Fq(\))p Fj(j)p 1844 650 388 4 v 1992 733 a Fp(l)2019 747 y Fo(0)2058 733 y Fq(!)2266 671 y(=)25 b Fj(j)p Fp(b)p Fj(j)2451 686 y Fo(1)p Fn(;l)q(;l)2569 695 y Fl(0)2670 671 y Fp(;)143 b Fq(sup)2772 749 y Fn(t)p Fk(2)p Fn(G)p Fk([)p Fn(D)3004 757 y Fd(\032)3065 608 y Fj(j)p Fp(@)3143 567 y Fn(l)3164 576 y Fl(0)3138 632 y Fn(t)3204 608 y Fp(b)3243 623 y Fo(2)p Fn(;l)3324 608 y Fq(\()p Fp(t)p Fq(\))p Fj(j)p 3065 650 V 3213 733 a Fp(l)3240 747 y Fo(0)3280 733 y Fq(!)3488 671 y(=)25 b Fj(j)p Fp(b)p Fj(j)3673 686 y Fo(2)p Fn(;l)q(;l)3791 695 y Fl(0)3829 671 y Fp(;)75 939 y Fg(exists)33 b(and)h(that)75 1143 y Fj(j)p Fp(b)p Fj(j)164 1158 y Fn(k)201 1167 y Fl(0)236 1158 y Fn(;k)293 1167 y Fl(1)331 1143 y Fq(\()p Fp(t;)15 b(x)p Fq(\))26 b(=)676 1057 y Fm(X)648 1254 y Fn(l)669 1263 y Fl(0)704 1254 y Fn(;l)q Fk(\025)p Fo(0)850 1143 y Fj(j)p Fp(b)p Fj(j)939 1158 y Fn(k)976 1167 y Fl(0)1011 1158 y Fn(;k)1068 1167 y Fl(1)1102 1158 y Fn(;l)q(;l)1185 1167 y Fl(0)1223 1143 y Fp(t)1256 1105 y Fn(l)1277 1114 y Fl(0)1316 1143 y Fp(x)1368 1105 y Fn(l)1456 1143 y Fp(;)78 b Fj(j)p Fp(b)p Fj(j)1648 1157 y Fo(1)1688 1143 y Fq(\()p Fp(t;)15 b(x)p Fq(\))26 b(=)2033 1057 y Fm(X)2005 1254 y Fn(l)2026 1263 y Fl(0)2061 1254 y Fn(;l)q Fk(\025)p Fo(0)2208 1143 y Fj(j)p Fp(b)p Fj(j)2297 1158 y Fo(1)p Fn(;l)q(;l)2415 1167 y Fl(0)2454 1143 y Fp(t)2487 1105 y Fn(l)2508 1114 y Fl(0)2546 1143 y Fp(x)2598 1105 y Fn(l)2687 1143 y Fp(;)78 b Fj(j)p Fp(b)p Fj(j)2879 1157 y Fo(2)2919 1143 y Fq(\()p Fp(t;)15 b(x)p Fq(\))26 b(=)3264 1057 y Fm(X)3236 1254 y Fn(l)3257 1263 y Fl(0)3291 1254 y Fn(;l)q Fk(\025)p Fo(0)3438 1143 y Fj(j)p Fp(b)p Fj(j)3527 1158 y Fo(2)p Fn(;l)q(;l)3645 1167 y Fl(0)3684 1143 y Fp(t)3717 1105 y Fn(l)3738 1114 y Fl(0)3777 1143 y Fp(x)3829 1105 y Fn(l)75 1452 y Fg(ar)-5 b(e)36 b(holomorphic)j(functions)d(on)g (a)g(neighb)-5 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1950 y Fj(jj)p Fp(c)p Fj(jj)1661 1965 y Fn(l)q(;l)1724 1974 y Fl(0)1789 1950 y Fq(=)1895 1887 y Fj(jj)p Fp(@)1998 1846 y Fn(l)2019 1855 y Fl(0)1993 1911 y Fn(t)2059 1887 y Fp(C)2124 1902 y Fn(l)2150 1887 y Fq(\()p Fp(t)p Fq(\))p Fj(jj)2303 1902 y Fn(d;\016)p 1895 1929 503 4 v 2100 2012 a Fp(l)2127 2026 y Fo(0)2167 2012 y Fq(!)75 2188 y Fg(and)1400 2301 y Fj(jj)p Fp(C)7 b Fj(jj)p Fq(\()p Fp(t;)15 b(x)p Fq(\))27 b(=)1918 2215 y Fm(X)1890 2413 y Fn(l)1911 2422 y Fl(0)1945 2413 y Fn(;l)q Fk(\025)p Fo(0)2092 2301 y Fj(jj)p Fp(c)p Fj(jj)2231 2316 y Fn(l)q(;l)2294 2325 y Fl(0)2334 2301 y Fp(t)2367 2264 y Fn(l)2388 2273 y Fl(0)2426 2301 y Fp(x)2478 2264 y Fn(l)2504 2301 y Fp(:)75 2660 y Fg(2\))36 b(Assume)f(that)h Fp(S)f Fj(\025)29 b Fp(k)962 2674 y Fo(0)1024 2660 y Fq(+)22 b Fp(k)1164 2674 y Fo(1)1239 2660 y Fg(and)36 b Fp(S)f(>)29 b(k)1656 2674 y Fo(1)1731 2660 y Fg(for)36 b(al)5 b(l)35 b Fq(\()p Fp(k)2093 2674 y Fo(0)2134 2660 y Fp(;)15 b(k)2221 2674 y Fo(1)2261 2660 y Fq(\))30 b Fj(2)f(S)7 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Fn(t)1462 3269 y Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\)\))22 b(+)e Fj(j)p Fp(b)p Fj(j)1966 3283 y Fo(2)2005 3269 y Fq(\()p Fp(t;)15 b(x)p Fq(\)\()p Fj(jj)p Fp(C)7 b Fj(jj)p Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))24 b(+)c Fj(jj)p Fp(C)7 b Fj(jj)p Fq(\(0)p Fp(;)15 b(x)p Fq(\))p Fp(@)3411 3283 y Fn(t)3443 3269 y Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\)\))p Fp(:)75 3474 y Fg(Assume)32 b(that)360 3678 y Fj(jjj)p Fp(b)p Fj(j)499 3693 y Fn(k)536 3702 y Fl(0)571 3693 y Fn(;k)628 3702 y Fl(1)666 3678 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)911 3696 y Fo(\()p Fn(T)979 3705 y Fl(0)1016 3696 y Fn(;X)1094 3705 y Fl(0)1128 3696 y Fo(\))1185 3678 y Fj(\024)25 b Fp(\016)n(;)81 b Fj(jjj)p Fp(b)p Fj(j)1564 3692 y Fo(1)1605 3678 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)1850 3696 y Fo(\()p Fn(T)1918 3705 y Fl(0)1954 3696 y Fn(;X)2032 3705 y Fl(0)2066 3696 y Fo(\))2123 3678 y Fj(\024)25 b Fp(\016)n(;)81 b Fj(jjj)p Fp(b)p Fj(j)2502 3692 y Fo(2)2543 3678 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)p Fp(C)7 b Fj(jj)p Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fj(jj)3155 3696 y Fo(\()p Fn(T)3223 3705 y Fl(0)3261 3696 y Fn(;X)3339 3705 y Fl(0)3373 3696 y Fo(\))3430 3678 y Fj(\024)25 b Fp(\016)75 3882 y Fg(for)33 b(the)g(r)-5 b(e)g(al)34 b(numb)-5 b(er)33 b Fp(\016)c(>)c Fq(0)33 b Fg(c)-5 b(onstructe)g(d)35 b(fr)-5 b(om)34 b(The)-5 b(or)g(em)34 b(\(CK\))f(applie)-5 b(d)35 b(to)e(the)g(e)-5 b(quation)33 b(\(17\).)75 4045 y(3\))38 b(We)f(make)g(the)h(fol)5 b(lowing)38 b(assumption)h(on)f(initial)g(c)-5 b(onditions.)57 b(L)-5 b(et)38 b Fp(V)2743 4059 y Fn(\026)2789 4045 y Fq(\()p Fp(t)p Fq(\))p Fg(,)h(for)f Fq(0)c Fj(\024)f Fp(\026)h Fj(\024)f Fp(S)c Fj(\000)23 b Fq(1)p Fg(,)39 b(b)-5 b(e)75 4158 y(given)32 b(holomorphic)k(functions)d(which)g(ar)-5 b(e)34 b(weighte)-5 b(d)34 b(with)f(r)-5 b(esp)g(e)g(ct)34 b(to)f Fp(t)g Fg(on)g Fp(G)21 b Fj([)e Fp(D)3044 4172 y Fn(\032)3085 4158 y Fg(.)75 4321 y(Then,)33 b(the)g(formal)h(series) 1540 4476 y Fp(V)20 b Fq(\()p Fp(t;)15 b(z)t Fq(\))27 b(=)1926 4390 y Fm(X)1925 4586 y Fn(n)p Fk(\025)p Fo(0)2083 4415 y Fp(V)2136 4429 y Fn(n)2183 4415 y Fq(\()p Fp(t)p Fq(\))p 2083 4455 204 4 v 2145 4539 a Fp(n)p Fq(!)2296 4476 y Fp(z)2342 4439 y Fn(n)75 4741 y Fg(solution)38 b(of)e(\(14\))i(with)f(initial)g(c)-5 b(onditions)38 b Fq(\()p Fp(@)1735 4697 y Fn(\026)1730 4753 y(z)1782 4741 y Fp(V)20 b Fq(\)\()p Fp(t;)15 b Fq(0\))34 b(=)d Fp(V)2267 4755 y Fn(\026)2314 4741 y Fq(\()p Fp(t)p Fq(\))p Fg(,)38 b Fq(0)32 b Fj(\024)g Fp(\026)f Fj(\024)h Fp(S)c Fj(\000)22 b Fq(1)p Fg(,)38 b(c)-5 b(onstructe)g(d)38 b(in)e(the)75 4854 y(pr)-5 b(op)g(osition)36 b(2)d(is)g(c)-5 b(onver)g(gent)33 b(for)h(al)5 b(l)33 b Fp(z)c Fj(2)c Fp(D)1677 4868 y Fn(r)1748 4854 y Fg(for)33 b Fp(r)i Fg(smal)5 b(l)34 b(enough.)42 b(Mor)-5 b(e)g(over,)34 b(the)f(function)f Fp(V)21 b Fq(\()p Fp(t;)15 b(z)t Fq(\))33 b Fg(is)75 4967 y(weighte)-5 b(d)34 b(with)f(r)-5 b(esp)g(e)g(ct)34 b(to)g Fp(t)e Fg(on)h Fq(\()p Fp(G)21 b Fj([)f Fp(D)1521 4981 y Fn(\032)1561 4967 y Fq(\))h Fj(\002)f Fp(D)1783 4981 y Fn(r)1821 4967 y Fg(.)75 5204 y Fh(Pro)s(of)40 b Fq(In)f(the)h(\014rst)e(part)i(of)f(the)h(pro)s(of,)h(w)m(e)f(sho)m (w)f(that)h(the)g(functions)f Fp(V)2815 5218 y Fn(n)2862 5204 y Fq(\()p Fp(t)p Fq(\))h(constructed)f(from)g(the)75 5317 y(prop)s(osition)e(2)g(satisfy)h(the)f(follo)m(wing)i(estimates:) 56 b(F)-8 b(or)38 b(all)g(closed)g(sector)g(of)f(in\014nite)g(radius) 3505 5294 y(\026)3490 5317 y Fp(S)3546 5332 y Fn(d;\016)3676 5317 y Fj(\032)f Fp(G)75 5430 y Fq(cen)m(tered)29 b(at)g(0)f(with)g (bisecting)h(direction)g Fp(d)p Fq(,)g(there)f(exist)h Fp(C)q(;)15 b(X)2300 5444 y Fo(0)2340 5430 y Fp(;)g(T)2433 5444 y Fo(0)2498 5430 y Fp(>)25 b Fq(0,)30 b(indep)s(enden)m(t)c(of)j Fp(n;)15 b(l)r Fq(,)28 b(suc)m(h)g(that)75 5634 y(\(18\))935 b Fj(jj)p Fp(@)1273 5597 y Fn(l)1268 5657 y(t)1300 5634 y Fp(V)1353 5648 y Fn(n)1400 5634 y Fq(\()p Fp(t)p Fq(\))p Fj(jj)1553 5649 y Fn(d;\016)1673 5634 y Fj(\024)25 b Fp(C)7 b Fq(\(1)p Fp(=X)2041 5648 y Fo(0)2081 5634 y Fq(\))2116 5597 y Fn(n)2164 5634 y Fq(\(1)p Fp(=T)2342 5648 y Fo(0)2382 5634 y Fq(\))2417 5597 y Fn(l)2444 5634 y Fq(\()p Fp(l)22 b Fq(+)e Fp(n)p Fq(\)!)p Fp(;)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 3764 100 a Fq(10)75 399 y(for)33 b(all)h Fp(l)r(;)15 b(n)29 b Fj(\025)h Fq(0.)49 b(F)-8 b(rom)34 b(\(18\),)h(w)m(e)f(deduce)e(that)i Fp(V)20 b Fq(\()p Fp(t;)15 b(z)t Fq(\))34 b(con)m(v)m(erges)h(for)e(all)h Fp(z)g Fj(2)c Fp(D)3042 413 y Fn(r)3113 399 y Fq(for)j Fp(r)i Fq(small)f(enough)75 511 y(and)c(is)g(w)m(eigh)m(ted)i(with)e (resp)s(ect)g(to)i Fp(t)e Fq(on)g(\()p Fp(G)21 b Fj([)f Fp(D)1824 525 y Fn(\032)1864 511 y Fq(\))h Fj(\002)f Fp(D)2086 525 y Fn(r)2124 511 y Fq(.)75 664 y(W)-8 b(e)32 b(consider)e(no)m(w)g(the)h(sequence)g(of)f(quan)m(tities)1466 837 y Fp(w)1531 851 y Fn(n)1574 860 y Fl(0)1609 851 y Fn(;m)1691 860 y Fl(1)1755 837 y Fq(=)25 b Fj(jj)p Fp(@)1954 797 y Fn(n)1997 806 y Fl(0)1949 861 y Fn(t)2036 837 y Fp(V)2089 851 y Fn(m)2151 860 y Fl(1)2190 837 y Fq(\()p Fp(t)p Fq(\))p Fj(jj)2343 852 y Fn(d;\016)2438 837 y Fp(;)75 1010 y Fq(for)36 b Fp(n)275 1024 y Fo(0)314 1010 y Fp(;)15 b(m)434 1024 y Fo(1)508 1010 y Fj(\025)35 b Fq(0.)58 b(W)-8 b(e)37 b(sho)m(w)f(that)g(this)g(sequence)h(is)f(in)f (fact)i(real)g(v)-5 b(alued)36 b(and)g(satis\014es)g(a)g(m)m(ultiv)-5 b(ariate)75 1123 y(recursion)30 b(inequalit)m(y)-8 b(.)216 1236 y(F)g(rom)31 b(the)g(Leibniz)f(form)m(ula,)h(w)m(e)g(ha)m(v)m(e)g (that)525 1471 y Fp(@)578 1430 y Fn(n)621 1439 y Fl(0)573 1495 y Fn(t)660 1471 y Fq(\()p Fp(b)734 1486 y Fn(k)771 1495 y Fl(0)806 1486 y Fn(;k)863 1495 y Fl(1)896 1486 y Fn(;l)942 1471 y Fq(\()p Fp(t)p Fq(\))1055 1409 y Fp(@)1108 1369 y Fn(k)1145 1378 y Fl(0)1103 1433 y Fn(t)1184 1409 y Fp(V)1237 1424 y Fn(n)p Fo(+)p Fn(k)1372 1433 y Fl(1)1411 1409 y Fq(\()p Fp(t)p Fq(\))p 1055 1450 459 4 v 1245 1533 a Fp(n)p Fq(!)1524 1471 y(\))26 b(=)1784 1384 y Fm(X)1681 1582 y Fn(h)1722 1591 y Fl(1)1756 1582 y Fo(+)p Fn(h)1852 1591 y Fl(2)1886 1582 y Fo(=)p Fn(n)1984 1591 y Fl(0)2101 1409 y Fp(n)2156 1423 y Fo(0)2195 1409 y Fq(!)p 2043 1450 235 4 v 2043 1533 a Fp(h)2095 1547 y Fo(2)2135 1533 y Fq(!)p Fp(h)2212 1547 y Fo(1)2252 1533 y Fq(!)2288 1471 y Fp(@)2341 1430 y Fn(h)2382 1439 y Fl(2)2336 1495 y Fn(t)2420 1471 y Fp(b)2459 1486 y Fn(k)2496 1495 y Fl(0)2531 1486 y Fn(;k)2588 1495 y Fl(1)2622 1486 y Fn(;l)2667 1471 y Fq(\()p Fp(t)p Fq(\))2780 1409 y Fp(@)2833 1369 y Fn(h)2874 1378 y Fl(1)2909 1369 y Fo(+)p Fn(k)3001 1378 y Fl(0)2828 1433 y Fn(t)3040 1409 y Fp(V)3093 1424 y Fn(n)p Fo(+)p Fn(k)3228 1433 y Fl(1)3266 1409 y Fq(\()p Fp(t)p Fq(\))p 2780 1450 589 4 v 3035 1533 a Fp(n)p Fq(!)3379 1471 y Fp(:)75 1737 y Fq(F)-8 b(rom)31 b(the)f(assumption)g(1\),)i(w)m(e)e(deduce)408 1978 y Fj(jj)p Fp(@)511 1938 y Fn(n)554 1947 y Fl(0)506 2002 y Fn(t)593 1978 y Fq(\()p Fp(b)667 1993 y Fn(k)704 2002 y Fl(0)739 1993 y Fn(;k)796 2002 y Fl(1)830 1993 y Fn(;l)875 1978 y Fq(\()p Fp(t)p Fq(\))988 1917 y Fp(@)1041 1876 y Fn(k)1078 1885 y Fl(0)1036 1941 y Fn(t)1118 1917 y Fp(V)1171 1932 y Fn(n)p Fo(+)p Fn(k)1306 1941 y Fl(1)1344 1917 y Fq(\()p Fp(t)p Fq(\))p 988 1957 459 4 v 1178 2041 a Fp(n)p Fq(!)1457 1978 y(\))p Fj(jj)1542 1993 y Fn(d;\016)1662 1978 y Fj(\024)1861 1892 y Fm(X)1758 2090 y Fn(h)1799 2099 y Fl(1)1833 2090 y Fo(+)p Fn(h)1929 2099 y Fl(2)1964 2090 y Fo(=)p Fn(n)2062 2099 y Fl(0)2111 1978 y Fp(n)2166 1992 y Fo(0)2205 1978 y Fq(!)p Fj(j)p Fp(b)p Fj(j)2319 1993 y Fn(k)2356 2002 y Fl(0)2391 1993 y Fn(;k)2448 2002 y Fl(1)2482 1993 y Fn(;l)q(;h)2585 2002 y Fl(2)2632 1915 y Fj(jj)p Fq(\()p Fp(@)2770 1875 y Fn(h)2811 1884 y Fl(1)2847 1875 y Fo(+)p Fn(k)2939 1884 y Fl(0)2765 1939 y Fn(t)2977 1915 y Fp(V)3030 1930 y Fn(n)p Fo(+)p Fn(k)3165 1939 y Fl(1)3204 1915 y Fq(\)\()p Fp(t)p Fq(\))p Fj(jj)3392 1930 y Fn(d;\016)p 2632 1957 855 4 v 2961 2041 a Fp(h)3013 2055 y Fo(1)3053 2041 y Fq(!)p Fp(n)p Fq(!)3497 1978 y Fp(:)216 2240 y Fq(On)g(the)g(other)h(hand,)e(w)m(e)i(ha)m(v)m(e)101 2481 y Fp(@)154 2441 y Fn(n)197 2450 y Fl(0)149 2505 y Fn(t)236 2481 y Fq(\()281 2419 y Fp(b)320 2433 y Fo(1)p Fn(;q)407 2442 y Fl(3)445 2419 y Fq(\()p Fp(t)p Fq(\))p 281 2460 268 4 v 310 2543 a Fp(q)351 2557 y Fo(1)390 2543 y Fq(!)p Fp(q)456 2557 y Fo(2)495 2543 y Fq(!)559 2481 y Fp(@)607 2495 y Fn(t)652 2357 y Fm(Z)743 2384 y Fn(t)702 2564 y Fo(0)787 2481 y Fp(V)840 2495 y Fn(q)872 2504 y Fl(1)911 2481 y Fq(\()p Fp(t)20 b Fj(\000)g Fp(\034)10 b Fq(\))p Fp(V)1228 2495 y Fn(q)1260 2504 y Fl(2)1299 2481 y Fq(\()p Fp(\034)g Fq(\))p Fp(d\034)g Fq(\))26 b(=)1776 2395 y Fm(X)1673 2592 y Fn(h)1714 2601 y Fl(1)1748 2592 y Fo(+)p Fn(h)1844 2601 y Fl(2)1879 2592 y Fo(=)p Fn(n)1977 2601 y Fl(0)2093 2420 y Fp(n)2148 2434 y Fo(0)2187 2420 y Fq(!)p 2036 2460 235 4 v 2036 2543 a Fp(h)2088 2557 y Fo(2)2128 2543 y Fq(!)p Fp(h)2205 2557 y Fo(1)2245 2543 y Fq(!)2280 2481 y Fp(@)2333 2441 y Fn(h)2374 2450 y Fl(2)2328 2505 y Fn(t)2413 2481 y Fp(b)2452 2495 y Fo(1)p Fn(;q)2539 2504 y Fl(3)2577 2481 y Fq(\()p Fp(t)p Fq(\))2690 2414 y Fp(@)2743 2373 y Fn(h)2784 2382 y Fl(1)2819 2373 y Fo(+1)2738 2438 y Fn(t)2913 2414 y Fq(\()2948 2341 y Fm(R)3009 2367 y Fn(t)2992 2446 y Fo(0)3054 2414 y Fp(V)3107 2428 y Fn(q)3139 2437 y Fl(1)3177 2414 y Fq(\()p Fp(t)21 b Fj(\000)f Fp(\034)10 b Fq(\))p Fp(V)3495 2428 y Fn(q)3527 2437 y Fl(2)3565 2414 y Fq(\()p Fp(\034)g Fq(\))p Fp(d\034)g Fq(\))p 2690 2460 1129 4 v 3149 2543 a Fp(q)3190 2557 y Fo(1)3229 2543 y Fq(!)p Fp(q)3295 2557 y Fo(2)3335 2543 y Fq(!)75 2748 y(F)-8 b(rom)31 b(the)f(assumption)g(1\),)i(w)m(e)e(deduce)158 3002 y Fj(jj)p Fp(@)261 2962 y Fn(n)304 2971 y Fl(0)256 3026 y Fn(t)343 3002 y Fq(\()388 2940 y Fp(b)427 2954 y Fo(1)p Fn(;q)514 2963 y Fl(3)553 2940 y Fq(\()p Fp(t)p Fq(\))p 388 2981 268 4 v 417 3064 a Fp(q)458 3078 y Fo(1)497 3064 y Fq(!)p Fp(q)563 3078 y Fo(2)603 3064 y Fq(!)666 3002 y Fp(@)714 3016 y Fn(t)759 2878 y Fm(Z)850 2905 y Fn(t)810 3085 y Fo(0)895 3002 y Fp(V)948 3016 y Fn(q)980 3025 y Fl(1)1018 3002 y Fq(\()p Fp(t)21 b Fj(\000)f Fp(\034)10 b Fq(\))p Fp(V)1336 3016 y Fn(q)1368 3025 y Fl(2)1406 3002 y Fq(\()p Fp(\034)g Fq(\))p Fp(d\034)g Fq(\))p Fj(jj)1708 3017 y Fn(d;\016)1829 3002 y Fj(\024)1725 3187 y Fm(X)1622 3384 y Fn(h)1663 3393 y Fl(1)1697 3384 y Fo(+)p Fn(h)1793 3393 y Fl(2)1827 3384 y Fo(=)p Fn(n)1925 3393 y Fl(0)1974 3273 y Fp(n)2029 3287 y Fo(0)2068 3273 y Fq(!)p Fj(j)p Fp(b)p Fj(j)2182 3288 y Fo(1)p Fn(;q)2269 3297 y Fl(3)2304 3288 y Fn(;h)2365 3297 y Fl(2)2413 3206 y Fj(jj)p Fp(@)2516 3165 y Fn(h)2557 3174 y Fl(1)2592 3165 y Fo(+1)2511 3230 y Fn(t)2687 3206 y Fq(\()2722 3132 y Fm(R)2783 3159 y Fn(t)2765 3237 y Fo(0)2827 3206 y Fp(V)2880 3220 y Fn(q)2912 3229 y Fl(1)2951 3206 y Fq(\()p Fp(t)20 b Fj(\000)g Fp(\034)10 b Fq(\))p Fp(V)3268 3220 y Fn(q)3300 3229 y Fl(2)3339 3206 y Fq(\()p Fp(\034)g Fq(\))p Fp(d\034)g Fq(\))p Fj(jj)3641 3221 y Fn(d;\016)p 2413 3252 1324 4 v 2911 3335 a Fp(h)2963 3349 y Fo(1)3003 3335 y Fq(!)p Fp(q)3069 3349 y Fo(1)3108 3335 y Fq(!)p Fp(q)3174 3349 y Fo(2)3213 3335 y Fq(!)3746 3273 y Fp(:)75 3534 y Fq(W)-8 b(e)32 b(ha)m(v)m(e)158 3741 y Fp(@)211 3701 y Fn(h)252 3710 y Fl(1)286 3701 y Fo(+1)206 3765 y Fn(t)381 3741 y Fq(\()416 3618 y Fm(Z)507 3644 y Fn(t)466 3824 y Fo(0)552 3741 y Fp(V)605 3755 y Fn(q)637 3764 y Fl(1)675 3741 y Fq(\()p Fp(t)20 b Fj(\000)g Fp(\034)10 b Fq(\))p Fp(V)992 3755 y Fn(q)1024 3764 y Fl(2)1063 3741 y Fq(\()p Fp(\034)g Fq(\))p Fp(d\034)g Fq(\))27 b(=)1520 3655 y Fm(X)1438 3853 y Fn(l)1459 3862 y Fl(1)1493 3853 y Fo(+)p Fn(l)1569 3862 y Fl(2)1604 3853 y Fo(=)p Fn(h)1700 3862 y Fl(1)1734 3741 y Fq(\()p Fp(@)1822 3701 y Fn(l)1843 3710 y Fl(1)1817 3765 y Fn(t)1882 3741 y Fp(V)1935 3755 y Fn(q)1967 3764 y Fl(1)2006 3741 y Fq(\)\(0\))p Fp(@)2209 3701 y Fn(l)2230 3710 y Fl(2)2204 3765 y Fn(t)2270 3741 y Fp(V)2323 3755 y Fn(q)2355 3764 y Fl(2)2394 3741 y Fq(\()p Fp(t)p Fq(\))20 b(+)2608 3618 y Fm(Z)2699 3644 y Fn(t)2659 3824 y Fo(0)2729 3741 y Fq(\()p Fp(@)2817 3701 y Fn(h)2858 3710 y Fl(1)2893 3701 y Fo(+1)2812 3765 y Fn(t)2987 3741 y Fp(V)3040 3755 y Fn(q)3072 3764 y Fl(1)3110 3741 y Fq(\)\()p Fp(t)h Fj(\000)f Fp(\034)10 b Fq(\))p Fp(V)3463 3755 y Fn(q)3495 3764 y Fl(2)3534 3741 y Fq(\()p Fp(\034)g Fq(\))p Fp(d\034)5 b(:)75 4003 y Fq(F)-8 b(rom)31 b(the)f(inequalities)1121 4153 y Fj(j)p Fq(\()p Fp(@)1234 4113 y Fn(l)1255 4122 y Fl(1)1229 4177 y Fn(t)1295 4153 y Fp(V)1348 4167 y Fn(q)1380 4176 y Fl(1)1418 4153 y Fq(\)\(0\))p Fj(j)d(\024)e(jj)p Fp(@)1819 4113 y Fn(l)1840 4122 y Fl(1)1814 4177 y Fn(t)1880 4153 y Fp(V)1933 4167 y Fn(q)1965 4176 y Fl(1)2003 4153 y Fq(\()p Fp(t)p Fq(\))p Fj(jj)2156 4168 y Fn(d;\016)2312 4153 y Fp(;)76 b Fq(1)25 b Fj(\024)2623 4092 y Fp(h)2675 4106 y Fo(1)2714 4092 y Fq(!)p 2589 4132 184 4 v 2589 4216 a Fp(l)2616 4230 y Fo(1)2656 4216 y Fq(!)p Fp(l)2708 4230 y Fo(2)2748 4216 y Fq(!)2783 4153 y Fp(;)75 4339 y Fq(for)30 b Fp(l)241 4353 y Fo(1)281 4339 y Fp(;)15 b(l)348 4353 y Fo(2)413 4339 y Fj(\025)25 b Fq(0)31 b(with)f Fp(l)819 4353 y Fo(1)878 4339 y Fq(+)20 b Fp(l)996 4353 y Fo(2)1061 4339 y Fq(=)25 b Fp(h)1209 4353 y Fo(1)1249 4339 y Fq(,)31 b(and)e(the)i(lemma)g(4,)g(w)m(e)g(get)158 4589 y Fj(jj)p Fp(@)261 4548 y Fn(h)302 4557 y Fl(1)337 4548 y Fo(+1)256 4613 y Fn(t)431 4589 y Fq(\()466 4465 y Fm(Z)558 4491 y Fn(t)517 4671 y Fo(0)602 4589 y Fp(V)655 4603 y Fn(q)687 4612 y Fl(1)726 4589 y Fq(\()p Fp(t)20 b Fj(\000)g Fp(\034)10 b Fq(\))p Fp(V)1043 4603 y Fn(q)1075 4612 y Fl(2)1114 4589 y Fq(\()p Fp(\034)g Fq(\))p Fp(d\034)g Fq(\))p Fj(jj)1416 4604 y Fn(d;\016)1537 4589 y Fj(\024)939 4829 y Fq(\()1057 4743 y Fm(X)974 4940 y Fn(l)995 4949 y Fl(1)1030 4940 y Fo(+)p Fn(l)1106 4949 y Fl(2)1141 4940 y Fo(=)p Fn(h)1237 4949 y Fl(1)1329 4767 y Fp(h)1381 4781 y Fo(1)1421 4767 y Fq(!)p 1296 4808 V 1296 4891 a Fp(l)1323 4905 y Fo(1)1362 4891 y Fq(!)p Fp(l)1414 4905 y Fo(2)1454 4891 y Fq(!)1489 4829 y Fj(jj)p Fp(@)1592 4788 y Fn(l)1613 4797 y Fl(1)1587 4853 y Fn(t)1653 4829 y Fp(V)1706 4843 y Fn(q)1738 4852 y Fl(1)1776 4829 y Fq(\()p Fp(t)p Fq(\))p Fj(jj)1929 4844 y Fn(d;\016)2024 4829 y Fj(jj)p Fp(@)2127 4788 y Fn(l)2148 4797 y Fl(2)2122 4853 y Fn(t)2188 4829 y Fp(V)2241 4843 y Fn(q)2273 4852 y Fl(2)2311 4829 y Fq(\()p Fp(t)p Fq(\))p Fj(jj)2464 4844 y Fn(d;\016)2559 4829 y Fq(\))20 b(+)g Fj(jj)p Fp(@)2808 4788 y Fn(h)2849 4797 y Fl(1)2884 4788 y Fo(+1)2803 4853 y Fn(t)2979 4829 y Fp(V)3032 4843 y Fn(q)3064 4852 y Fl(1)3102 4829 y Fq(\()p Fp(t)p Fq(\))p Fj(jj)3255 4844 y Fn(d;\016)3350 4829 y Fj(jj)p Fp(V)3453 4843 y Fn(q)3485 4852 y Fl(2)3524 4829 y Fq(\()p Fp(t)p Fq(\))p Fj(jj)3677 4844 y Fn(d;\016)75 5090 y Fq(In)30 b(the)g(same)h(w)m(a)m(y)g(as)g(ab) s(o)m(v)m(e,)h(one)e(can)h(c)m(hec)m(k)h(that)158 5340 y Fj(jj)p Fp(@)261 5299 y Fn(n)304 5308 y Fl(0)256 5364 y Fn(t)343 5340 y Fq(\()388 5278 y Fp(b)427 5292 y Fo(2)p Fn(;q)514 5301 y Fl(3)553 5278 y Fq(\()p Fp(t)p Fq(\))p 388 5319 268 4 v 470 5402 a Fp(q)511 5416 y Fo(1)550 5402 y Fq(!)666 5340 y Fp(@)714 5354 y Fn(t)759 5216 y Fm(Z)850 5242 y Fn(t)810 5422 y Fo(0)895 5340 y Fp(V)948 5354 y Fn(q)980 5363 y Fl(1)1018 5340 y Fq(\()p Fp(t)21 b Fj(\000)f Fp(\034)10 b Fq(\))p Fp(C)1348 5354 y Fn(q)1380 5363 y Fl(2)1418 5340 y Fq(\()p Fp(\034)g Fq(\))p Fp(d\034)g Fq(\))p Fj(jj)1720 5355 y Fn(d;\016)1841 5340 y Fj(\024)1713 5524 y Fm(X)1610 5722 y Fn(h)1651 5731 y Fl(1)1685 5722 y Fo(+)p Fn(h)1781 5731 y Fl(2)1815 5722 y Fo(=)p Fn(n)1913 5731 y Fl(0)1962 5611 y Fp(n)2017 5625 y Fo(0)2057 5611 y Fq(!)p Fj(j)p Fp(b)p Fj(j)2171 5626 y Fo(2)p Fn(;q)2258 5635 y Fl(3)2292 5626 y Fn(;h)2353 5635 y Fl(2)2401 5543 y Fj(jj)p Fp(@)2504 5503 y Fn(h)2545 5512 y Fl(1)2580 5503 y Fo(+1)2499 5567 y Fn(t)2675 5543 y Fq(\()2710 5470 y Fm(R)2771 5496 y Fn(t)2753 5575 y Fo(0)2815 5543 y Fp(V)2868 5557 y Fn(q)2900 5566 y Fl(1)2939 5543 y Fq(\()p Fp(t)20 b Fj(\000)g Fp(\034)10 b Fq(\))p Fp(C)3268 5557 y Fn(q)3300 5566 y Fl(2)3339 5543 y Fq(\()p Fp(\034)g Fq(\))p Fp(d\034)g Fq(\))p Fj(jj)3641 5558 y Fn(d;\016)p 2401 5590 1336 4 v 2958 5673 a Fp(h)3010 5687 y Fo(1)3049 5673 y Fq(!)p Fp(q)3115 5687 y Fo(1)3155 5673 y Fq(!)3746 5611 y Fp(:)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 3764 100 a Fq(11)75 399 y(and)30 b(that)158 652 y Fj(jj)p Fp(@)261 612 y Fn(h)302 621 y Fl(1)337 612 y Fo(+1)256 676 y Fn(t)431 652 y Fq(\()466 528 y Fm(Z)558 555 y Fn(t)517 734 y Fo(0)602 652 y Fp(V)655 666 y Fn(q)687 675 y Fl(1)726 652 y Fq(\()p Fp(t)20 b Fj(\000)g Fp(\034)10 b Fq(\))p Fp(C)1055 666 y Fn(q)1087 675 y Fl(2)1126 652 y Fq(\()p Fp(\034)g Fq(\))p Fp(d\034)g Fq(\))p Fj(jj)1428 667 y Fn(d;\016)1549 652 y Fj(\024)915 892 y Fq(\()1033 806 y Fm(X)950 1004 y Fn(l)971 1013 y Fl(1)1006 1004 y Fo(+)p Fn(l)1082 1013 y Fl(2)1117 1004 y Fo(=)p Fn(h)1213 1013 y Fl(1)1305 831 y Fp(h)1357 845 y Fo(1)1397 831 y Fq(!)p 1272 871 184 4 v 1272 955 a Fp(l)1299 969 y Fo(1)1339 955 y Fq(!)p Fp(l)1391 969 y Fo(2)1430 955 y Fq(!)1466 892 y Fj(jj)p Fp(@)1569 852 y Fn(l)1590 861 y Fl(1)1564 916 y Fn(t)1629 892 y Fp(V)1682 906 y Fn(q)1714 915 y Fl(1)1752 892 y Fq(\()p Fp(t)p Fq(\))p Fj(jj)1905 907 y Fn(d;\016)2000 892 y Fj(jj)p Fp(@)2103 852 y Fn(l)2124 861 y Fl(2)2098 916 y Fn(t)2164 892 y Fp(C)2229 906 y Fn(q)2261 915 y Fl(2)2299 892 y Fq(\()p Fp(t)p Fq(\))p Fj(jj)2452 907 y Fn(d;\016)2547 892 y Fq(\))21 b(+)e Fj(jj)p Fp(@)2796 852 y Fn(h)2837 861 y Fl(1)2873 852 y Fo(+1)2791 916 y Fn(t)2967 892 y Fp(V)3020 906 y Fn(q)3052 915 y Fl(1)3090 892 y Fq(\()p Fp(t)p Fq(\))p Fj(jj)3243 907 y Fn(d;\016)3338 892 y Fj(jj)p Fp(C)3453 906 y Fn(q)3485 915 y Fl(2)3524 892 y Fq(\()p Fp(t)p Fq(\))p Fj(jj)3677 907 y Fn(d;\016)216 1175 y Fq(Finally)-8 b(,)32 b(from)e(the)h(latter)g(estimates,)i(w)m(e) d(get)i(the)f(follo)m(wing)g(inequalities)h(for)e(the)h(n)m(um)m(b)s (ers)e Fp(w)3619 1189 y Fn(n)3662 1198 y Fl(0)3696 1189 y Fn(;n)3759 1198 y Fl(1)3797 1175 y Fq(,)75 1394 y(\(19\))1200 1331 y Fp(w)1265 1345 y Fn(n)1308 1354 y Fl(0)1342 1345 y Fn(;m)1424 1354 y Fl(1)1459 1345 y Fo(+)p Fn(S)p 1200 1373 365 4 v 1250 1456 a Fp(n)1305 1470 y Fo(0)1344 1456 y Fq(!)p Fp(m)1449 1470 y Fo(1)1489 1456 y Fq(!)1600 1394 y Fj(\024)c Fp(A)1764 1408 y Fn(n)1807 1417 y Fl(0)1841 1408 y Fn(;m)1923 1417 y Fl(1)1982 1394 y Fq(+)20 b Fp(B)2142 1408 y Fn(n)2185 1417 y Fl(0)2219 1408 y Fn(;m)2301 1417 y Fl(1)2360 1394 y Fq(+)g Fp(C)2516 1408 y Fn(n)2559 1417 y Fl(0)2594 1408 y Fn(;m)2676 1417 y Fl(1)2715 1394 y Fp(;)75 1623 y Fq(where)839 1736 y Fp(A)907 1750 y Fn(n)950 1759 y Fl(0)985 1750 y Fn(;m)1067 1759 y Fl(1)1131 1736 y Fq(=)1317 1650 y Fm(X)1227 1851 y Fo(\()p Fn(k)1291 1860 y Fl(0)1326 1851 y Fn(;k)1383 1860 y Fl(1)1417 1851 y Fo(\))p Fk(2S)1624 1650 y Fm(X)1554 1848 y Fn(l)q Fo(+)p Fn(n)p Fo(=)p Fn(m)1791 1857 y Fl(1)1944 1650 y Fm(X)1841 1848 y Fn(h)1882 1857 y Fl(1)1916 1848 y Fo(+)p Fn(h)2012 1857 y Fl(2)2046 1848 y Fo(=)p Fn(n)2144 1857 y Fl(0)2193 1736 y Fj(j)p Fp(b)p Fj(j)2282 1751 y Fn(k)2319 1760 y Fl(0)2354 1751 y Fn(;k)2411 1760 y Fl(1)2445 1751 y Fn(;l)q(;h)2548 1760 y Fl(2)2596 1673 y Fp(w)2661 1688 y Fn(h)2702 1697 y Fl(1)2736 1688 y Fo(+)p Fn(k)2828 1697 y Fl(0)2862 1688 y Fn(;n)p Fo(+)p Fn(k)3017 1697 y Fl(1)p 2596 1715 460 4 v 2727 1799 a Fp(h)2779 1813 y Fo(1)2819 1799 y Fq(!)p Fp(n)p Fq(!)3065 1736 y Fp(;)75 1998 y Fq(and)457 2111 y Fp(B)526 2125 y Fn(n)569 2134 y Fl(0)603 2125 y Fn(;m)685 2134 y Fl(1)749 2111 y Fq(=)1009 2024 y Fm(X)845 2216 y Fn(q)877 2225 y Fl(1)911 2216 y Fo(+)p Fn(q)998 2225 y Fl(2)1032 2216 y Fo(+)p Fn(q)1119 2225 y Fl(3)1153 2216 y Fo(=)p Fn(m)1270 2225 y Fl(1)1423 2024 y Fm(X)1320 2222 y Fn(h)1361 2231 y Fl(1)1395 2222 y Fo(+)p Fn(h)1491 2231 y Fl(2)1526 2222 y Fo(=)p Fn(n)1624 2231 y Fl(0)1673 2111 y Fj(j)p Fp(b)p Fj(j)1762 2126 y Fo(1)p Fn(;q)1849 2135 y Fl(3)1884 2126 y Fn(;h)1945 2135 y Fl(2)1983 2111 y Fq(\()2100 2024 y Fm(X)2018 2222 y Fn(l)2039 2231 y Fl(1)2074 2222 y Fo(+)p Fn(l)2150 2231 y Fl(2)2184 2222 y Fo(=)p Fn(h)2280 2231 y Fl(1)2339 2047 y Fp(w)2404 2062 y Fn(l)2425 2071 y Fl(1)2460 2062 y Fn(;q)2512 2071 y Fl(1)2550 2047 y Fp(w)2615 2062 y Fn(l)2636 2071 y Fl(2)2671 2062 y Fn(;q)2723 2071 y Fl(2)p 2339 2090 422 4 v 2353 2173 a Fp(l)2380 2187 y Fo(1)2419 2173 y Fq(!)p Fp(q)2485 2187 y Fo(1)2525 2173 y Fq(!)p Fp(l)2577 2187 y Fo(2)2617 2173 y Fq(!)p Fp(q)2683 2187 y Fo(2)2722 2173 y Fq(!)2791 2111 y(+)2892 2047 y Fp(w)2957 2062 y Fn(h)2998 2071 y Fl(1)3032 2062 y Fo(+1)p Fn(;q)3174 2071 y Fl(1)3212 2047 y Fp(w)3277 2061 y Fo(0)p Fn(;q)3364 2070 y Fl(2)p 2892 2090 511 4 v 2983 2173 a Fp(h)3035 2187 y Fo(1)3075 2173 y Fq(!)p Fp(q)3141 2187 y Fo(1)3180 2173 y Fq(!)p Fp(q)3246 2187 y Fo(2)3285 2173 y Fq(!)3412 2111 y(\))p Fp(;)75 2361 y Fq(and)383 2583 y Fp(C)448 2597 y Fn(n)491 2606 y Fl(0)526 2597 y Fn(;m)608 2606 y Fl(1)672 2583 y Fq(=)932 2496 y Fm(X)768 2688 y Fn(q)800 2697 y Fl(1)834 2688 y Fo(+)p Fn(q)921 2697 y Fl(2)955 2688 y Fo(+)p Fn(q)1042 2697 y Fl(3)1076 2688 y Fo(=)p Fn(m)1193 2697 y Fl(1)1346 2496 y Fm(X)1243 2694 y Fn(h)1284 2703 y Fl(1)1318 2694 y Fo(+)p Fn(h)1414 2703 y Fl(2)1448 2694 y Fo(=)p Fn(n)1546 2703 y Fl(0)1596 2583 y Fj(j)p Fp(b)p Fj(j)1685 2598 y Fo(2)p Fn(;q)1772 2607 y Fl(3)1806 2598 y Fn(;h)1867 2607 y Fl(2)1905 2583 y Fq(\()2023 2496 y Fm(X)1940 2694 y Fn(l)1961 2703 y Fl(1)1996 2694 y Fo(+)p Fn(l)2072 2703 y Fl(2)2107 2694 y Fo(=)p Fn(h)2203 2703 y Fl(1)2262 2519 y Fp(w)2327 2534 y Fn(l)2348 2543 y Fl(1)2383 2534 y Fn(;q)2435 2543 y Fl(1)2473 2519 y Fj(jj)p Fp(c)p Fj(jj)2612 2534 y Fn(q)2644 2543 y Fl(2)2679 2534 y Fn(;l)2720 2543 y Fl(2)p 2262 2562 497 4 v 2412 2645 a Fp(l)2439 2659 y Fo(1)2478 2645 y Fq(!)p Fp(q)2544 2659 y Fo(1)2584 2645 y Fq(!)2789 2583 y(+)2890 2519 y Fp(w)2955 2534 y Fn(h)2996 2543 y Fl(1)3030 2534 y Fo(+1)p Fn(;q)3172 2543 y Fl(1)3210 2519 y Fj(jj)p Fp(c)p Fj(jj)3349 2533 y Fn(q)3381 2542 y Fl(2)3416 2533 y Fn(;)p Fo(0)p 2890 2562 586 4 v 3071 2645 a Fp(h)3123 2659 y Fo(1)3163 2645 y Fq(!)p Fp(q)3229 2659 y Fo(1)3268 2645 y Fq(!)3485 2583 y(\))p Fp(;)75 2866 y Fq(for)30 b(all)h Fp(n)395 2880 y Fo(0)434 2866 y Fp(;)15 b(m)554 2880 y Fo(1)619 2866 y Fj(\025)25 b Fq(0.)216 3025 y(Consider)30 b(no)m(w)g(the)h(auxiliary)g(partial)g(di\013eren)m(tial)h(equation)f (\(17\))158 3259 y Fp(@)211 3221 y Fn(S)206 3281 y(z)262 3259 y Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))26 b(=)741 3172 y Fm(X)651 3374 y Fo(\()p Fn(k)715 3383 y Fl(0)750 3374 y Fn(;k)807 3383 y Fl(1)841 3374 y Fo(\))p Fk(2S)978 3259 y Fj(j)p Fp(b)p Fj(j)1067 3274 y Fn(k)1104 3283 y Fl(0)1139 3274 y Fn(;k)1196 3283 y Fl(1)1234 3259 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(@)1482 3218 y Fn(k)1519 3227 y Fl(0)1477 3283 y Fn(t)1559 3259 y Fp(@)1612 3221 y Fn(k)1649 3230 y Fl(1)1607 3281 y Fn(x)1688 3259 y Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\)+)280 3513 y Fj(j)p Fp(b)p Fj(j)369 3527 y Fo(1)409 3513 y Fq(\()p Fp(t;)g(x)p Fq(\)\(\()p Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\)\))976 3475 y Fo(2)1038 3513 y Fq(+)20 b Fp(U)10 b Fq(\(0)p Fp(;)15 b(x)p Fq(\))p Fp(@)1456 3527 y Fn(t)1487 3513 y Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\)\))22 b(+)e Fj(j)p Fp(b)p Fj(j)1991 3527 y Fo(2)2031 3513 y Fq(\()p Fp(t;)15 b(x)p Fq(\)\()p Fj(jj)p Fp(C)7 b Fj(jj)p Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))23 b(+)d Fj(jj)p 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y(0)26 b Fj(\024)f Fp(k)j Fj(\024)d Fp(S)g Fj(\000)20 b Fq(1,)31 b(for)f(giv)m(en)i Fp(X)1138 4174 y Fo(0)1177 4160 y Fp(;)15 b(T)1270 4174 y Fo(0)1336 4160 y Fp(>)25 b Fq(0.)216 4273 y(If)30 b(w)m(e)h(expand)f Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))31 b(in)m(to)g(a)g(T)-8 b(a)m(ylor)31 b(series)g(in)f(the)h(v)-5 b(ariables)31 b Fp(t;)15 b(x)p Fq(,)1363 4509 y Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))27 b(=)1819 4422 y Fm(X)1753 4618 y Fn(n)1796 4627 y Fl(0)1830 4618 y Fn(;n)1893 4627 y Fl(1)1927 4618 y Fk(\025)p Fo(0)2032 4509 y Fp(U)2094 4523 y Fn(n)2137 4532 y Fl(0)2172 4523 y Fn(;n)2235 4532 y Fl(1)2283 4447 y Fp(t)2316 4414 y Fn(n)2359 4423 y Fl(0)2397 4447 y Fp(x)2449 4414 y Fn(n)2492 4423 y Fl(1)p 2283 4488 248 4 v 2288 4571 a Fp(n)2343 4585 y Fo(0)2382 4571 y Fq(!)p Fp(n)2462 4585 y Fo(1)2501 4571 y Fq(!)2541 4509 y Fp(;)75 4794 y Fq(w)m(e)k(get)g(that)1029 4915 y Fj(j)p Fp(b)p Fj(j)1118 4930 y Fn(k)1155 4939 y Fl(0)1190 4930 y Fn(;k)1247 4939 y Fl(1)1285 4915 y Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(@)1533 4875 y Fn(k)1570 4884 y Fl(0)1528 4939 y Fn(t)1610 4915 y Fp(@)1663 4878 y Fn(k)1700 4887 y Fl(1)1658 4938 y Fn(x)1739 4915 y Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))26 b(=)2159 4829 y Fm(X)2128 5020 y Fn(n)2171 5029 y Fl(0)2205 5020 y Fn(;m)2287 5029 y Fl(1)2337 4915 y Fj(A)2410 4929 y Fn(n)2453 4938 y Fl(0)2487 4929 y Fn(;m)2569 4938 y Fl(1)2608 4915 y Fp(t)2641 4878 y Fn(n)2684 4887 y Fl(0)2722 4915 y Fp(x)2774 4878 y Fn(m)2836 4887 y Fl(1)2875 4915 y Fp(;)75 5162 y Fq(with)968 5296 y Fj(A)1041 5310 y Fn(n)1084 5319 y Fl(0)1118 5310 y Fn(;m)1200 5319 y Fl(1)1264 5296 y Fq(=)1463 5210 y Fm(X)1360 5407 y Fn(h)1401 5416 y Fl(1)1435 5407 y Fo(+)p Fn(h)1531 5416 y Fl(2)1565 5407 y Fo(=)p Fn(n)1663 5416 y Fl(0)1800 5210 y Fm(X)1712 5407 y Fn(l)q Fo(+)p Fn(n)1832 5416 y Fl(1)1867 5407 y Fo(=)p Fn(m)1984 5416 y Fl(1)2034 5296 y Fj(j)p Fp(b)p Fj(j)2123 5311 y Fn(k)2160 5320 y Fl(0)2194 5311 y Fn(;k)2251 5320 y Fl(1)2285 5311 y Fn(;l)q(;h)2388 5320 y Fl(2)2436 5233 y Fp(U)2498 5248 y Fn(h)2539 5257 y Fl(1)2573 5248 y Fo(+)p Fn(k)2665 5257 y Fl(0)2699 5248 y Fn(;n)2762 5257 y Fl(1)2796 5248 y Fo(+)p Fn(k)2888 5257 y Fl(1)p 2436 5275 491 4 v 2563 5358 a Fp(h)2615 5372 y Fo(1)2655 5358 y Fq(!)p Fp(n)2735 5372 y Fo(1)2774 5358 y Fq(!)2937 5296 y Fp(;)75 5546 y Fq(and)769 5659 y Fj(j)p Fp(b)p Fj(j)858 5673 y Fo(1)898 5659 y Fq(\()p Fp(t;)15 b(x)p Fq(\)\(\()p Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\)\))1465 5622 y Fo(2)1527 5659 y Fq(+)20 b Fp(U)10 b Fq(\(0)p Fp(;)15 b(x)p Fq(\))p Fp(@)1945 5673 y Fn(t)1976 5659 y Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\)\))27 b(=)2432 5573 y Fm(X)2401 5764 y Fn(n)2444 5773 y Fl(0)2478 5764 y Fn(;m)2560 5773 y Fl(1)2610 5659 y Fj(B)2670 5673 y Fn(n)2713 5682 y Fl(0)2747 5673 y Fn(;m)2829 5682 y Fl(1)2868 5659 y Fp(t)2901 5622 y Fn(n)2944 5631 y Fl(0)2982 5659 y Fp(x)3034 5622 y Fn(m)3096 5631 y Fl(1)3136 5659 y Fp(;)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 3764 100 a Fq(12)75 399 y(with)480 532 y Fj(B)540 546 y Fn(n)583 555 y Fl(0)617 546 y Fn(;m)699 555 y Fl(1)763 532 y Fq(=)1023 446 y Fm(X)859 637 y Fn(q)891 646 y Fl(1)925 637 y Fo(+)p Fn(q)1012 646 y Fl(2)1046 637 y Fo(+)p Fn(q)1133 646 y Fl(3)1167 637 y Fo(=)p Fn(m)1284 646 y Fl(1)1437 446 y Fm(X)1334 643 y Fn(h)1375 652 y Fl(1)1410 643 y Fo(+)p Fn(h)1506 652 y Fl(2)1540 643 y Fo(=)p Fn(n)1638 652 y Fl(0)1687 532 y Fj(j)p Fp(b)p Fj(j)1776 547 y Fo(1)p Fn(;q)1863 556 y Fl(3)1898 547 y Fn(;h)1959 556 y Fl(2)1997 532 y Fq(\()2114 446 y Fm(X)2032 643 y Fn(l)2053 652 y Fl(1)2088 643 y Fo(+)p Fn(l)2164 652 y Fl(2)2198 643 y Fo(=)p Fn(h)2294 652 y Fl(1)2353 469 y Fp(U)2415 484 y Fn(l)2436 493 y Fl(1)2471 484 y Fn(;q)2523 493 y Fl(1)2561 469 y Fp(U)2623 484 y Fn(l)2644 493 y Fl(2)2679 484 y Fn(;q)2731 493 y Fl(2)p 2353 511 416 4 v 2364 594 a Fp(l)2391 608 y Fo(1)2430 594 y Fq(!)p Fp(q)2496 608 y Fo(1)2536 594 y Fq(!)p Fp(l)2588 608 y Fo(2)2628 594 y Fq(!)p Fp(q)2694 608 y Fo(2)2733 594 y Fq(!)2799 532 y(+)2900 469 y Fp(U)2962 484 y Fn(h)3003 493 y Fl(1)3037 484 y Fo(+1)p Fn(;q)3179 493 y Fl(1)3217 469 y Fp(U)3279 483 y Fo(0)p Fn(;q)3366 492 y Fl(2)p 2900 511 505 4 v 2988 594 a Fp(h)3040 608 y Fo(1)3080 594 y Fq(!)p Fp(q)3146 608 y Fo(1)3185 594 y Fq(!)p Fp(q)3251 608 y Fo(2)3290 594 y Fq(!)3414 532 y(\))75 779 y(for)30 b(all)h Fp(n)395 793 y Fo(0)434 779 y Fp(;)15 b(m)554 793 y Fo(1)619 779 y Fj(\025)25 b Fq(0.)41 b(In)30 b(the)h(same)f(w)m (a)m(y)-8 b(,)32 b(w)m(e)f(ha)m(v)m(e)595 975 y Fj(j)p Fp(b)p Fj(j)684 989 y Fo(2)724 975 y Fq(\()p Fp(t;)15 b(x)p Fq(\)\()p Fj(jj)p Fp(C)7 b Fj(jj)p Fq(\()p Fp(t;)15 b(x)p Fq(\))p Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))24 b(+)c Fj(jj)p Fp(C)7 b Fj(jj)p Fq(\(0)p Fp(;)15 b(x)p Fq(\))p Fp(@)2130 989 y Fn(t)2162 975 y Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\)\))26 b(=)2618 889 y Fm(X)2586 1080 y Fn(n)2629 1089 y Fl(0)2664 1080 y Fn(;m)2746 1089 y Fl(1)2795 975 y Fj(C)2843 989 y Fn(n)2886 998 y Fl(0)2921 989 y Fn(;m)3003 998 y Fl(1)3041 975 y Fp(t)3074 937 y Fn(n)3117 946 y Fl(0)3156 975 y Fp(x)3208 937 y Fn(m)3270 946 y Fl(1)3309 975 y Fp(;)75 1248 y Fq(with)408 1462 y Fj(C)456 1476 y Fn(n)499 1485 y Fl(0)533 1476 y Fn(;m)615 1485 y Fl(1)679 1462 y Fq(=)939 1376 y Fm(X)775 1568 y Fn(q)807 1577 y Fl(1)841 1568 y Fo(+)p Fn(q)928 1577 y Fl(2)962 1568 y Fo(+)p Fn(q)1049 1577 y Fl(3)1083 1568 y Fo(=)p Fn(m)1200 1577 y Fl(1)1353 1376 y Fm(X)1250 1574 y Fn(h)1291 1583 y Fl(1)1325 1574 y Fo(+)p Fn(h)1421 1583 y Fl(2)1455 1574 y Fo(=)p Fn(n)1553 1583 y Fl(0)1603 1462 y Fj(j)p Fp(b)p Fj(j)1692 1477 y Fo(2)p Fn(;q)1779 1486 y Fl(3)1813 1477 y Fn(;h)1874 1486 y Fl(2)1912 1462 y Fq(\()2030 1376 y Fm(X)1947 1574 y Fn(l)1968 1583 y Fl(1)2003 1574 y Fo(+)p Fn(l)2079 1583 y Fl(2)2114 1574 y Fo(=)p Fn(h)2210 1583 y Fl(1)2269 1399 y Fp(U)2331 1414 y Fn(l)2352 1423 y Fl(1)2387 1414 y Fn(;q)2439 1423 y Fl(1)2477 1399 y Fj(jj)p Fp(c)p Fj(jj)2616 1414 y Fn(q)2648 1423 y Fl(2)2683 1414 y Fn(;l)2724 1423 y Fl(2)p 2269 1441 494 4 v 2417 1525 a Fp(l)2444 1539 y Fo(1)2484 1525 y Fq(!)p Fp(q)2550 1539 y 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y Fl(0)2237 2099 y Fn(;m)2319 2108 y Fl(1)2378 2085 y Fq(+)g Fj(C)2517 2099 y Fn(n)2560 2108 y Fl(0)2594 2099 y Fn(;m)2676 2108 y Fl(1)75 2312 y Fq(for)31 b(all)g Fp(n)396 2326 y Fo(0)435 2312 y Fp(;)15 b(m)555 2326 y Fo(1)621 2312 y Fj(\025)26 b Fq(0.)42 b(F)-8 b(rom)32 b(the)f(fact)g(that)h Fp(U)10 b Fq(\()p Fp(t;)15 b(x)p Fq(\))27 b Fj(2)e Fp(G)p Fq(\()p Fp(T)8 b(;)15 b(X)7 b Fq(\),)33 b(w)m(e)e(deduce)g(the)g(follo)m(wing)h (estimates)h(for)75 2425 y(the)e(co)s(e\016cien)m(ts)h Fp(U)754 2439 y Fn(n)797 2448 y Fl(0)831 2439 y Fn(;n)894 2448 y Fl(1)932 2425 y Fq(.)41 b(There)30 b(exists)h Fp(C)h(>)25 b Fq(0)30 b(suc)m(h)g(that)75 2613 y(\(21\))960 b Fj(j)p Fp(U)1282 2627 y Fn(n)1325 2636 y Fl(0)1360 2627 y Fn(;n)1423 2636 y Fl(1)1461 2613 y Fj(j)26 b(\024)f Fp(C)7 b Fq(\(1)p Fp(=X)g Fq(\))1922 2575 y Fn(n)1965 2584 y Fl(1)2005 2613 y Fq(\(1)p Fp(=T)13 b Fq(\))2231 2575 y Fn(n)2274 2584 y Fl(0)2314 2613 y Fq(\()p Fp(n)2404 2627 y Fo(0)2463 2613 y Fq(+)20 b Fp(n)2609 2627 y Fo(1)2648 2613 y Fq(\)!)p Fp(;)75 2801 y Fq(for)39 b(all)i Fp(n)414 2815 y Fo(0)453 2801 y Fp(;)15 b(n)548 2815 y Fo(1)628 2801 y Fj(\025)40 b Fq(0.)68 b(F)-8 b(rom)40 b(Theorem)f(\(CK\),)h(w)m (e)g(see)g(that)g(the)g(sequence)g(\()p Fp(U)2944 2815 y Fn(n)2987 2824 y Fl(0)3022 2815 y Fn(;m)3104 2824 y Fl(1)3143 2801 y Fq(\))3178 2815 y Fn(n)3221 2824 y Fl(0)3255 2815 y Fn(;m)3337 2824 y Fl(1)3416 2801 y Fq(is)f(uniquely)75 2914 y(determined)24 b(b)m(y)g(the)g(sequences)h(\()p Fp(U)1318 2928 y Fn(n)1361 2937 y Fl(0)1396 2928 y Fn(;m)1478 2937 y Fl(1)1516 2914 y Fq(\))1551 2928 y Fn(n)1594 2937 y Fl(0)1629 2928 y Fn(;m)1711 2937 y Fl(1)1750 2914 y Fq(,)h(0)f Fj(\024)g Fp(m)2047 2928 y Fo(1)2112 2914 y Fj(\024)g Fp(S)13 b Fj(\000)8 b Fq(1.)38 b(W)-8 b(e)26 b(c)m(ho)s(ose)f(no)m(w)f(the)h(latter)g(sequences)75 3026 y(as)31 b(follo)m(ws.)1617 3139 y Fp(U)1679 3153 y Fn(n)1722 3162 y Fl(0)1757 3153 y Fn(;m)1839 3162 y Fl(1)1903 3139 y Fq(:=)25 b Fp(w)2089 3153 y Fn(n)2132 3162 y Fl(0)2166 3153 y Fn(;m)2248 3162 y Fl(1)2287 3139 y Fp(;)75 3298 y Fq(for)30 b(all)h(0)26 b Fj(\024)f Fp(m)587 3312 y Fo(1)652 3298 y Fj(\024)g Fp(S)g Fj(\000)20 b Fq(1,)31 b Fp(n)1076 3312 y Fo(0)1140 3298 y Fj(\025)25 b Fq(0.)216 3411 y(F)-8 b(rom)31 b(the)g(assumption)e(3\),)j(w)m(e)f (deduce)f(that)h(the)f(series)1470 3599 y Fp(U)10 b Fq(\()p Fp(t;)15 b Fq(0\))p Fp(;)g(:)g(:)g(:)j(;)d Fq(\()p Fp(@)2021 3561 y Fn(S)t Fk(\000)p Fo(1)2016 3621 y Fn(z)2163 3599 y Fp(U)10 b Fq(\)\()p Fp(t;)15 b Fq(0\))75 3786 y(are)31 b(con)m(v)m(ergen)m(t,)j(whic)m(h)c(means)h(in)f(particular)h(that)g (they)g(b)s(elong)g(to)g Fp(G)p Fq(\()p Fp(T)2736 3800 y Fo(0)2777 3786 y Fp(;)15 b(X)2892 3800 y Fo(0)2932 3786 y Fq(\))31 b(for)f Fp(X)3212 3800 y Fo(0)3252 3786 y Fp(;)15 b(T)3345 3800 y Fo(0)3411 3786 y Fp(>)25 b Fq(0.)42 b(More-)75 3899 y(o)m(v)m(er,)32 b(due)e(to)h(the)f(relations) i(\(19\))g(and)d(\(20\),)k(w)m(e)d(deduce)h(that)75 4087 y(\(22\))1395 b Fp(w)1695 4101 y Fn(n)1738 4110 y Fl(0)1772 4101 y Fn(;m)1854 4110 y Fl(1)1918 4087 y Fj(\024)25 b Fp(U)2076 4101 y Fn(n)2119 4110 y Fl(0)2154 4101 y Fn(;m)2236 4110 y Fl(1)2274 4087 y Fp(;)75 4275 y Fq(for)30 b(all)h Fp(n)395 4289 y Fo(0)434 4275 y Fp(;)15 b(m)554 4289 y Fo(1)619 4275 y Fj(\025)25 b Fq(0.)216 4387 y(F)-8 b(rom)35 b(the)f(estimates)i(\(21\),)h(w)m(e)e(\014nally)f(deduce)g (that)h(there)g(exist)g Fp(C)j(>)32 b Fq(0,)k(suc)m(h)e(that)h(the)f (sequence)75 4500 y Fp(w)140 4514 y Fn(n)183 4523 y Fl(0)217 4514 y Fn(;n)280 4523 y Fl(1)349 4500 y Fq(satis\014es)1219 4613 y Fp(w)1284 4627 y Fn(n)1327 4636 y Fl(0)1362 4627 y Fn(;n)1425 4636 y Fl(1)1488 4613 y Fj(\024)25 b Fp(C)7 b Fq(\(1)p Fp(=X)g Fq(\))1898 4576 y Fn(n)1941 4585 y Fl(1)1981 4613 y Fq(\(1)p Fp(=T)13 b Fq(\))2207 4576 y Fn(n)2250 4585 y Fl(0)2290 4613 y Fq(\()p Fp(n)2380 4627 y Fo(0)2440 4613 y Fq(+)20 b Fp(n)2586 4627 y Fo(1)2625 4613 y Fq(\)!)p Fp(;)75 4772 y Fq(for)30 b(all)h Fp(n)395 4786 y Fo(0)434 4772 y Fp(;)15 b(n)529 4786 y Fo(1)594 4772 y Fj(\025)25 b Fq(0,)31 b(whic)m(h)f(yields)h(the)f(result.)2045 b Ff(2)75 5100 y Fr(4)135 b(Analytic)55 b(solutions)g(in)g(a)f(small)i (parameter)g(of)f(a)f(singularly)i(p)t(er-)277 5249 y(turb)t(ed)44 b(Cauc)l(h)l(y)h(problem)75 5456 y Fc(4.1)112 b(Laplace)39 b(transform)f(and)h(asymptotic)f(expansions)75 5627 y Fq(W)-8 b(e)26 b(recall)f(the)g(de\014nition)f(of)g(Borel)i(summabilit) m(y)e(of)h(formal)g(series)f(with)g(co)s(e\016cien)m(ts)i(in)e(a)h (Banac)m(h)h(space,)75 5740 y(see)31 b([2)q(].)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 3764 100 a Fq(13)75 399 y Fh(De\014nition)35 b(4)46 b Fg(A)32 b(formal)i(series)1677 553 y Fq(^)1651 576 y Fp(X)7 b Fq(\()p Fp(t)p Fq(\))26 b(=)1989 462 y Fk(1)1958 489 y Fm(X)1963 685 y Fn(j)t Fo(=0)2115 514 y Fp(a)2163 528 y Fn(j)p 2115 555 85 4 v 2123 638 a Fp(j)5 b Fq(!)2209 576 y Fp(t)2242 538 y Fn(j)75 837 y Fg(with)36 b(c)-5 b(o)g(e\016cients)36 b(in)e(a)i(Banach)g(sp)-5 b(ac)g(e)36 b Fq(\()p Fi(E)p Fp(;)15 b Fj(jj)p Fp(:)p Fj(jj)1767 851 y Fb(E)1817 837 y Fq(\))36 b Fg(is)e(said)i(to)g(b)-5 b(e)35 b Fq(1)p Fj(\000)p Fg(summable)h(with)f(r)-5 b(esp)g(e)g(ct)37 b(to)f Fp(t)e Fg(in)h(the)75 950 y(dir)-5 b(e)g(ction)34 b Fp(d)26 b Fj(2)f Fq([0)p Fp(;)15 b Fq(2)p Fp(\031)s Fq(\))34 b Fg(if)216 1113 y Fh(i\))h Fg(Ther)-5 b(e)35 b(exists)h Fp(\032)29 b Fj(2)g Fi(R)1068 1127 y Fo(+)1162 1113 y Fg(such)35 b(that)h(the)g(fol)5 b(lowing)36 b(formal)g(series,)g (c)-5 b(al)5 b(le)-5 b(d)36 b(formal)h(Bor)-5 b(el)36 b(tr)-5 b(ansform)75 1226 y(with)33 b(r)-5 b(esp)g(e)g(ct)35 b(to)e(the)g(variable)g Fp(\034)43 b Fg(of)33 b(or)-5 b(der)34 b Fq(1)p Fg(,)1610 1494 y Fp(W)13 b Fq(\()p Fp(\034)d Fq(\))25 b(=)1981 1381 y Fk(1)1950 1408 y Fm(X)1955 1604 y Fn(j)t Fo(=0)2110 1432 y Fp(a)2158 1446 y Fn(j)2195 1432 y Fp(\034)2245 1399 y Fn(j)p 2107 1474 179 4 v 2107 1557 a Fq(\()p Fp(j)5 b Fq(!\))2244 1531 y Fo(2)2295 1494 y Fp(;)75 1793 y Fg(is)33 b(absolutely)g(c)-5 b(onver)g(gent)34 b(for)f Fj(j)p Fp(\034)10 b Fj(j)26 b Fp(<)f(\032)p Fg(,)216 1956 y Fh(ii\))30 b Fg(Ther)-5 b(e)31 b(exists)f Fp(\016)f(>)c Fq(0)31 b Fg(such)f(that)h(the)g(series)f Fp(W)13 b Fq(\()p Fp(\034)d Fq(\))30 b Fg(c)-5 b(an)31 b(b)-5 b(e)30 b(analytic)-5 b(al)5 b(ly)33 b(c)-5 b(ontinue)g(d)31 b(with)g(r)-5 b(esp)g(e)g(ct)32 b(to)75 2069 y Fp(\034)42 b Fg(in)32 b(a)g(se)-5 b(ctor)33 b Fp(S)665 2084 y Fn(d;\016)784 2069 y Fq(=)25 b Fj(f)p Fp(\034)35 b Fj(2)25 b Fi(C)g Fq(:)h Fj(j)p Fp(d)19 b Fj(\000)f Fq(arg)r(\()p Fp(\034)10 b Fq(\))p Fj(j)26 b Fp(<)f(\016)s Fj(g)p Fg(.)43 b(Mor)-5 b(e)g(over,)33 b(ther)-5 b(e)33 b(exist)f Fp(\016)2866 2083 y Fo(1)2931 2069 y Fp(<)25 b(\016)s Fg(,)32 b Fp(C)g(>)25 b Fq(0)p Fg(,)32 b(and)h Fp(K)f(>)25 b Fq(0)75 2182 y Fg(such)33 b(that)1590 2295 y Fj(jj)p Fp(W)13 b Fq(\()p Fp(\034)d Fq(\))p Fj(jj)1909 2309 y Fb(E)1984 2295 y Fj(\024)25 b Fp(C)7 b(e)2194 2257 y Fn(K)e Fk(j)p Fn(\034)j Fk(j)75 2462 y Fg(for)33 b(al)5 b(l)33 b Fp(\034)j Fj(2)24 b Fp(S)567 2477 y Fn(d;\016)654 2486 y Fl(1)693 2462 y Fg(.)75 2674 y Fq(If)j(this)h(is)g(so,)h(the)f(v)m(ector)h(v)-5 b(alued)28 b(Laplace)h(transform)e(of)h(order)g(1)g(of)g Fp(W)13 b Fq(\()p Fp(\034)d Fq(\))28 b(in)f(the)h(direction)h Fp(d)f Fq(is)g(de\014ned)75 2787 y(b)m(y)1211 2960 y Fj(L)1274 2923 y Fn(d)1314 2960 y Fq(\()p Fp(W)13 b Fq(\)\()p Fp(t)p Fq(\))26 b(=)f Fp(t)1741 2923 y Fk(\000)p Fo(1)1850 2837 y Fm(Z)1941 2863 y Fk(1)p Fo(\()p Fn(\015)t Fo(\))1901 3043 y(0)2126 2960 y Fp(W)13 b Fq(\()p Fp(\034)d Fq(\))p Fp(e)2387 2923 y Fk(\000)p Fo(\()p Fn(\034)e(=t)p Fo(\))2601 2960 y Fp(d\034)d(;)75 3173 y Fq(where)34 b(the)g(in)m(tegration)i(is)e (made)g(along)i(the)e(ra)m(y)g(arg)r(\()p Fp(\034)10 b Fq(\))32 b(=)g Fp(\015)39 b Fq(with)34 b Fj(j)p Fp(d)23 b Fj(\000)f Fp(\015)5 b Fj(j)32 b Fp(<)f(\016)s Fq(,)36 b(is)f(called)g(the)f(1)p Fj(\000)p Fq(sum)75 3286 y(of)29 b(the)g(formal)g(series)884 3263 y(^)858 3286 y Fp(X)7 b Fq(\()p Fp(t)p Fq(\).)41 b(The)28 b(function)g Fj(L)1711 3253 y Fn(d)1751 3286 y Fq(\()p Fp(W)13 b Fq(\)\()p Fp(t)p Fq(\))30 b(is)e(an)h(holomorphic)g(and)f(b)s(ounded)e(function)j(on)f (a)75 3399 y(sector)1025 3512 y Fp(S)1081 3527 y Fn(d;\022)r(;R)1274 3512 y Fq(=)d Fj(f)p Fp(t)g Fj(2)g Fi(C)g Fq(:)h Fj(j)p Fp(t)p Fj(j)f Fp(<)g(R)61 b(;)77 b Fj(j)p Fp(d)20 b Fj(\000)g Fq(arg)r(\()p Fp(t)p Fq(\))p Fj(j)26 b Fp(<)f(\022)s(=)p Fq(2)p Fj(g)p Fp(;)75 3679 y Fq(where)38 b Fp(\022)j(>)d(\031)k Fq(and)c Fp(R)h(>)g Fq(0.)65 b(Moreo)m(v)m(er,)43 b(for)38 b(all)i Fp(l)g Fj(\025)f Fq(0,)i(the)e(function)f Fp(@)2758 3646 y Fn(l)2753 3701 y(t)2784 3679 y Fj(L)2847 3646 y Fn(d)2887 3679 y Fq(\()p Fp(W)13 b Fq(\)\()p Fp(t)p Fq(\))39 b(is)g(asymptotic)h(to)75 3792 y Fp(@)128 3759 y Fn(l)123 3814 y(t)180 3769 y Fq(^)154 3792 y Fp(X)8 b Fq(\()p Fp(t)p Fq(\))37 b(with)f(Gevrey)h(\014ltration)g(of)f(order)g (1,)i(with)f(resp)s(ect)f(to)h Fp(t)f Fq(on)g Fp(S)2661 3807 y Fn(d;\022)r(;R)2829 3792 y Fq(.)59 b(This)35 b(means)i(that)f (for)h(all)75 3905 y Fp(\022)118 3919 y Fo(1)182 3905 y Fp(<)25 b(\022)s Fq(,)30 b(all)h Fp(l)d Fj(\025)c Fq(0,)31 b(there)g(exist)g Fp(C)q(;)15 b(M)36 b(>)25 b Fq(0,)31 b(suc)m(h)f(that)1130 4188 y Fj(jj)p Fp(@)1233 4150 y Fn(l)1228 4210 y(t)1260 4188 y Fj(L)1323 4150 y Fn(d)1363 4188 y Fq(\()p Fp(W)13 b Fq(\)\()p Fp(t)p Fq(\))21 b Fj(\000)1747 4074 y Fn(n)p Fk(\000)p Fo(1)1748 4101 y Fm(X)1751 4295 y Fn(p)p Fo(=0)1905 4124 y Fp(a)1953 4139 y Fn(p)p Fo(+)p Fn(l)p 1905 4167 165 4 v 1952 4250 a Fp(p)p Fq(!)2080 4188 y Fp(t)2113 4150 y Fn(p)2152 4188 y Fj(jj)2202 4202 y Fb(E)2276 4188 y Fj(\024)k Fp(C)7 b(M)2542 4150 y Fn(n)2589 4188 y Fp(n)p Fq(!)p Fj(j)p Fp(t)p Fj(j)2752 4150 y Fn(n)75 4480 y Fq(for)30 b(all)h Fp(n)25 b Fj(\025)g Fq(1)31 b(and)f Fp(t)25 b Fj(2)g Fp(S)969 4495 y Fn(d;\022)1058 4504 y Fl(1)1092 4495 y Fn(;R)1169 4480 y Fq(.)75 4643 y(In)34 b(the)h(next)g(prop)s (osition,)h(w)m(e)f(giv)m(e)i(a)e(criterion)g(for)g(a)g(formal)g (series)g(to)h(b)s(e)e(1)p Fj(\000)p Fq(summable.)54 b(This)34 b(is)h(an)75 4756 y(application)d(of)e(a)h(general)g(momen)m (t)g(summabilit)m(y)g(result)f(obtained)h(in)f(Theorem)g(38,)i(p.)40 b(108)32 b(in)e([2].)75 4968 y Fh(Prop)s(osition)36 b(3)46 b Fg(L)-5 b(et)915 4945 y Fq(^)889 4968 y Fp(X)8 b Fq(\()p Fp(t)p Fq(\))25 b(=)1196 4900 y Fm(P)1292 4927 y Fk(1)1292 4995 y Fn(j)t Fo(=0)1419 4968 y Fq(\()p Fp(a)1502 4982 y Fn(j)1539 4968 y Fp(=j)5 b Fq(!\))p Fp(t)1719 4935 y Fn(j)1784 4968 y Fg(b)-5 b(e)27 b(a)g(formal)i(series)e(with)h(c)-5 b(o)g(e\016cients)28 b(in)f(a)g(Banach)h(sp)-5 b(ac)g(e)75 5081 y Fq(\()p Fi(E)p Fp(;)15 b Fj(jj)p Fp(:)p Fj(jj)336 5095 y Fb(E)386 5081 y Fq(\))p Fg(.)42 b(L)-5 b(et)1473 5213 y Fj(B)s Fq(\()1597 5190 y(^)1571 5213 y Fp(X)7 b Fq(\)\()p Fp(\034)j Fq(\))27 b(=)1941 5127 y Fm(X)1931 5323 y Fn(m)p Fk(\025)p Fo(0)2162 5151 y Fp(a)2210 5165 y Fn(m)p 2109 5192 222 4 v 2109 5275 a Fq(\(2)p Fp(m)p Fq(\)!)2340 5213 y Fp(\034)2390 5175 y Fn(m)75 5468 y Fg(the)37 b(mo)-5 b(di\014e)g(d)39 b(Bor)-5 b(el)38 b(tr)-5 b(ansform)39 b(of)e(or)-5 b(der)38 b(1)f(which)h(is)e(a)h(formal)i (series)d(with)i(c)-5 b(o)g(e\016cients)37 b(in)g Fi(E)p Fg(.)53 b(Then,)101 5558 y Fq(^)75 5581 y Fp(X)7 b Fq(\()p Fp(t)p Fq(\))34 b Fg(is)e Fq(1)p Fj(\000)p Fg(summable)i(with)f(r)-5 b(esp)g(e)g(ct)34 b(to)g Fp(t)e Fg(in)g(the)h(dir)-5 b(e)g(ction)35 b Fp(d)25 b Fj(2)g Fq([0)p Fp(;)15 b Fq(2)p Fp(\031)s Fq(\))35 b Fg(if)75 5694 y Fh(i\))d Fg(Ther)-5 b(e)34 b(exists)f Fp(\032)25 b Fj(2)g Fi(R)912 5708 y Fo(+)1003 5694 y Fg(such)33 b(that)g Fq(\()p Fj(B)1519 5671 y Fq(^)1493 5694 y Fp(X)8 b Fq(\)\()p Fp(\034)i Fq(\))33 b Fg(is)g(absolutely)h(c)-5 b(onver)g(gent)33 b(for)h(al)5 b(l)33 b Fp(\034)i Fj(2)25 b Fp(D)3239 5708 y Fn(\032)3279 5694 y Fg(.)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 3764 100 a Fq(14)75 399 y Fh(ii\))31 b Fg(Ther)-5 b(e)33 b(exists)f Fp(\016)d(>)c Fq(0)32 b Fg(such)g(that)h(the)f(series)g Fq(\()p Fj(B)1877 376 y Fq(^)1851 399 y Fp(X)7 b Fq(\)\()p Fp(\034)j Fq(\))33 b Fg(c)-5 b(an)32 b(b)-5 b(e)32 b(analytic)-5 b(al)5 b(ly)34 b(c)-5 b(ontinue)g(d)33 b(with)g(r)-5 b(esp)g(e)g(ct)33 b(to)75 511 y Fp(\034)42 b Fg(in)32 b(a)g(se)-5 b(ctor)33 b Fp(S)665 526 y Fn(d;\016)784 511 y Fq(=)25 b Fj(f)p Fp(\034)35 b Fj(2)25 b Fi(C)g Fq(:)h Fj(j)p Fp(d)19 b Fj(\000)f Fq(arg)r(\()p Fp(\034)10 b Fq(\))p Fj(j)26 b Fp(<)f(\016)s Fj(g)p Fg(.)43 b(Mor)-5 b(e)g(over,)33 b(ther)-5 b(e)33 b(exist)f Fp(\016)2866 525 y Fo(1)2931 511 y Fp(<)25 b(\016)s Fg(,)32 b Fp(C)g(>)25 b Fq(0)p Fg(,)32 b(and)h Fp(K)f(>)25 b Fq(0)75 624 y Fg(such)33 b(that)1531 737 y Fj(jj)p Fq(\()p Fj(B)1705 714 y Fq(^)1679 737 y Fp(X)8 b Fq(\)\()p Fp(\034)i Fq(\))p Fj(jj)1967 751 y Fb(E)2042 737 y Fj(\024)25 b Fp(C)7 b(e)2252 700 y Fn(K)e Fk(j)p Fn(\034)j Fk(j)75 904 y Fg(for)33 b(al)5 b(l)33 b Fp(\034)j Fj(2)24 b Fp(S)567 919 y Fn(d;\016)654 928 y Fl(1)693 904 y Fg(.)75 1075 y(Mor)-5 b(e)g(over,)39 b(if)d Fp(X)7 b Fq(\()p Fp(t)p Fq(\))38 b Fg(is)f(the)g Fq(1)p Fj(\000)p Fg(sum)h(of)1531 1053 y Fq(^)1505 1075 y Fp(X)44 b Fg(de\014ne)-5 b(d)38 b(on)f(some)h(se)-5 b(ctor)38 b Fp(S)2630 1090 y Fn(d;\022)r(;R)2797 1075 y Fg(,)g(with)g Fp(\022)d(>)e(\031)s Fg(,)k Fp(R)d(>)f Fq(0)p Fg(,)38 b(then)75 1188 y(the)g(fol)5 b(lowing)38 b(asymptotic)i(exp)-5 b(ansion)39 b(hold:)53 b(F)-7 b(or)39 b(al)5 b(l)38 b Fp(\022)2102 1202 y Fo(1)2175 1188 y Fp(<)c(\022)s Fg(,)j(al)5 b(l)38 b(p)-5 b(ositive)39 b(inte)-5 b(ger)37 b Fp(l)f Fj(\025)e Fq(0)p Fg(,)39 b(ther)-5 b(e)38 b(exist)75 1301 y Fp(C)q(;)15 b(M)36 b(>)25 b Fq(0)p Fg(,)33 b(such)g(that)75 1583 y Fq(\(23\))990 b Fj(jj)p Fp(@)1328 1546 y Fn(l)1323 1606 y(t)1355 1583 y Fp(X)7 b Fq(\()p Fp(t)p Fq(\))21 b Fj(\000)1652 1470 y Fn(n)p Fk(\000)p Fo(1)1653 1497 y Fm(X)1656 1691 y Fn(p)p Fo(=0)1810 1520 y Fp(a)1858 1535 y Fn(p)p Fo(+)p Fn(l)p 1810 1563 165 4 v 1857 1646 a Fp(p)p Fq(!)1985 1583 y Fp(t)2018 1546 y Fn(p)2057 1583 y Fj(jj)2107 1597 y Fb(E)2182 1583 y Fj(\024)j Fp(C)7 b(M)2447 1546 y Fn(n)2494 1583 y Fp(n)p Fq(!)p Fj(j)p Fp(t)p Fj(j)2657 1546 y Fn(n)75 1876 y Fg(for)33 b(al)5 b(l)33 b Fp(n)25 b Fj(\025)g Fq(1)p Fg(,)33 b(al)5 b(l)33 b Fp(t)25 b Fj(2)g Fp(S)962 1891 y Fn(d;\022)1051 1900 y Fl(1)1085 1891 y Fn(;R)1162 1876 y Fg(.)75 2087 y Fh(Remark:)41 b Fq(The)29 b(1)p Fj(\000)p Fq(sum)g Fp(X)7 b Fq(\()p Fp(t)p Fq(\))30 b(can)g(b)s(e)f (expressed)g(in)h(term)f(of)h(the)g(mo)s(di\014ed)f(Borel)h(transform)f (using)h(an)75 2200 y(in)m(tegral)i(op)s(erator)1325 2374 y Fp(X)7 b Fq(\()p Fp(t)p Fq(\))26 b(=)1632 2250 y Fm(Z)1723 2276 y Fk(1)p Fo(\()p Fn(d)p Fo(\))1682 2456 y(0)1914 2312 y Fp(e)p Fq(\()p Fp(u=t)p Fq(\))p 1914 2353 244 4 v 2009 2436 a Fp(u)2167 2374 y Fj(B)s Fq(\()2291 2351 y(^)2265 2374 y Fp(X)7 b Fq(\)\()p Fp(u)p Fq(\))p Fp(du)75 2586 y Fq(where)28 b Fp(e)p Fq(\()p Fp(z)t Fq(\))h(is)f(some)h (holomorphic)f(function)g(with)f(exp)s(onen)m(tial)i(deca)m(y)h(on)e (the)g(half-plane)g Fj(f)p Fq(Re)q(\()p Fp(z)t Fq(\))e Fp(>)f Fq(0)p Fj(g)p Fq(,)75 2699 y(see)31 b(Theorem)f(38)h(in)f([2)q (].)75 2942 y Fc(4.2)112 b(Analytic)35 b(solutions)h(of)f(a)h(singular) g(Cauc)m(h)m(y)g(problem)g(with)e(irregular)i(singular-)330 3059 y(it)m(y)75 3230 y Fq(Let)31 b Fp(S)f Fj(\025)25 b Fq(1)31 b(b)s(e)f(an)g(in)m(teger.)42 b(Let)31 b Fj(S)37 b Fq(b)s(e)30 b(a)g(\014nite)h(subset)f(of)g Fi(N)2217 3197 y Fo(2)2287 3230 y Fq(and)f(let)908 3442 y Fp(b)947 3457 y Fn(k)984 3466 y Fl(0)1018 3457 y Fn(;k)1075 3466 y Fl(1)1113 3442 y Fq(\()p Fp(z)t(;)15 b(\017)p Fq(\))27 b(=)1429 3356 y Fm(X)1429 3552 y Fn(n)p Fk(\025)p Fo(0)1577 3442 y Fp(b)1616 3457 y Fn(k)1653 3466 y Fl(0)1687 3457 y Fn(;k)1744 3466 y Fl(1)1778 3457 y Fn(;n)1845 3442 y Fq(\()p Fp(\017)p Fq(\))p Fp(z)1998 3405 y Fn(n)2106 3442 y Fp(;)76 b(b)p Fq(\()p Fp(z)t(;)15 b(\017)p Fq(\))27 b(=)2562 3356 y Fm(X)2562 3552 y Fn(n)p Fk(\025)p Fo(0)2710 3442 y Fp(b)2749 3456 y Fn(n)2796 3442 y Fq(\()p Fp(\017)p Fq(\))p Fp(z)2949 3405 y Fn(n)2997 3442 y Fp(;)75 3747 y Fq(b)s(e)j(holomorphic)g(functions)g(on)g(a)h(neigh)m(b)s(orho)s(o)s (d)e(of)i(the)f(origin)h(in)f Fi(C)2574 3714 y Fo(2)2613 3747 y Fq(,)h(for)f(all)h(\()p Fp(k)3016 3761 y Fo(0)3056 3747 y Fp(;)15 b(k)3143 3761 y Fo(1)3183 3747 y Fq(\))26 b Fj(2)f(S)7 b Fq(.)75 3934 y Fh(De\014nition)35 b(5)46 b Fg(L)-5 b(et)31 b Fp(V)20 b Fq(\()p Fp(\034)5 b(;)15 b(\017)p Fq(\))32 b Fg(b)-5 b(e)31 b(a)h(holomorphic)i(function)d(on)h (some)f(p)-5 b(olydisc)33 b Fp(D)s Fq(\(0)p Fp(;)15 b(\034)3109 3948 y Fo(0)3149 3934 y Fq(\))i Fj(\002)g Fp(D)s Fq(\(0)p Fp(;)e(\017)3524 3948 y Fo(0)3564 3934 y Fq(\))31 b Fg(wher)-5 b(e)75 4047 y Fq(0)39 b Fp(<)g(\034)309 4061 y Fo(0)387 4047 y Fp(<)f Fq(1)j Fg(and)g Fp(\017)803 4061 y Fo(0)881 4047 y Fp(>)e Fq(0)p Fg(.)64 b(L)-5 b(et)41 b Fp(d)e Fj(2)f Fq([0)p Fp(;)15 b Fq(2)p Fp(\031)s Fq(\))42 b Fg(with)f Fp(d)f Fj(6)p Fq(=)e Fp(\031)s Fg(.)64 b(L)-5 b(et)41 b Fp(U)2540 4062 y Fn(d)2620 4047 y Fg(b)-5 b(e)40 b(a)h(se)-5 b(ctor)41 b(c)-5 b(enter)g(e)g(d)42 b(at)e(0,)j(with)75 4160 y(bise)-5 b(cting)36 b(dir)-5 b(e)g(ction)38 b Fp(d)p Fg(,)f(with)g(in\014nite)g(r)-5 b(adius)37 b(and)g(with)h(smal)5 b(l)37 b(op)-5 b(ening)37 b(such)f(that)i Fj(\000)p Fq(1)k Fp(=)-55 b Fj(2)31 b Fp(U)3401 4175 y Fn(d)3442 4160 y Fg(.)53 b(L)-5 b(et)36 b Fj(E)45 b Fg(b)-5 b(e)75 4273 y(a)34 b(se)-5 b(ctor)36 b(c)-5 b(enter)g(e)g(d)35 b(at)f(0)h(such)f (that)h Fj(E)h(\032)28 b Fp(D)s Fq(\(0)p Fp(;)15 b(\017)1778 4287 y Fo(0)1818 4273 y Fq(\))p Fg(.)46 b(Assume)34 b(that)i(the)e (function)g Fq(\()p Fp(\034)5 b(;)15 b(\017)p Fq(\))29 b Fj(7!)f Fp(V)20 b Fq(\()p Fp(\034)5 b(;)15 b(\017)p Fq(\))35 b Fg(c)-5 b(an)35 b(b)-5 b(e)75 4386 y(extende)g(d)33 b(to)f(an)g(analytic)h(and)g(b)-5 b(ounde)g(d)33 b(function)f Fq(\()p Fp(\034)5 b(;)15 b(\017)p Fq(\))26 b Fj(7!)f Fp(V)2289 4400 y Fn(U)2337 4412 y Fd(d)2373 4400 y Fn(;)p Fk(E)2440 4386 y Fq(\()p Fp(\034)5 b(;)15 b(\017)p Fq(\))32 b Fg(on)g Fp(U)2855 4401 y Fn(d)2914 4386 y Fj(\002)18 b(E)8 b Fg(.)41 b(Assume)32 b(mor)-5 b(e)g(over,)75 4498 y(that)34 b(ther)-5 b(e)33 b(exist)g(two)h(c)-5 b(onstants)34 b Fp(C)q(;)15 b(T)1427 4512 y Fo(0)1493 4498 y Fp(>)25 b Fq(0)33 b Fg(such)g(that)75 4702 y Fq(\(24\))564 b(sup)808 4780 y Fn(\017)p Fk(2E)1119 4702 y Fq(sup)951 4786 y Fn(t)976 4767 y Fl(2)1011 4786 y Fk(2)p Fn(D)r Fo(\(0)p Fn(;\034)1231 4795 y Fl(0)1266 4786 y Fo(\))p Fk([)p Fn(U)1388 4798 y Fd(d)1424 4702 y Fq(\(1)21 b(+)f Fj(j)p Fp(t)p Fj(j)1699 4664 y Fo(2)1739 4702 y Fq(\))p Fp(e)1816 4664 y Fk(\000)p Fn(\027)t Fk(j)p Fn(t)p Fk(j)1979 4702 y Fj(j)p Fp(@)2057 4664 y Fn(l)2052 4724 y(t)2084 4702 y Fq(\()p Fp(V)2172 4716 y Fn(U)2220 4728 y Fd(d)2256 4716 y Fn(;)p Fk(E)2323 4702 y Fq(\()p Fp(t)2391 4664 y Fo(2)2431 4702 y Fp(;)15 b(\017)p Fq(\)\))p Fj(j)26 b(\024)f Fp(C)7 b Fq(\(1)p Fp(=T)2975 4716 y Fo(0)3015 4702 y Fq(\))3050 4664 y Fn(l)3077 4702 y Fp(l)r Fq(!)75 4979 y Fg(for)33 b(al)5 b(l)33 b Fp(l)27 b Fj(\025)e Fq(0)p Fg(.)43 b(We)32 b(say)h(that)h(the)f(set)g Fj(f)p Fp(V)5 b(;)15 b(V)1616 4993 y Fn(U)1664 5005 y Fd(d)1700 4993 y Fn(;)p Fk(E)1767 4979 y Fp(;)g(T)1860 4993 y Fo(0)1900 4979 y Fp(;)g(\034)1980 4993 y Fo(0)2020 4979 y Fp(;)g(\017)2097 4993 y Fo(0)2137 4979 y Fj(g)32 b Fg(is)h(admissible.)216 5166 y Fq(W)-8 b(e)32 b(consider)e(the)h(follo)m(wing)g(singular)g (Cauc)m(h)m(y)f(problem)75 5394 y(\(25\))85 b(4)p Fp(t)398 5357 y Fo(2)437 5394 y Fp(@)485 5408 y Fn(t)515 5394 y Fp(@)568 5357 y Fn(S)563 5417 y(z)619 5394 y Fp(Y)672 5408 y Fn(U)720 5420 y Fd(d)756 5408 y Fn(;)p Fk(E)823 5394 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))21 b(+)f(\(2)p Fp(t)h Fq(+)f(1\))p Fp(@)1559 5357 y Fn(S)1554 5417 y(z)1611 5394 y Fp(Y)1664 5408 y Fn(U)1712 5420 y Fd(d)1748 5408 y Fn(;)p Fk(E)1814 5394 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))27 b(=)514 5513 y Fm(X)423 5715 y Fo(\()p Fn(k)487 5724 y Fl(0)522 5715 y Fn(;k)579 5724 y Fl(1)613 5715 y Fo(\))p Fk(2S)750 5600 y Fp(b)789 5615 y Fn(k)826 5624 y Fl(0)861 5615 y Fn(;k)918 5624 y Fl(1)956 5600 y Fq(\()p Fp(z)t(;)15 b(\017)p Fq(\))p Fp(\017)1186 5562 y Fn(k)1223 5571 y Fl(0)1262 5600 y Fp(@)1315 5559 y Fn(k)1352 5568 y Fl(0)1310 5624 y Fn(t)1391 5600 y Fp(@)1444 5562 y Fn(k)1481 5571 y Fl(1)1439 5622 y Fn(z)1520 5600 y Fp(Y)1573 5614 y Fn(U)1621 5626 y Fd(d)1657 5614 y Fn(;)p Fk(E)1723 5600 y Fq(\()p Fp(t;)g(z)t(;)g(\017)p Fq(\))22 b(+)e Fp(b)p Fq(\()p Fp(z)t(;)15 b(\017)p Fq(\))p Fp(@)2382 5614 y Fn(t)2428 5476 y Fm(Z)2519 5502 y Fn(t)2479 5682 y Fo(0)2564 5600 y Fp(Y)2617 5614 y Fn(U)2665 5626 y Fd(d)2701 5614 y Fn(;)p Fk(E)2767 5600 y Fq(\()p Fp(t)21 b Fj(\000)f Fp(\034)5 b(;)15 b(z)t(;)g(\017)p Fq(\))p Fp(Y)3243 5614 y Fn(U)3291 5626 y Fd(d)3328 5614 y Fn(;)p Fk(E)3394 5600 y Fq(\()p Fp(\034)5 b(;)15 b(z)t(;)g(\017)p Fq(\))p Fp(d\034)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 3764 100 a Fq(15)75 399 y(for)30 b(giv)m(en)i(initial)f(conditions)75 591 y(\(26\))781 b(\()p Fp(@)1104 554 y Fn(j)1099 614 y(z)1141 591 y Fp(Y)1194 605 y Fn(U)1242 617 y Fd(d)1278 605 y Fn(;)p Fk(E)1344 591 y Fq(\)\()p Fp(t;)15 b Fq(0)p Fp(;)g(\017)p Fq(\))28 b(=)d Fp(Y)1821 605 y Fn(U)1869 617 y Fd(d)1904 605 y Fn(;)p Fk(E)6 b Fn(;j)2023 591 y Fq(\()p Fp(t;)15 b(\017)p Fq(\))62 b Fp(;)76 b Fq(0)25 b Fj(\024)g Fp(j)31 b Fj(\024)25 b Fp(S)g Fj(\000)20 b Fq(1)75 784 y(The)48 b(initial)i(conditions)f Fp(Y)1067 798 y Fn(U)1115 810 y Fd(d)1151 798 y Fn(;)p Fk(E)6 b Fn(;j)1270 784 y Fq(\()p Fp(t;)15 b(\017)p Fq(\))49 b(are)g(constructed)g(as)g(follo)m(ws:)79 b(for)48 b(all)i(0)56 b Fj(\024)f Fp(j)62 b Fj(\024)55 b Fp(S)38 b Fj(\000)32 b Fq(1,)54 b(let)75 897 y Fj(f)p Fp(V)173 911 y Fn(j)210 897 y Fp(;)15 b(V)303 911 y Fn(U)351 923 y Fd(d)387 911 y Fn(;)p Fk(E)6 b Fn(;j)506 897 y Fp(;)15 b(T)599 911 y Fo(0)639 897 y Fp(;)g(\034)719 911 y Fo(0)759 897 y Fp(;)g(\017)836 911 y Fo(0)875 897 y Fj(g)31 b Fq(b)s(e)f(an)g (admissible)g(set.)42 b(Let)1469 1143 y Fp(V)1522 1157 y Fn(j)1559 1143 y Fq(\()p Fp(\034)5 b(;)15 b(\017)p Fq(\))26 b(=)1884 1056 y Fm(X)1873 1252 y Fn(m)p Fk(\025)p Fo(0)2051 1081 y Fp(\037)2108 1095 y Fn(m;j)2226 1081 y Fq(\()p Fp(\017)p Fq(\))p 2051 1122 284 4 v 2082 1205 a(\(2)p Fp(m)p Fq(\)!)2344 1143 y Fp(\034)2394 1105 y Fn(m)75 1429 y Fq(b)s(e)k(it's)h(T)-8 b(a)m(ylor)31 b(expansion)g(for)f (\()p Fp(\034)5 b(;)15 b(\017)p Fq(\))26 b Fj(2)f Fp(D)s Fq(\(0)p Fp(;)15 b(\034)1740 1443 y Fo(0)1780 1429 y Fq(\))21 b Fj(\002)e Fp(D)s Fq(\(0)p Fp(;)c(\017)2161 1443 y Fo(0)2202 1429 y Fq(\).)41 b(W)-8 b(e)31 b(consider)g(the)f (formal)h(series)1485 1647 y(^)1471 1670 y Fp(Y)1524 1684 y Fn(j)1561 1670 y Fq(\()p Fp(t;)15 b(\017)p Fq(\))26 b(:=)1899 1584 y Fm(X)1888 1780 y Fn(m)p Fk(\025)p Fo(0)2066 1608 y Fp(\037)2123 1622 y Fn(m;j)2241 1608 y Fq(\()p Fp(\017)p Fq(\))p 2066 1649 V 2155 1732 a Fp(m)p Fq(!)2359 1670 y Fp(t)2392 1632 y Fn(m)75 1974 y Fq(for)37 b(all)i Fp(\017)e Fj(2)g Fp(D)s Fq(\(0)p Fp(;)15 b(\017)762 1988 y Fo(0)802 1974 y Fq(\).)63 b(W)-8 b(e)39 b(de\014ne)e Fp(Y)1412 1988 y Fn(U)1460 2000 y Fd(d)1495 1988 y Fn(;)p Fk(E)6 b Fn(;j)1614 1974 y Fq(\()p Fp(t;)15 b(\017)p Fq(\))39 b(as)e(the)h(1)p Fj(\000)p Fq(sum)f(of)2556 1951 y(^)2542 1974 y Fp(Y)2595 1988 y Fn(j)2631 1974 y Fq(\()p Fp(t;)15 b(\017)p Fq(\))38 b(in)g(the)g(direction)g Fp(d)p Fq(,)i(whic)m(h)75 2087 y(exists)31 b(from)f(Prop)s(osition)g(3) h(using)f(the)h(estimates)h(\(24\).)75 2246 y(W)-8 b(e)32 b(ha)m(v)m(e)f(the)g(follo)m(wing)h(result.)75 2447 y Fh(Theorem)j(2)46 b Fg(L)-5 b(et)34 b(the)h(fol)5 b(lowing)35 b(assumption)i(hold:)46 b(for)35 b(al)5 b(l)35 b Fq(\()p Fp(k)2392 2461 y Fo(0)2432 2447 y Fp(;)15 b(k)2519 2461 y Fo(1)2559 2447 y Fq(\))29 b Fj(2)f(S)41 b Fg(we)35 b(have)f Fp(S)g Fj(\025)28 b Fq(2)p Fp(k)3436 2461 y Fo(0)3497 2447 y Fq(+)21 b Fp(k)3636 2461 y Fo(1)3710 2447 y Fg(and)75 2560 y Fp(S)30 b(>)25 b(k)304 2574 y Fo(1)344 2560 y Fg(.)42 b(Mor)-5 b(e)g(over,)33 b(we)g(put)813 2761 y Fj(j)p Fp(b)p Fj(j)902 2776 y Fn(k)939 2785 y Fl(0)974 2776 y Fn(;k)1031 2785 y Fl(1)1069 2761 y Fq(\()p Fp(r)m(;)15 b(\017)p Fq(\))26 b(=)1377 2675 y Fm(X)1376 2871 y Fn(n)p Fk(\025)p Fo(0)1525 2761 y Fj(j)p Fp(b)1589 2776 y Fn(k)1626 2785 y Fl(0)1660 2776 y Fn(;k)1717 2785 y Fl(1)1751 2776 y Fn(;n)1818 2761 y Fq(\()p Fp(\017)p Fq(\))p Fj(j)p Fp(r)1994 2724 y Fn(n)2106 2761 y Fp(;)81 b Fj(j)p Fp(b)p Fj(j)p Fq(\()p Fp(r)m(;)15 b(\017)p Fq(\))27 b(=)2610 2675 y Fm(X)2609 2871 y Fn(n)p Fk(\025)p Fo(0)2757 2761 y Fj(j)p Fp(b)2821 2775 y Fn(n)2868 2761 y Fq(\()p Fp(\017)p Fq(\))p Fj(j)p Fp(r)3044 2724 y Fn(n)3092 2761 y Fp(;)75 3043 y Fg(for)33 b Fp(r)28 b(>)d Fq(0)33 b Fg(and)g(we)g(assume)h(that)f(ther)-5 b(e)34 b(exists)f Fp(r)28 b(>)d Fq(0)33 b Fg(such)f(that)1303 3236 y Fq(lim)1299 3294 y Fn(\017)p Fk(!)p Fo(0)1449 3236 y Fj(j)p Fp(b)p Fj(j)1538 3251 y Fn(k)1575 3260 y Fl(0)1610 3251 y Fn(;k)1667 3260 y Fl(1)1705 3236 y Fq(\()p Fp(r)m(;)15 b(\017)p Fq(\))27 b(=)i(lim)2013 3294 y Fn(\017)p Fk(!)p Fo(0)2162 3236 y Fj(j)p Fp(b)p Fj(j)p Fq(\()p Fp(r)m(;)15 b(\017)p Fq(\))27 b(=)e(0)p Fp(;)75 3459 y Fg(for)33 b(al)5 b(l)33 b Fq(\()p Fp(k)432 3473 y Fo(0)472 3459 y Fp(;)15 b(k)559 3473 y Fo(1)600 3459 y Fq(\))25 b Fj(2)g(S)7 b Fg(.)42 b(L)-5 b(et)33 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a(\(2)p Fp(m)p Fq(\)!)2401 1546 y Fp(t)2434 1509 y Fo(2)p Fn(m)75 1873 y Fq(it's)31 b(T)-8 b(a)m(ylor)32 b(expansion)e(with)g(resp)s(ect)g(to)h Fp(t)g Fq(on)f Fp(D)s Fq(\(0)p Fp(;)15 b(\034)2002 1825 y Fo(1)p Fn(=)p Fo(2)1992 1899 y(0)2113 1873 y Fq(\).)41 b(No)m(w,)31 b(w)m(e)g(sho)m(w)f(that)h(the)g(formal)f(series)1445 2106 y(^)1432 2129 y Fp(Y)19 b Fq(\()p Fp(t;)c(z)t(;)g(\017)p Fq(\))27 b(=)1904 2042 y Fm(X)1893 2238 y Fn(m)p Fk(\025)p Fo(0)2071 2067 y Fp(\037)2128 2081 y Fn(m)2194 2067 y Fq(\()p Fp(z)t(;)15 b(\017)p Fq(\))p 2071 2108 V 2177 2191 a Fp(m)p Fq(!)2399 2129 y Fp(t)2432 2091 y Fn(m)75 2426 y Fq(satis\014es)31 b(the)f(equation)i(\(25\).)42 b(Indeed,)30 b(from)g(the)g(classical)j(iden)m(tit)m(y)1025 2563 y Fm(Z)1116 2590 y Fn(t)1076 2769 y Fo(0)1146 2687 y Fq(\()p Fp(t)20 b Fj(\000)g Fp(\034)10 b Fq(\))1410 2649 y Fn(m)1472 2658 y Fl(1)1511 2687 y Fp(\034)1561 2649 y Fn(m)1623 2658 y Fl(2)1663 2687 y Fp(d\034)35 b Fq(=)2048 2626 y Fp(m)2128 2640 y Fo(1)2167 2626 y Fq(!)p 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Fp(t)3177 2998 y Fn(j)p 3144 3071 70 4 v 3145 3154 a Fp(j)5 b Fq(!)75 3349 y(and)30 b(also)555 3587 y Fp(@)603 3601 y Fn(t)648 3463 y Fm(Z)739 3490 y Fn(t)698 3669 y Fo(0)783 3587 y Fp(V)21 b Fq(\(\()p Fp(t)f Fj(\000)g Fp(\034)10 b Fq(\))1156 3549 y Fo(2)1196 3587 y Fp(;)15 b(z)t(;)g(\017)p Fq(\))p Fp(V)22 b Fq(\()p Fp(\034)1554 3549 y Fo(2)1593 3587 y Fp(;)15 b(z)t(;)g(\017)p Fq(\))p Fp(d\034)37 b Fq(=)2011 3501 y Fm(X)2016 3697 y Fn(j)t Fk(\025)p Fo(0)2143 3587 y Fq(\()2250 3501 y Fm(X)2178 3696 y Fn(q)2210 3705 y Fl(1)2244 3696 y Fo(+)p Fn(q)2331 3705 y Fl(2)2365 3696 y Fo(=)p Fn(j)2467 3587 y Fp(\037)2524 3601 y Fn(q)2556 3610 y Fl(1)2595 3587 y Fq(\()p Fp(z)t(;)15 b(\017)p Fq(\))p Fp(\037)2845 3601 y Fn(q)2877 3610 y Fl(2)2916 3587 y Fq(\()p Fp(z)t(;)g(\017)p Fq(\)\))3195 3526 y Fp(t)3228 3493 y Fo(2)p Fn(j)p 3154 3566 185 4 v 3154 3649 a Fq(\(2)p Fp(j)5 b Fq(\)!)3350 3587 y Fp(:)75 3881 y Fq(On)30 b(the)g(other)h(hand,)970 4137 y(\(4)p Fp(t)1083 4100 y Fo(2)1123 4137 y Fp(@)1171 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Fn(d)202 5336 y Fj(\002)23 b Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\))26 b Fj(\002)d(E)8 b Fq(.)58 b(Moreo)m(v)m(er,)40 b(the)c(function)g Fp(V)1844 5350 y Fn(U)1892 5362 y Fd(d)1928 5350 y Fn(;)p Fk(E)2030 5336 y Fq(satis\014es)h(in)f(particular)g(the)g(follo)m(wing)i (estimates:)75 5449 y(there)31 b(exists)g(a)f(constan)m(t)i Fp(C)g(>)25 b Fq(0)30 b(suc)m(h)h(that)75 5654 y(\(30\))651 b(sup)810 5736 y Fn(z)s Fk(2)p Fn(D)r Fo(\(0)p Fn(;\032)p Fo(\))1114 5654 y Fq(sup)1123 5732 y Fn(\017)p Fk(2E)1434 5654 y Fq(sup)1266 5737 y Fn(t)1291 5719 y Fl(2)1326 5737 y Fk(2)p Fn(D)r Fo(\(0)p Fn(;\034)1546 5746 y Fl(0)1581 5737 y Fo(\))p Fk([)p Fn(U)1703 5749 y Fd(d)1740 5654 y Fq(\(1)21 b(+)e Fj(j)p Fp(t)p Fj(j)2014 5616 y Fo(2)2054 5654 y Fq(\))p Fp(e)2131 5616 y Fk(\000)p Fn(\027)t Fk(j)p Fn(t)p Fk(j)2295 5654 y Fj(j)p Fq(\()p Fp(V)2408 5668 y Fn(U)2456 5680 y Fd(d)2492 5668 y Fn(;)p Fk(E)2559 5654 y Fq(\()p Fp(t)2627 5616 y Fo(2)2667 5654 y Fp(;)c(z)t(;)g(\017)p Fq(\)\))p Fj(j)27 b(\024)e Fp(C)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 3764 100 a Fq(17)75 399 y(F)-8 b(rom)41 b(these)f(latter)i(estimates)g(and)d(the)i(prop)s(osition)f (3,)j(w)m(e)e(deduce)e(that)i(the)g(function)f Fp(Y)3437 413 y Fn(U)3485 425 y Fd(d)3520 413 y Fn(;)p Fk(E)3587 399 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))75 520 y(de\014ned)29 b(as)i(the)f(1)p Fj(\000)p Fq(sum)g(of)h(the)f(formal)h (series)1768 497 y(^)1755 520 y Fp(Y)20 b Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))32 b(in)e(the)g(direction)h Fp(d)g Fq(exists)g(and)f(is)g(holomorphic)75 633 y(and)35 b(b)s(ounded)f (function)h(on)h Fp(U)1189 648 y Fn(d;\022)r(;R)1381 633 y Fj(\002)23 b Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\))26 b Fj(\002)d(E)8 b Fq(,)38 b(for)d(some)h Fp(\022)h(>)d(\031)s Fq(,)j Fp(R)q(;)15 b(\032)35 b(>)f Fq(0.)57 b(F)-8 b(rom)36 b(the)g(fact)h(that)89 732 y(^)75 755 y Fp(Y)20 b Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))44 b(is)f(a)h(formal)f(solution)h(of)f (the)g(equation)h(\(25\),)k(w)m(e)43 b(get)h(from)f(the)g(general)h Fp(\024)p Fj(\000)p Fq(summabilit)m(y)75 868 y(theory)-8 b(,)34 b(see)f([2)q(],)g(c)m(hapter)g(6.3,)h(that)f(the)g(1)p Fj(\000)p Fq(sum)f Fp(Y)1908 882 y Fn(U)1956 894 y Fd(d)1991 882 y Fn(;)p Fk(E)2058 868 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))34 b(is)e(also)i(a)f(solution)g(of)f(\(25\).)49 b(Finally)-8 b(,)34 b(the)75 981 y(asymptotic)e(expansion)e(\(27\))i (follo)m(ws)f(from)f(the)h(estimates)h(\(23\).)1344 b Ff(2)75 1274 y Fc(4.3)112 b(Analytic)29 b(solutions)h(in)e(a)i (parameter)f(of)g(a)g(singularly)h(p)s(erturb)s(ed)g(Cauc)m(h)m(y)f (prob-)330 1390 y(lem)75 1562 y Fq(W)-8 b(e)32 b(de\014ne)d(what)h (will)h(b)s(e)f(called)h(a)g(go)s(o)s(d)f(co)m(v)m(ering.)75 1774 y Fh(De\014nition)35 b(6)46 b Fg(F)-7 b(or)34 b(dir)-5 b(e)g(ctions)34 b Fq(0)26 b Fj(\024)f Fp(d)1457 1788 y Fo(0)1522 1774 y Fp(<)g Fj(\001)15 b(\001)g(\001)26 b Fp(<)f(d)1892 1788 y Fn(\027)t Fk(\000)p Fo(1)2051 1774 y Fp(<)g Fq(2)p Fp(\031)s Fg(,)33 b Fp(d)2355 1788 y Fn(\027)2423 1774 y Fq(=)25 b Fp(d)2566 1788 y Fo(0)2606 1774 y Fg(,)33 b(we)f(c)-5 b(onsider)34 b(se)-5 b(ctors)1441 1979 y Fj(E)1489 1993 y Fn(i)1542 1979 y Fq(=)25 b Fp(S)1694 1997 y Fo(\()p Fn(d)1757 2007 y Fd(i)p Fl(+1)1861 1997 y Fo(+)p Fn(d)1952 2007 y Fd(i)1978 1997 y Fo(\))p Fn(=)p Fo(2)p Fn(;)p Fo(\()p Fn(\031)r Fo(+)p Fn(\016)2251 2007 y Fd(i)2279 1997 y Fo(\))p Fn(=)p Fo(2)p Fn(;\017)2425 2006 y Fl(0)2464 1979 y Fp(;)75 2183 y Fg(c)-5 b(enter)g(e)g(d)39 b(at)e(0,)i(with)f(bise)-5 b(cting)37 b(dir)-5 b(e)g(ction)39 b Fq(\()p Fp(d)1696 2197 y Fn(i)p Fo(+1)1839 2183 y Fq(+)23 b Fp(d)1980 2197 y Fn(i)2008 2183 y Fq(\))p Fp(=)p Fq(2)p Fg(,)40 b(r)-5 b(adius)39 b Fp(\017)2516 2197 y Fo(0)2592 2183 y Fg(and)f(op)-5 b(ening)38 b Fp(\031)27 b Fq(+)c Fp(\016)3325 2197 y Fn(i)3354 2183 y Fg(,)38 b(wher)-5 b(e)39 b Fp(\016)3722 2197 y Fn(i)3784 2183 y Fq(=)75 2296 y Fp(d)122 2310 y Fn(i)p Fo(+1)255 2296 y Fj(\000)14 b Fp(d)387 2310 y Fn(i)415 2296 y Fg(,)31 b Fq(0)25 b Fj(\024)g Fp(i)h Fj(\024)f 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Fn(;k)1744 4014 y Fl(1)1778 4005 y Fn(;n)1845 3990 y Fq(\()p Fp(\017)p Fq(\))p Fp(z)1998 3953 y Fn(n)2106 3990 y Fp(;)76 b(b)p Fq(\()p Fp(z)t(;)15 b(\017)p Fq(\))27 b(=)2562 3904 y Fm(X)2562 4100 y Fn(n)p Fk(\025)p Fo(0)2710 3990 y Fp(b)2749 4004 y Fn(n)2796 3990 y Fq(\()p Fp(\017)p Fq(\))p Fp(z)2949 3953 y Fn(n)2997 3990 y Fp(;)75 4296 y Fq(b)s(e)43 b(holomorphic)h(functions)g(on)f(a)i(neigh)m(b)s(orho)s (o)s(d)d(of)i(the)g(origin)h(in)e Fi(C)2709 4263 y Fo(2)2748 4296 y Fq(,)48 b(for)43 b(all)i(\()p Fp(k)3195 4310 y Fo(0)3235 4296 y Fp(;)15 b(k)3322 4310 y Fo(1)3362 4296 y Fq(\))49 b Fj(2)e(S)7 b Fq(.)81 b(Let)75 4409 y Fj(fE)168 4423 y Fn(i)197 4409 y Fj(g)242 4423 y Fo(0)p Fk(\024)p Fn(i)p Fk(\024)p Fn(\027)t Fk(\000)p Fo(1)585 4409 y Fq(b)s(e)40 b(a)h(go)s(o)s(d)g(co)m(v)m(ering)h(of)f(the)g(origin)g(in) g Fi(C)p Fq(.)71 b(F)-8 b(or)41 b(all)h(0)h Fj(\024)f Fp(i)h Fj(\024)f Fp(\027)33 b Fj(\000)27 b Fq(1,)43 b(w)m(e)f(consider) e(the)75 4522 y(follo)m(wing)32 b(singularly)e(p)s(erturb)s(ed)e(Cauc)m (h)m(y)j(problem)75 4751 y(\(31\))85 b(4)p Fp(\017t)435 4713 y Fo(2)474 4751 y Fp(@)522 4765 y Fn(t)552 4751 y Fp(@)605 4713 y Fn(S)600 4773 y(z)656 4751 y Fp(X)731 4765 y Fn(i)760 4751 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))21 b(+)f(\(2)p Fp(t\017)h Fq(+)f(1\))p Fp(@)1533 4713 y Fn(S)1528 4773 y(z)1585 4751 y Fp(X)1660 4765 y Fn(i)1688 4751 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))719 4956 y(=)905 4870 y Fm(X)815 5071 y Fo(\()p Fn(k)879 5080 y Fl(0)914 5071 y Fn(;k)971 5080 y Fl(1)1005 5071 y Fo(\))p Fk(2S)1142 4956 y Fp(b)1181 4971 y Fn(k)1218 4980 y Fl(0)1253 4971 y Fn(;k)1310 4980 y Fl(1)1348 4956 y Fq(\()p Fp(z)t(;)g(\017)p Fq(\))p Fp(@)1594 4916 y Fn(k)1631 4925 y Fl(0)1589 4980 y Fn(t)1671 4956 y Fp(@)1724 4919 y Fn(k)1761 4928 y Fl(1)1719 4979 y Fn(z)1800 4956 y Fp(X)1875 4970 y Fn(i)1903 4956 y Fq(\()p Fp(t;)g(z)t(;)g(\017)p Fq(\))22 b(+)e Fp(b)p Fq(\()p Fp(z)t(;)15 b(\017)p Fq(\))p Fp(@)2562 4970 y Fn(t)2608 4832 y Fm(Z)2699 4859 y Fn(t)2658 5039 y Fo(0)2744 4956 y Fp(X)2819 4970 y Fn(i)2847 4956 y Fq(\()p Fp(t)21 b Fj(\000)e Fp(\034)5 b(;)15 b(z)t(;)g(\017)p Fq(\))p Fp(X)3344 4970 y Fn(i)3374 4956 y Fq(\()p Fp(\034)5 b(;)15 b(z)t(;)g(\017)p Fq(\))p Fp(d\034)5 b(;)75 5260 y Fq(for)30 b(giv)m(en)i(initial)f(conditions)75 5464 y(\(32\))889 b(\()p Fp(@)1212 5426 y Fn(j)1207 5486 y(z)1249 5464 y Fp(X)1324 5478 y Fn(i)1352 5464 y Fq(\)\()p Fp(t;)15 b Fq(0)p Fp(;)g(\017)p Fq(\))28 b(=)d Fp(')1835 5478 y Fn(i;j)1915 5464 y Fq(\()p Fp(t;)15 b(\017)p Fq(\))62 b Fp(;)76 b Fq(0)25 b Fj(\024)g Fp(j)31 b Fj(\024)25 b Fp(S)g Fj(\000)20 b Fq(1)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 3764 100 a Fq(18)75 399 y(where)42 b(the)h(functions)g Fp(')983 413 y Fn(i;j)1063 399 y Fq(\()p Fp(t;)15 b(\017)p Fq(\))44 b(are)f(constructed)g(as)g(follo)m (ws.)79 b(Let)44 b Fj(ff)p Fp(U)2777 414 y Fn(f)2811 424 y Fd(i)2838 414 y Fn(;\022)r(;R)2970 399 y Fj(g)3015 413 y Fo(0)p Fk(\024)p Fn(i)p Fk(\024)p Fn(\027)t Fk(\000)p Fo(1)3318 399 y Fp(;)15 b Fj(T)23 b(g)43 b Fq(a)g(family)75 511 y(of)k(op)s(en)g(sets)h(asso)s(ciated)g(to)g Fj(fE)1297 525 y Fn(i)1326 511 y Fj(g)1371 525 y Fo(0)p Fk(\024)p Fn(i)p Fk(\024)p Fn(\027)t Fk(\000)p Fo(1)1674 511 y Fq(.)91 b(F)-8 b(or)48 b(all)g(0)54 b Fj(\024)f Fp(j)59 b Fj(\024)53 b Fp(S)36 b Fj(\000)31 b Fq(1,)53 b(all)48 b(0)53 b Fj(\024)g Fp(i)h Fj(\024)f Fp(\027)37 b Fj(\000)31 b Fq(1,)53 b(let)75 624 y Fj(f)p Fp(V)173 638 y Fn(j)210 624 y Fp(;)15 b(V)303 638 y Fn(U)351 650 y Fd(f)381 666 y(i)412 638 y Fn(;)p Fk(E)469 648 y Fd(i)495 638 y Fn(;j)551 624 y Fp(;)g(T)644 638 y Fo(0)684 624 y Fp(;)g(\034)764 638 y Fo(0)803 624 y Fp(;)g(\017)880 638 y Fo(0)920 624 y Fj(g)31 b Fq(b)s(e)f(an)g(admissible)g(set.)42 b(Let)1469 890 y Fp(V)1522 904 y Fn(j)1559 890 y Fq(\()p Fp(\034)5 b(;)15 b(\017)p Fq(\))26 b(=)1884 804 y Fm(X)1873 1000 y Fn(m)p Fk(\025)p Fo(0)2051 828 y Fp(\037)2108 842 y Fn(m;j)2226 828 y Fq(\()p Fp(\017)p Fq(\))p 2051 869 284 4 v 2082 952 a(\(2)p Fp(m)p Fq(\)!)2344 890 y Fp(\034)2394 853 y Fn(m)75 1182 y Fq(b)s(e)k(it's)h(T)-8 b(a)m(ylor)31 b(expansion)g(for)f(\()p Fp(\034)5 b(;)15 b(\017)p Fq(\))26 b Fj(2)f Fp(D)s Fq(\(0)p Fp(;)15 b(\034)1740 1196 y Fo(0)1780 1182 y Fq(\))21 b Fj(\002)e Fp(D)s Fq(\(0)p Fp(;)c(\017)2161 1196 y Fo(0)2202 1182 y Fq(\).)41 b(W)-8 b(e)31 b(consider)g(the)f (formal)h(series)1485 1406 y(^)1471 1429 y Fp(Y)1524 1443 y Fn(j)1561 1429 y Fq(\()p Fp(t;)15 b(\017)p Fq(\))26 b(:=)1899 1343 y Fm(X)1888 1538 y Fn(m)p Fk(\025)p Fo(0)2066 1367 y Fp(\037)2123 1381 y Fn(m;j)2241 1367 y Fq(\()p Fp(\017)p Fq(\))p 2066 1408 V 2155 1491 a Fp(m)p Fq(!)2359 1429 y Fp(t)2392 1391 y Fn(m)75 1739 y Fq(for)34 b(all)h Fp(\017)d Fj(2)f Fp(D)s Fq(\(0)p Fp(;)15 b(\017)744 1753 y Fo(0)784 1739 y Fq(\).)53 b(W)-8 b(e)35 b(de\014ne)e Fp(Y)1376 1753 y Fn(U)1424 1765 y Fd(f)1454 1780 y(i)1484 1753 y Fn(;)p Fk(E)1541 1763 y Fd(i)1568 1753 y Fn(;j)1624 1739 y Fq(\()p Fp(t;)15 b(\017)p Fq(\))35 b(as)f(the)h(1)p Fj(\000)p Fq(sum)e(of)2548 1716 y(^)2534 1739 y Fp(Y)2587 1753 y Fn(j)2624 1739 y Fq(\()p Fp(t;)15 b(\017)p Fq(\))35 b(in)e(the)i(direction)g Fp(f)3537 1753 y Fn(i)3564 1739 y Fq(,)h(whic)m(h)75 1852 y(exists)31 b(from)f(Prop)s(osition)g(3)h (using)f(the)h(estimates)h(\(24\).)42 b(Then,)29 b(w)m(e)i(put)1460 2050 y Fp(')1519 2064 y Fn(i;j)1599 2050 y Fq(\()p Fp(t;)15 b(\017)p Fq(\))26 b(:=)g Fp(Y)1980 2064 y Fn(U)2028 2076 y Fd(f)2058 2092 y(i)2088 2064 y Fn(;)p Fk(E)2145 2074 y Fd(i)2171 2064 y Fn(;j)2227 2050 y Fq(\()p Fp(\017t;)15 b(\017)p Fq(\))p Fp(;)75 2249 y Fq(for)29 b(all)i(0)25 b Fj(\024)g Fp(j)31 b Fj(\024)25 b Fp(S)e Fj(\000)18 b Fq(1,)31 b(all)f(0)c Fj(\024)f Fp(i)g Fj(\024)g Fp(\027)f Fj(\000)18 b Fq(1.)41 b(By)30 b(construction,)g(the)g(function)f Fp(')2908 2263 y Fn(i;j)2989 2249 y Fq(\()p Fp(t;)15 b(\017)p Fq(\))30 b(is)g(a)f(holomorphic)75 2362 y(and)h(b)s(ounded)e (function)i(on)g Fj(T)43 b(\002)20 b(E)1337 2376 y Fn(i)1365 2362 y Fq(,)31 b(for)f(all)h(0)26 b Fj(\024)f Fp(j)31 b Fj(\024)25 b Fp(S)g Fj(\000)20 b Fq(1,)31 b(all)g(0)26 b Fj(\024)f Fp(i)g Fj(\024)g Fp(\027)h Fj(\000)20 b Fq(1.)75 2569 y Fh(Prop)s(osition)36 b(4)46 b Fg(L)-5 b(et)36 b(the)h(fol)5 b(lowing)37 b(assumption)h(hold:)50 b(F)-7 b(or)37 b(al)5 b(l)36 b Fq(\()p Fp(k)2551 2583 y Fo(0)2591 2569 y Fp(;)15 b(k)2678 2583 y Fo(1)2718 2569 y Fq(\))32 b Fj(2)f(S)43 b Fg(we)36 b(have)g Fp(S)h Fj(\025)31 b Fq(2)p Fp(k)3612 2583 y Fo(0)3674 2569 y Fq(+)23 b Fp(k)3815 2583 y Fo(1)75 2681 y Fg(and)34 b Fp(S)c(>)25 b(k)481 2695 y Fo(1)521 2681 y Fg(.)41 b(Mor)-5 b(e)g(over,)34 b(we)f(put)813 2889 y Fj(j)p Fp(b)p Fj(j)902 2904 y Fn(k)939 2913 y Fl(0)974 2904 y Fn(;k)1031 2913 y Fl(1)1069 2889 y Fq(\()p Fp(r)m(;)15 b(\017)p Fq(\))26 b(=)1377 2802 y Fm(X)1376 2998 y Fn(n)p Fk(\025)p Fo(0)1525 2889 y Fj(j)p Fp(b)1589 2904 y Fn(k)1626 2913 y Fl(0)1660 2904 y Fn(;k)1717 2913 y Fl(1)1751 2904 y Fn(;n)1818 2889 y Fq(\()p Fp(\017)p Fq(\))p Fj(j)p Fp(r)1994 2851 y Fn(n)2106 2889 y Fp(;)81 b Fj(j)p Fp(b)p Fj(j)p Fq(\()p Fp(r)m(;)15 b(\017)p Fq(\))27 b(=)2610 2802 y Fm(X)2609 2998 y Fn(n)p Fk(\025)p Fo(0)2757 2889 y Fj(j)p Fp(b)2821 2903 y Fn(n)2868 2889 y Fq(\()p Fp(\017)p Fq(\))p Fj(j)p Fp(r)3044 2851 y Fn(n)3092 2889 y Fp(;)75 3175 y Fg(for)33 b Fp(r)28 b(>)d Fq(0)33 b Fg(and)g(we)g(assume)h(that)f(ther)-5 b(e)34 b(exists)f Fp(r)28 b(>)d Fq(0)33 b Fg(such)f(that)1303 3374 y Fq(lim)1299 3432 y Fn(\017)p Fk(!)p Fo(0)1449 3374 y Fj(j)p Fp(b)p Fj(j)1538 3389 y Fn(k)1575 3398 y Fl(0)1610 3389 y Fn(;k)1667 3398 y Fl(1)1705 3374 y Fq(\()p Fp(r)m(;)15 b(\017)p Fq(\))27 b(=)i(lim)2013 3432 y Fn(\017)p Fk(!)p Fo(0)2162 3374 y Fj(j)p Fp(b)p Fj(j)p Fq(\()p Fp(r)m(;)15 b(\017)p Fq(\))27 b(=)e(0)p Fp(;)75 3603 y Fg(for)33 b(al)5 b(l)33 b Fq(\()p Fp(k)432 3617 y Fo(0)472 3603 y Fp(;)15 b(k)559 3617 y Fo(1)600 3603 y Fq(\))25 b Fj(2)g(S)7 b Fg(.)42 b(L)-5 b(et)33 b(the)g(initial)g(data)h(\(32\))g(c)-5 b(onstructe)g(d)34 b(as)f(ab)-5 b(ove.)216 3716 y(Then,)47 b(for)d(al)5 b(l)44 b Fq(0)h Fj(\024)g Fp(i)g Fj(\024)f Fp(\027)34 b Fj(\000)28 b Fq(1)p Fg(,)46 b(the)e(pr)-5 b(oblem)45 b(\(31\),)j(\(32\))c(has)h(a)f(solution)g Fp(X)3144 3730 y Fn(i)3173 3716 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))45 b Fg(such)e(that)75 3829 y Fp(@)128 3796 y Fn(l)123 3851 y(t)154 3829 y Fp(X)229 3843 y Fn(i)258 3829 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))41 b Fg(is)f(holomorphic)j(and)e(b)-5 b(ounde)g(d)42 b(on)e Fj(T)48 b(\002)26 b Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\))26 b Fj(\002)g(E)2515 3843 y Fn(i)2543 3829 y Fg(,)41 b(for)g(some)g Fp(\032)d(>)g Fq(0)p Fg(,)k(for)f(al)5 b(l)40 b Fp(l)h Fj(\025)d Fq(0)p Fg(.)75 3942 y(Mor)-5 b(e)g(over,)34 b(for)f(al)5 b(l)33 b Fp(l)27 b Fj(\025)e Fq(0)p Fg(,)33 b(ther)-5 b(e)33 b(exist)g(two)h(c)-5 b(onstants)34 b Fp(C)q(;)15 b(A)27 b(>)e Fq(0)33 b Fg(such)f(that)75 4140 y Fq(\(33\))817 b(sup)902 4223 y Fn(t)p Fk(2T)17 b Fn(;z)s Fk(2)p Fn(D)r Fo(\(0)p Fn(;\032)p Fo(\))1353 4140 y Fj(j)p Fp(@)1431 4103 y Fn(l)1426 4163 y(t)1458 4140 y Fp(X)1533 4154 y Fn(i)p Fo(+1)1652 4140 y Fq(\()p Fp(t;)e(z)t(;)g(\017)p Fq(\))21 b Fj(\000)f Fp(@)2083 4103 y Fn(l)2078 4163 y(t)2110 4140 y Fp(X)2185 4154 y Fn(i)2213 4140 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))p Fj(j)27 b(\024)e Fp(C)7 b(A)2767 4103 y Fn(n)2814 4140 y Fp(n)p Fq(!)p Fj(j)p Fp(\017)p Fj(j)2981 4103 y Fn(n)75 4411 y Fg(for)40 b(al)5 b(l)39 b Fp(n)e Fj(\025)g Fq(0)p Fg(,)k(for)f(al)5 b(l)40 b Fp(\017)d Fj(2)g(E)1186 4425 y Fn(i)1239 4411 y Fj(\\)24 b(E)1372 4425 y Fn(i)p Fo(+1)1491 4411 y Fg(,)40 b(for)g(al)5 b(l)40 b Fq(0)e Fj(\024)e Fp(i)i Fj(\024)f Fp(\027)31 b Fj(\000)24 b Fq(1)40 b Fg(\(wher)-5 b(e)40 b(by)f(c)-5 b(onvention)41 b Fp(X)3436 4425 y Fn(\027)3479 4411 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))39 b(=)75 4524 y Fp(X)150 4538 y Fo(0)190 4524 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))34 b Fg(and)f Fp(S)722 4538 y Fn(\027)791 4524 y Fq(=)24 b Fp(S)942 4538 y Fo(0)982 4524 y Fg(\).)75 4779 y Fh(Pro)s(of)j Fq(F)-8 b(or)27 b(all)g(0)f Fj(\024)f Fp(i)h Fj(\024)e Fp(\027)18 b Fj(\000)12 b Fq(1,)28 b(w)m(e)e(consider)g(the)h(Cauc)m(h) m(y)g(problem)e(\(25\),)k(\(26\))f(for)e(the)g(initial)i(conditions)75 4892 y Fp(Y)128 4906 y Fn(U)176 4918 y Fd(f)206 4933 y(i)236 4906 y Fn(;)p Fk(E)293 4916 y Fd(i)319 4906 y Fn(;j)375 4892 y Fq(\()p Fp(t;)15 b(\017)p Fq(\),)40 b(for)c(0)h Fj(\024)e Fp(j)42 b Fj(\024)35 b Fp(S)30 b Fj(\000)24 b Fq(1.)60 b(F)-8 b(rom)37 b(Theorem)g(2,)i(this)e (problem)f(has)g(a)h(solution)h Fp(Y)3392 4906 y Fn(U)3440 4918 y Fd(f)3470 4933 y(i)3500 4906 y Fn(;)p Fk(E)3557 4916 y Fd(i)3587 4892 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))75 5004 y(whic)m(h)30 b(is)h(holomorphic)f(and)g(b)s(ounded)e(on) i Fp(U)1683 5019 y Fn(f)1717 5029 y Fd(i)1744 5019 y Fn(;\022)r(;R)1896 5004 y Fj(\002)19 b Fp(D)s Fq(\(0)p Fp(;)c(\032)p Fq(\))22 b Fj(\002)e(E)2427 5018 y Fn(i)2455 5004 y Fq(,)31 b(for)f Fp(\022)d(>)e Fq(0,)31 b Fp(R)q(;)15 b(\032)26 b(>)f Fq(0.)41 b(No)m(w,)31 b(w)m(e)g(put)1430 5203 y Fp(X)1505 5217 y Fn(i)1533 5203 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))27 b(:=)e Fp(Y)2000 5217 y Fn(U)2048 5229 y Fd(f)2078 5245 y(i)2108 5217 y Fn(;)p Fk(E)2165 5227 y Fd(i)2195 5203 y Fq(\()p Fp(\017t;)15 b(z)t(;)g(\017)p Fq(\))75 5402 y(for)29 b(all)g(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))27 b Fj(2)e(T)39 b(\002)17 b Fp(D)s Fq(\(0)p Fp(;)e(\032)p Fq(\))j Fj(\002)f(E)1327 5416 y Fn(i)1355 5402 y Fq(.)40 b(By)30 b(construction,)f(one)h(can)f(c)m(hec)m(k)h(that)f Fp(X)2950 5416 y Fn(i)2979 5402 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))30 b(is)f(a)g(solution)g(of)75 5514 y(the)24 b(Cauc)m(h)m(y)g (problem)g(\(31\),)j(\(32\))e(whic)m(h)f(is)g(holomorphic)f(and)h(b)s (ounded)d(on)j Fj(T)30 b(\002)7 b Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\))7 b Fj(\002)g(E)3367 5528 y Fn(i)3397 5514 y Fq(.)38 b(Moreo)m(v)m(er,)75 5627 y(all)32 b(the)g(partial)g (deriv)-5 b(ativ)m(es)32 b Fp(@)1160 5594 y Fn(l)1155 5650 y(t)1187 5627 y Fp(X)1262 5641 y Fn(i)1290 5627 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))28 b(=)f Fp(\017)1719 5594 y Fn(l)1745 5627 y Fq(\()p Fp(@)1833 5594 y Fn(l)1828 5650 y(t)1859 5627 y Fp(Y)1912 5641 y Fn(U)1960 5653 y Fd(f)1990 5669 y(i)2020 5641 y Fn(;)p Fk(E)2077 5651 y Fd(i)2108 5627 y Fq(\)\()p Fp(\017t;)15 b(z)t(;)g(\017)p Fq(\))33 b(are)e(also)i(holomorphic)e(and)g(b)s(ounded)75 5740 y(functions)f(on)g Fj(T)43 b(\002)20 b Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\))21 b Fj(\002)f(E)1217 5754 y Fn(i)1246 5740 y Fq(,)30 b(for)g(all)h Fp(l)d Fj(\025)d Fq(0.)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 3764 100 a Fq(19)216 399 y(In)29 b(the)g(rest)h(of)f(the)h(pro)s(of,)f(w)m(e)g(sho)m(w)h(the)f (estimates)i(\(33\).)42 b(F)-8 b(rom)30 b(the)f(estimates)i(\(27\),)h (there)d(exists)h(a)75 511 y(sequence)k(\()p Fp(\037)547 525 y Fn(m)613 511 y Fq(\()p Fp(z)t(;)15 b(\017)p Fq(\)\))841 525 y Fn(m)p Fk(\025)p Fo(0)1033 511 y Fq(of)33 b(holomorphic)h(and)e (b)s(ounded)f(functions)i(on)g Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\))23 b Fj(\002)f Fp(D)s Fq(\(0)p Fp(;)15 b(\017)3371 525 y Fo(0)3412 511 y Fq(\))33 b(suc)m(h)g(that)75 624 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b(v)m(ector)i(space)e Fi(E)859 2822 y Fn(K)927 2808 y Fq(\()p Fj(T)44 b(\002)20 b Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\)\))32 b(will)e(b)s(e)g(equipp)s(ed)f (with)h(the)h(norm)984 3098 y Fj(jj)p Fp(f)10 b Fq(\()p Fp(t;)15 b(z)t Fq(\))p Fj(jj)1328 3117 y Fb(E)1372 3128 y Fd(K)1432 3117 y Fo(\()p Fk(T)j(\002)p Fn(D)r Fo(\(0)p Fn(;\032)p Fo(\)\))1833 3098 y Fq(=)1963 2984 y Fn(K)1929 3012 y Fm(X)1929 3209 y Fn(h)p Fo(=0)2226 3098 y Fq(sup)2076 3181 y Fn(t)p Fk(2T)f Fn(;z)s Fk(2)p Fn(D)r Fo(\(0)p Fn(;\032)p Fo(\))2528 3098 y Fj(j)p Fp(@)2606 3061 y Fn(h)2601 3121 y(t)2651 3098 y Fp(f)10 b Fq(\()p Fp(t;)15 b(z)t Fq(\))p Fj(j)p Fp(:)75 3388 y Fq(The)36 b(space)i(\()p Fi(E)614 3402 y Fn(K)682 3388 y Fq(\()p Fj(T)48 b(\002)24 b Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\)\))p Fp(;)g Fj(jj)p Fp(:)p Fj(jj)1390 3406 y Fb(E)1434 3417 y Fd(K)1496 3406 y Fo(\()p Fk(T)j(\002)p Fn(D)r Fo(\(0)p Fn(;\032)p Fo(\)\))1872 3388 y Fq(\))37 b(is)g(a)g(Banac)m(h)h(space.)61 b(Moreo)m(v)m(er,)41 b(for)36 b Fp(K)3415 3355 y Fk(0)3474 3388 y Fp(<)g(K)7 b Fq(,)38 b(the)75 3501 y(Banac)m(h)31 b(space)g Fi(E)703 3515 y Fn(K)771 3501 y Fq(\()p Fj(T)44 b(\002)19 b Fp(D)s Fq(\(0)p Fp(;)c(\032)p Fq(\)\))32 b(is)f(con)m(tin)m(uously)g(em)m(b)s (edded)e(in)i Fi(E)2554 3516 y Fn(K)2619 3497 y Fe(0)2644 3501 y Fq(\()p Fj(T)43 b(\002)20 b Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\)\).)216 3664 y(In)27 b(this)h(section,)i(w)m(e)e (establish)g(the)g(existence)h(of)f(formal)g(p)s(o)m(w)m(er)g(series)g (solution)3126 3641 y(^)3100 3664 y Fp(X)7 b Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))27 b Fj(2)e Fi(E)3622 3678 y Fo(0)3661 3664 y Fq(\()p Fj(T)38 b(\002)75 3776 y Fp(D)s Fq(\(0)p Fp(;)15 b(\032)p Fq(\)\)[[)p Fp(\017)p Fq(]])32 b(of)d(the)h(equation)g(\(31\))g(whic)m(h)f(satis\014es)h(the) f(prop)s(ert)m(y)g(that)h(the)f(solutions)h Fp(X)3301 3790 y Fn(i)3329 3776 y Fq(\()p Fp(t;)15 b(z)t(;)g(\017)p Fq(\))31 b(of)e(the)75 3889 y(problem)h(\(31\),)i(\(32\))g(has)1026 3866 y(^)1000 3889 y Fp(X)38 b Fq(as)30 b(Gevrey)h(asymptotic)h (expansion)e(of)h(order)f(1)g(on)h Fj(E)3020 3903 y Fn(i)3048 3889 y Fq(.)216 4052 y(The)h(pro)s(of)g(is)h(based)f(on)g(a)h (cohomological)j(criterion)e(for)e(summabilit)m(y)h(of)g(formal)f (series)h(with)g(co)s(ef-)75 4165 y(\014cien)m(ts)f(in)g(a)g(Banac)m(h) g(space,)h(see)g([2],)g(page)f(121,)i(whic)m(h)d(is)h(kno)m(wn)f(as)h (the)g(Sibuy)m(a-Malgrange)h(theorem)75 4278 y(in)d(the)h(literature.) 75 4441 y Fh(Theorem)h(\(CC\))26 b Fg(L)-5 b(et)30 b Fq(\()p Fi(E)p Fp(;)15 b Fj(jj)p Fp(:)p Fj(jj)1191 4455 y Fb(E)1242 4441 y Fq(\))30 b Fg(b)-5 b(e)30 b(a)h(Banach)g(sp)-5 b(ac)g(e)32 b(over)e Fi(C)p Fg(.)41 b(Assume)30 b(that)h(a)g(c)-5 b(ol)5 b(le)-5 b(ction)31 b Fq(\()p Fp(d)3465 4455 y Fn(i)3494 4441 y Fp(;)15 b Fj(E)3582 4455 y Fn(i)3611 4441 y Fp(;)g(f)3696 4455 y Fn(i)3724 4441 y Fq(\))30 b Fg(is)75 4554 y(given)i(with)211 4741 y Fj(\017)46 b Fq(Directions)32 b(0)26 b Fj(\024)f Fp(d)950 4755 y Fo(0)1015 4741 y Fp(<)g Fj(\001)15 b(\001)g(\001)26 b Fp(<)f(d)1385 4755 y Fn(\027)t Fk(\000)p Fo(1)1544 4741 y Fp(<)g Fq(2)p Fp(\031)s Fq(,)31 b Fp(d)1843 4755 y Fn(\027)1912 4741 y Fq(=)25 b Fp(d)2055 4755 y Fo(0)2094 4741 y Fq(.)211 4929 y Fj(\017)46 b Fq(F)-8 b(or)31 b(all)h(0)25 b Fj(\024)g Fp(i)h Fj(\024)f Fp(\027)h Fj(\000)20 b Fq(1,)31 b(sectors)g Fj(E)1521 4943 y Fn(i)1574 4929 y Fq(=)25 b Fp(S)1726 4947 y Fo(\()p Fn(d)1789 4957 y Fd(i)p Fl(+1)1893 4947 y Fo(+)p Fn(d)1984 4957 y Fd(i)2011 4947 y Fo(\))p Fn(=)p Fo(2)p Fn(;)p Fo(\()p Fn(\031)r Fo(+)p Fn(\016)2284 4957 y Fd(i)2311 4947 y Fo(\))p Fn(=)p Fo(2)p Fn(;\017)2457 4956 y Fl(0)2526 4929 y Fq(where)30 b Fp(\016)2829 4943 y Fn(i)2883 4929 y Fq(=)25 b Fp(d)3026 4943 y Fn(i)p Fo(+1)3165 4929 y Fj(\000)20 b Fp(d)3303 4943 y Fn(i)3331 4929 y Fq(.)211 5116 y Fj(\017)46 b Fq(F)-8 b(or)31 b(all)h(0)25 b Fj(\024)g Fp(i)h Fj(\024)f Fp(\027)h Fj(\000)20 b Fq(1,)31 b(holomorphic)f(functions)g Fp(f)2131 5130 y Fn(i)2189 5116 y Fq(from)g Fj(E)2452 5130 y Fn(i)2510 5116 y Fq(in)m(to)i(the)e 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