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T\"oplitz matrix, Hankel matrices, asymptotically T\"oplitz matrices, algebra of matrices, non-commutative exponent
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 1 /periodcentered put
dup 2 /multiply put
dup 3 /asteriskmath put
dup 6 /plusminus put
dup 8 /circleplus put
dup 10 /circlemultiply put
dup 17 /equivalence put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 26 /propersubset put
dup 29 /greatermuch put
dup 33 /arrowright put
dup 41 /arrowdblright put
dup 49 /infinity put
dup 50 /element put
dup 54 /negationslash put
dup 55 /mapsto put
dup 56 /universal put
dup 59 /emptyset put
dup 65 /A put
dup 77 /M put
dup 79 /O put
dup 83 /S put
dup 91 /union put
dup 92 /intersection put
dup 94 /logicaland put
dup 95 /logicalor put
dup 100 /ceilingleft put
dup 101 /ceilingright put
dup 102 /braceleft put
dup 103 /braceright put
dup 104 /angbracketleft put
dup 105 /angbracketright put
dup 106 /bar put
dup 107 /bardbl put
dup 110 /backslash put
dup 112 /radical put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI5
%!PS-AdobeFont-1.1: CMMI5 1.100
%%CreationDate: 1996 Aug 02 08:21:10
% 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 (CMMI5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI5 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 13 /gamma put
dup 28 /tau put
dup 97 /a put
dup 100 /d put
dup 116 /t put
readonly def
/FontBBox{37 -250 1349 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: MSBM7
%!PS-AdobeFont-1.1: MSBM7 2.1
%%CreationDate: 1992 Oct 17 08:30:50
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM7) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 67 /C put
dup 82 /R put
dup 90 /Z put
readonly def
/FontBBox{0 -504 2615 1004}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 12 /fi put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 46 /period put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 69 /E put
dup 72 /H put
dup 73 /I put
dup 76 /L put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 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{-36 -250 1070 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% 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 (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 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 77 /M put
dup 108 /l put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX12
%!PS-AdobeFont-1.1: CMBX12 1.0
%%CreationDate: 1991 Aug 20 16:34:54
% 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 (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX12 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 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 58 /colon put
dup 65 /A put
dup 68 /D put
dup 69 /E put
dup 73 /I put
dup 76 /L put
dup 77 /M put
dup 82 /R put
dup 84 /T put
dup 87 /W put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 123 /endash put
dup 127 /dieresis put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 46 /period 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 65 /A put
dup 67 /C put
dup 76 /L put
dup 78 /N put
dup 80 /P put
dup 82 /R put
dup 84 /T put
dup 97 /a put
dup 101 /e 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 121 /y put
dup 122 /z put
readonly def
/FontBBox{-301 -250 1164 946}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 16 /parenleftBig put
dup 17 /parenrightBig put
dup 18 /parenleftbigg put
dup 19 /parenrightbigg put
dup 40 /braceleftBigg put
dup 80 /summationtext put
dup 82 /integraltext put
dup 88 /summationdisplay put
dup 89 /productdisplay put
dup 90 /integraldisplay put
dup 91 /uniondisplay put
dup 108 /ceilingleftBig put
dup 109 /ceilingrightBig put
dup 112 /radicalbig put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR5
%!PS-AdobeFont-1.1: CMR5 1.00B
%%CreationDate: 1992 Feb 19 19:55:02
% 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 (CMR5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR5 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 1 /Delta put
dup 43 /plus put
dup 49 /one put
dup 50 /two put
readonly def
/FontBBox{-341 -250 1304 965}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 12 /fi put
dup 14 /ffi put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 55 /seven put
dup 58 /colon put
dup 65 /A put
dup 66 /B put
dup 69 /E put
dup 70 /F put
dup 72 /H put
dup 73 /I put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 80 /P put
dup 83 /S put
dup 84 /T 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 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{-163 -250 1146 969}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR7
%!PS-AdobeFont-1.1: CMR7 1.0
%%CreationDate: 1991 Aug 20 16:39:21
% 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 (CMR7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR7 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 1 /Delta put
dup 3 /Lambda put
dup 6 /Sigma put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 54 /six put
dup 61 /equal put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 94 /circumflex put
dup 103 /g put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 115 /s put
dup 126 /tilde put
readonly def
/FontBBox{-27 -250 1122 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 67 /C put
dup 82 /R put
dup 84 /T put
dup 90 /Z put
dup 123 /kappa put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY7
%!PS-AdobeFont-1.1: CMSY7 1.0
%%CreationDate: 1991 Aug 15 07:21:52
% 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 (CMSY7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY7 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 3 /asteriskmath put
dup 6 /plusminus put
dup 7 /minusplus 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 54 /negationslash put
dup 91 /union put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
dup 110 /backslash put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI7
%!PS-AdobeFont-1.1: CMMI7 1.100
%%CreationDate: 1996 Jul 23 07:53:53
% 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 (CMMI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI7 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 13 /gamma put
dup 14 /delta put
dup 18 /theta put
dup 27 /sigma put
dup 28 /tau put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 65 /A put
dup 70 /F put
dup 75 /K put
dup 76 /L put
dup 78 /N put
dup 79 /O put
dup 82 /R put
dup 84 /T put
dup 88 /X put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 105 /i put
dup 106 /j put
dup 109 /m put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{0 -250 1171 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 13 /gamma put
dup 14 /delta put
dup 18 /theta put
dup 22 /mu put
dup 27 /sigma put
dup 28 /tau put
dup 32 /psi put
dup 34 /epsilon 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 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 70 /F put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 79 /O put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 86 /V 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 105 /i put
dup 106 /j 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 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 1 /Delta put
dup 3 /Lambda put
dup 6 /Sigma put
dup 7 /Upsilon put
dup 8 /Phi put
dup 11 /ff put
dup 12 /fi put
dup 13 /fl put
dup 14 /ffi 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 59 /semicolon 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 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 91 /bracketleft put
dup 93 /bracketright put
dup 94 /circumflex put
dup 95 /dotaccent put
dup 96 /quoteleft put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 123 /endash put
dup 126 /tilde put
dup 127 /dieresis put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
currentdict end
currentfile eexec
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%%EndFont 
%%BeginFont: CMSY9
%!PS-AdobeFont-1.1: CMSY9 1.0
%%CreationDate: 1991 Aug 15 07:22:27
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
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/FullName (CMSY9) readonly def
/FamilyName (Computer Modern) readonly def
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end readonly def
/FontName /CMSY9 def
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/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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dup 33 /arrowright put
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readonly def
/FontBBox{-30 -958 1146 777}readonly def
currentdict end
currentfile eexec
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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: CMSY5
%!PS-AdobeFont-1.1: CMSY5 1.0
%%CreationDate: 1991 Aug 15 07:21:16
% 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 (CMSY5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY5 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 3 /asteriskmath put
dup 48 /prime put
readonly def
/FontBBox{21 -944 1448 791}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 100 /d put
dup 115 /s put
readonly def
/FontBBox{11 -250 1241 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI9
%!PS-AdobeFont-1.1: CMMI9 1.100
%%CreationDate: 1996 Jul 23 07:53:55
% 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 (CMMI9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI9 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 59 /comma put
dup 65 /A put
dup 115 /s put
readonly def
/FontBBox{-29 -250 1075 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR9
%!PS-AdobeFont-1.1: CMR9 1.0
%%CreationDate: 1991 Aug 20 16:39:59
% 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 (CMR9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR9 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 13 /fl put
dup 14 /ffi 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 72 /H put
dup 84 /T put
dup 87 /W put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 121 /y put
dup 122 /z put
dup 123 /endash put
dup 127 /dieresis put
readonly def
/FontBBox{-39 -250 1036 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX9
%!PS-AdobeFont-1.1: CMBX9 1.0
%%CreationDate: 1991 Aug 20 16:36:25
% 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 (CMBX9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX9 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 65 /A put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 114 /r put
dup 115 /s put
dup 116 /t put
readonly def
/FontBBox{-58 -250 1195 750}readonly def
currentdict end
currentfile eexec
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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 23 /ring put
dup 46 /period put
dup 50 /two put
dup 66 /B put
dup 69 /E put
dup 72 /H put
dup 75 /K put
dup 76 /L put
dup 83 /S put
dup 97 /a put
dup 100 /d put
dup 101 /e put
dup 103 /g put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 110 /n put
dup 111 /o put
dup 114 /r put
dup 115 /s put
dup 117 /u put
readonly def
/FontBBox{-34 -251 988 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR17
%!PS-AdobeFont-1.1: CMR17 1.0
%%CreationDate: 1991 Aug 20 16:38:24
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. 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 12 /fi put
dup 73 /I put
dup 76 /L put
dup 84 /T put
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 104 /h put
dup 105 /i 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 122 /z put
dup 123 /endash put
dup 127 /dieresis put
readonly def
/FontBBox{-33 -250 945 749}readonly def
currentdict end
currentfile eexec
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Fs(j)p Ft(s)p Fs(j)1698 523 y Fv(for)27 b(eac)n(h)g Fu(s)p
Fv(.)37 b(Therefore,)1209 736 y Fr(k)p Fu(p)18 b Fr(\003)g
Fu(q)s Fr(k)1453 701 y Fp(2)1453 756 y Ft(\015)1518 736
y Fv(=)1606 657 y Fm(X)1650 831 y Ft(s)1726 736 y Fr(h)p
Fu(s)p Fr(i)1829 701 y Fp(2)p Ft(m)1921 709 y FA(\003)1960
736 y Fu(e)1999 701 y Fp(2)p Ft(\015)t Fs(j)p Ft(s)p
Fs(j)2145 619 y Fm(\022)2220 657 y(X)2265 834 y Ft(j)2354
736 y Fu(p)2396 748 y Ft(j)2431 736 y Fu(q)2468 748 y
Ft(s)p Fs(\000)p Ft(j)2586 619 y Fm(\023)2647 636 y Fp(2)1184
1024 y Fv(=)1271 945 y Fm(X)1315 1119 y Ft(s)1391 1024
y Fr(h)p Fu(s)p Fr(i)1494 989 y Fp(2)p Ft(m)1586 997
y FA(\003)1625 907 y Fm(\022)1700 945 y(X)1745 1122 y
Ft(j)1834 1024 y Fu(e)1873 989 y Ft(\015)t Fp(\()p Fs(j)p
Ft(s)p Fs(j\000j)p Ft(j)s Fs(j\000j)p Ft(s)p Fs(\000)p
Ft(j)s Fs(j)2340 1024 y Fu(P)2393 1036 y Ft(j)2428 1024
y Fu(Q)2494 1036 y Ft(s)p Fs(\000)p Ft(j)2612 907 y Fm(\023)2673
924 y Fp(2)1191 1312 y Fr(\024)1279 1233 y Fm(X)1323
1407 y Ft(s)1398 1312 y Fr(h)p Fu(s)p Fr(i)1501 1278
y Fp(2)p Ft(m)1593 1286 y FA(\003)1633 1195 y Fm(\022)1708
1233 y(X)1752 1410 y Ft(j)1842 1312 y Fu(P)1895 1324
y Ft(j)1930 1312 y Fu(Q)1996 1324 y Ft(s)p Fs(\000)p
Ft(j)2114 1195 y Fm(\023)2175 1212 y Fp(2)2235 1312 y
Fv(=)23 b Fr(k)p Fu(P)30 b Fr(\003)17 b Fu(Q)p Fr(k)2615
1278 y Fp(2)2615 1332 y(+0)1075 1534 y Fr(\024)23 b Fu(C)1228
1500 y Fp(2)1265 1534 y Fv(\()p Fu(m)1370 1546 y Fs(\003)1408
1534 y Fu(;)14 b(d)p Fv(\))p Fr(k)p Fu(P)e Fr(k)1669
1500 y Fp(2)1669 1555 y(+0)1757 1534 y Fr(k)p Fu(Q)p
Fr(k)1907 1500 y Fp(2)1907 1555 y(+0)2017 1534 y Fv(=)22
b Fu(C)2169 1500 y Fp(2)2207 1534 y Fv(\()p Fu(m)2312
1546 y Fs(\003)2350 1534 y Fu(;)14 b(d)p Fv(\))p Fr(k)p
Fu(p)p Fr(k)2588 1500 y Fp(2)2588 1555 y Ft(\015)2630
1534 y Fr(k)p Fu(q)s Fr(k)2754 1500 y Fp(2)2754 1555
y Ft(\015)2796 1534 y Fu(;)515 1685 y Fv(as)27 b(stated.)p
3318 1685 4 57 v 3322 1632 50 4 v 3322 1685 V 3372 1685
4 57 v 515 1912 a Fj(1.2)112 b(Matrices)38 b(with)f(exp)s(onen)m(tial)i
(deca)m(y)f(o\013)g(the)f(diagonals.)515 2065 y Fv(In)21
b Fq(R)672 2035 y Ft(d)732 2065 y Fv(and)g Fq(Z)942 2035
y Ft(d)1003 2065 y Fv(w)n(e)g(in)n(tro)r(duce)f(a)h(quasidistance)g
(whic)n(h)g(\(with)h(some)f(abuse)f(of)i(notations\))515
2165 y(will)28 b(b)r(e)g(denoted)f([)p Fu(x)19 b Fr(\000)f
Fu(y)s Fv(]:)1393 2315 y([)p Fu(x)h Fr(\000)f Fu(y)s
Fv(])23 b(=)f(min)q Fr(fj)p Fu(x)c Fr(\000)g Fu(y)s Fr(j)p
Fu(;)c Fr(j)p Fu(x)19 b Fv(+)f Fu(y)s Fr(jg)p Fu(:)515
2466 y Fv(Ob)n(viously)-7 b(,)24 b([)p Fu(x)13 b Fr(\000)g
Fu(y)s Fv(])23 b(=)g([)p Fu(y)15 b Fr(\000)e Fu(x)p Fv(],)40
b([0)13 b Fr(\000)g Fu(x)p Fv(])23 b(=)f Fr(j)p Fu(x)p
Fr(j)p Fv(,)40 b([)p Fu(x)13 b Fr(\000)g Fu(y)s Fv(])23
b(=)g(0)h(if)i(and)e(only)h(if)g Fu(x)f Fv(=)e Fr(\006)p
Fu(y)s Fv(,)j(and)1350 2616 y([)p Fu(x)19 b Fr(\000)f
Fu(y)s Fv(])23 b(=)g Fr(j)p Fu(x)18 b Fr(\000)g Fu(y)s
Fr(j)83 b Fv(i\013)h Fr(h)p Fu(x;)14 b(y)s Fr(i)23 b(\025)g
Fv(0)p Fu(:)515 2767 y Fv(Moreo)n(v)n(er,)i(the)j(triangle)e(inequalit)
n(y)h(holds:)1474 2917 y([)p Fu(x)19 b Fr(\000)f Fu(y)s
Fv(])23 b Fr(\024)g Fv([)p Fu(x)c Fr(\000)f Fu(z)t Fv(])g(+)g([)p
Fu(z)j Fr(\000)d Fu(y)s Fv(])p Fu(:)789 b Fv(\(1.4\))515
3068 y(Indeed,)43 b([)p Fu(x)27 b Fr(\000)f Fu(y)s Fv(])40
b(equals)f(the)h(Hausdor\013)f(distance)h(b)r(et)n(w)n(een)f(the)h
(sets)g Fr(f)p Fu(x;)14 b Fr(\000)p Fu(x)p Fr(g)39 b
Fv(and)515 3167 y Fr(f)p Fu(y)s(;)14 b Fr(\000)p Fu(y)s
Fr(g)p Fv(,)23 b(so)i(\(1.4\))f(follo)n(ws)g(from)h(the)g(triangle)g
(inequalit)n(y)f(for)h(the)g(Hausdor\013)f(distance.)639
3304 y(Let)e Fu(A)h Fv(=)g(\()p Fu(A)1049 3316 y Ft(ss)1111
3300 y FA(0)1139 3304 y Fu(;)35 b(s;)14 b(s)1312 3274
y Fs(0)1358 3304 y Fr(2)24 b Fq(Z)1492 3274 y Ft(d)1531
3304 y Fv(\),)f(b)r(e)f(an)f(in\014nite)h(matrix)f(with)h(complex)f(or)
g(real)g(en)n(tries.)515 3404 y(F)-7 b(or)27 b Fr(j)p
Fu(\015)5 b Fr(j)22 b(\024)h Fv(1)k(w)n(e)h(set)1432
3515 y Fr(kj)p Fu(A)p Fr(kj)1624 3527 y Ft(\015)1689
3515 y Fv(=)23 b(sup)1787 3585 y Ft(s;s)1869 3568 y FA(0)1902
3515 y Fr(f)p Fu(e)1983 3481 y Ft(\015)t Fp([)p Ft(s)p
Fs(\000)p Ft(s)2155 3456 y FA(0)2176 3481 y Fp(])2199
3515 y Fr(j)p Fu(A)2284 3527 y Ft(ss)2346 3511 y FA(0)2374
3515 y Fr(jg)p Fu(;)515 3701 y Fv(and)k(denote)1242 3800
y Fu(M)1323 3812 y Ft(\015)1388 3800 y Fv(=)c Fr(f)p
Fu(A)g Fr(j)g Fu(A)g Fv(is)28 b(real)e(and)46 b Fr(kj)p
Fu(A)p Fr(kj)2351 3812 y Ft(\015)2416 3800 y Fu(<)23
b Fr(1g)p Fu(;)974 3931 y(M)1064 3896 y Ft(c)1055 3951
y(\015)1121 3931 y Fv(=)f Fu(M)1289 3943 y Ft(\015)1350
3931 y Fr(\012)c Fq(C)23 b Fv(=)g Fr(f)p Fu(A)f Fr(j)h
Fu(A)h Fv(is)j(complex)g(and)c Fr(kj)p Fu(A)p Fr(kj)2619
3943 y Ft(\015)2684 3931 y Fu(<)g Fr(1g)p Fu(:)515 4061
y Fv(Ob)n(viously)-7 b(,)25 b Fu(M)1001 4073 y Ft(\015)1069
4061 y Fv(and)h Fu(M)1319 4031 y Ft(c)1310 4081 y(\015)1378
4061 y Fv(are)f(Banac)n(h)g(spaces.)36 b(They)25 b(are)g(formed)h(b)n
(y)g(matrices)f(whic)n(h)515 4160 y(deca)n(y)31 b(exp)r(onen)n(tially)i
(outside)f(the)h(union)g(of)g(the)g(diagonal)e Fr(f)p
Fu(s)g Fv(=)g Fu(s)2775 4130 y Fs(0)2799 4160 y Fr(g)h
Fv(and)g(the)i(`an)n(ti-)515 4260 y(diagonal')g Fr(f)p
Fu(s)h Fv(=)g Fr(\000)p Fu(s)1198 4230 y Fs(0)1221 4260
y Fr(g)p Fv(.)60 b(Clearly)-7 b(,)36 b Fr(k)14 b Fv(id)g
Fr(kj)1864 4272 y Ft(\015)1941 4260 y Fv(=)36 b(1)f(for)f(an)n(y)h
Fu(\015)5 b Fv(.)60 b(So)35 b(the)g(iden)n(tit)n(y)h(matrix)515
4360 y(b)r(elongs)27 b(to)g(all)g(spaces)g Fu(M)1365
4372 y Ft(\015)1407 4360 y Fv(.)639 4459 y(Let)d Fu(Y)850
4429 y Ft(c)p Fp(0)832 4480 y Ft(\015)941 4459 y Fv(b)r(e)f(the)h(set)f
(of)g(v)n(ectors)f Fu(y)k Fr(2)d Fu(Y)1893 4429 y Ft(c)1874
4480 y(\015)1950 4459 y Fv(with)h(\014nitely)f(man)n(y)g(non-zero)e(co)
r(e\016cien)n(ts.)515 4559 y(Then)28 b(an)n(y)e Fu(A)e
Fr(2)f Fu(M)1142 4529 y Ft(c)1133 4582 y(\015)1172 4565
y FA(0)1225 4559 y Fv(de\014nes)28 b(a)f(linear)g(map)1097
4739 y Fu(A)c Fv(:)g Fu(Y)1295 4705 y Ft(c)p Fp(0)1276
4760 y Ft(\015)1385 4739 y Fr(!)g Fu(Y)1558 4705 y Ft(c)1539
4760 y(\015)1591 4739 y Fu(;)97 b(y)26 b Fr(!)d Fu(Ay)s(;)60
b Fv(\()p Fu(Ay)s Fv(\))2243 4751 y Ft(s)2302 4739 y
Fv(=)2390 4661 y Fm(X)2524 4739 y Fu(A)2586 4751 y Ft(ss)2648
4735 y FA(0)2675 4739 y Fu(y)2716 4751 y Ft(s)2747 4735
y FA(0)2774 4739 y Fu(:)515 4907 y Fv(The)26 b(main)g(result)g(of)h
(this)f(section)g(is)g(that)h(the)g(map)f Fu(A)g Fv(is)h(b)r(ounded)f
(in)h(the)g Fr(k)15 b(\001)h(k)3120 4919 y Ft(\015)3162
4907 y Fv(-norm)515 5006 y(if)28 b Fu(\015)639 4976 y
Fs(0)685 5006 y Fu(>)22 b Fr(j)p Fu(\015)5 b Fr(j)p Fv(.)37
b(Accordingly)-7 b(,)27 b(it)h(extends)f(to)h(a)f(b)r(ounded)h(map)f
Fu(A)d Fv(:)f Fu(Y)2682 4976 y Ft(c)2663 5027 y(\015)2738
5006 y Fr(!)h Fu(Y)2911 4976 y Ft(c)2893 5027 y(\015)2945
5006 y Fv(:)1926 5255 y(4)p eop end
%%Page: 5 5
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b(Then)1107 688 y Fr(k)p Fu(Ay)s Fr(k)1297 700 y Ft(\015)1362
688 y Fr(\024)23 b Fu(C)6 b Fv(\()p Fu(d;)14 b(m)1700
700 y Fs(\003)1738 688 y Fv(\)\()p Fu(\015)1850 653 y
Fs(0)1892 688 y Fr(\000)k(j)p Fu(\015)5 b Fr(j)p Fv(\))2101
653 y Fs(\000)p Ft(d)p Fs(\000)p Ft(m)2299 661 y FA(\003)2337
688 y Fr(kj)p Fu(A)p Fr(kj)2529 700 y Ft(\015)2568 683
y FA(0)2594 688 y Fr(k)p Fu(y)s Fr(k)2722 700 y Ft(\015)2763
688 y Fu(;)422 b Fv(\(1.5\))515 852 y Fo(for)34 b(any)g
Fu(y)e Fr(2)d Fu(Y)1038 822 y Ft(c)p Fp(0)1019 873 y
Ft(\015)1105 852 y Fo(.)49 b(Henc)l(e,)35 b Fu(A)e Fo(de\014nes)g(a)h
(b)l(ounde)l(d)g(line)l(ar)g(op)l(er)l(ator)g(in)f Fu(Y)2968
822 y Ft(c)2949 873 y(\015)3035 852 y Fo(and)42 b Fv(\(1.5\))515
952 y Fo(holds)31 b(for)g(e)l(ach)f Fu(y)c Fr(2)d Fu(Y)1258
922 y Ft(c)1239 972 y(\015)1292 952 y Fo(.)515 1103 y(Pr)l(o)l(of.)43
b Fv(Without)28 b(lost)g(of)f(generalit)n(y)f(w)n(e)h(assume)g(b)r(elo)
n(w)g(that)h Fr(kj)p Fu(A)p Fr(kj)2760 1115 y Ft(\015)2799
1099 y FA(0)2848 1103 y Fv(=)23 b(1.)639 1203 y(F)-7
b(or)27 b(an)n(y)g Fu(s)c Fr(2)g Fu(Z)1148 1173 y Ft(d)1214
1203 y Fv(let)28 b(us)g(denote)f(b)n(y)h Fu(l)1851 1173
y Ft(s)1913 1203 y Fv(the)g(v)n(ector)1476 1367 y Fu(l)1503
1333 y Ft(s)1561 1367 y Fv(=)23 b Fr(f)p Fu(l)1718 1333
y Ft(s)1716 1388 y(a)1778 1367 y Fv(=)g Fu(\016)1903
1379 y Ft(s;a)1994 1367 y Fu(;)97 b(a)23 b Fr(2)g Fq(Z)2314
1333 y Ft(d)2353 1367 y Fr(g)p Fu(:)790 b Fv(\(1.6\))515
1532 y(Then)28 b(the)f(v)n(ectors)1250 1696 y Fu(f)1300
1662 y Ft(s)1358 1696 y Fv(=)c Fu(e)1485 1662 y Fs(\000)p
Ft(\015)t Fs(j)p Ft(s)p Fs(j)1650 1696 y Fr(h)p Fu(s)p
Fr(i)1753 1662 y Fs(\000)p Ft(m)1864 1670 y Ff(\015)1906
1696 y Fu(l)1933 1662 y Ft(s)1968 1696 y Fu(;)97 b(m)2161
1708 y Ft(\015)2226 1696 y Fv(=)23 b Fu(m)2387 1708 y
Fs(\003)2439 1696 y Fv(sgn)13 b Fu(\015)5 b(;)515 1861
y Fv(form)26 b(a)g(Hilb)r(ert)h(basis)e(of)i(the)f(space)g
Fu(Y)1769 1873 y Ft(\015)1812 1861 y Fv(.)36 b(In)27
b(this)g(basis)e(the)i(matrix)f(of)g(the)h(op)r(erator)e
Fu(A)515 1961 y Fv(is)1322 2060 y Fr(f)p Fu(B)1427 2072
y Ft(ab)1519 2060 y Fv(=)e Fr(h)p Fu(a)p Fr(i)1715 2026
y Ft(m)1774 2034 y Ff(\015)1816 2060 y Fr(h)p Fu(b)p
Fr(i)1916 2026 y Fs(\000)p Ft(m)2027 2034 y Ff(\015)2069
2060 y Fu(e)2108 2026 y Ft(\015)t Fp(\()p Fs(j)p Ft(a)p
Fs(j\000j)p Ft(b)p Fs(j)p Fp(\))2399 2060 y Fu(A)2461
2072 y Ft(ab)2530 2060 y Fr(g)515 2199 y Fv(\(see)g([HS78)o(]\).)36
b(Let)23 b(us)g(\014rst)f(assume)g(that)i Fu(\015)j(>)c
Fv(0)f(and)h(denote)g Fu(\015)2560 2211 y Fp(\001)2642
2199 y Fv(=)f Fu(\015)2777 2169 y Fs(0)2809 2199 y Fr(\000)9
b Fu(\015)c Fv(,)24 b Fu(\015)3021 2211 y Fp(\001)3102
2199 y Fr(2)g Fv(\(0)p Fu(;)14 b Fv(1].)515 2298 y(F)-7
b(or)27 b(an)n(y)g Fu(a)22 b Fr(2)i Fq(Z)1021 2268 y
Ft(d)1088 2298 y Fv(w)n(e)j(ha)n(v)n(e:)1125 2427 y Fm(X)1171
2606 y Ft(b)1259 2506 y Fr(j)p Fu(B)1345 2518 y Ft(ab)1415
2506 y Fr(j)c(\024)f Fu(e)1587 2472 y Ft(\015)t Fs(j)p
Ft(a)p Fs(j)1733 2427 y Fm(X)1719 2611 y Ft(b)p Fs(2)p
Fg(Z)1832 2594 y Ff(d)1880 2414 y Fm(\020)1954 2450 y
Fr(h)p Fu(a)p Fr(i)p 1954 2487 109 4 v 1958 2563 a(h)p
Fu(b)p Fr(i)2086 2414 y Fm(\021)2135 2431 y Ft(m)2194
2439 y FA(\003)2233 2506 y Fu(e)2272 2472 y Fs(\000)p
Ft(\015)t Fs(j)p Ft(b)p Fs(j)2435 2506 y Fu(e)2474 2472
y Fs(\000)p Ft(\015)2565 2447 y FA(0)2586 2472 y Fp([)p
Ft(a)p Fs(\000)p Ft(b)p Fp(])2745 2506 y Fu(:)515 2755
y Fv(T)-7 b(o)27 b(b)r(ound)h(the)g(r.h.s.)36 b(w)n(e)28
b(need)515 2904 y Fl(Lemma)j(1.3.)40 b Fo(F)-6 b(or)30
b(any)g Fu(x;)14 b(y)26 b Fr(2)e Fq(Z)1642 2874 y Ft(d)1711
2904 y Fo(we)30 b(have)1392 3020 y Fm(\020)1467 3056
y Fr(h)p Fu(y)s Fr(i)p 1465 3093 113 4 v 1465 3169 a(h)p
Fu(x)p Fr(i)1601 3020 y Fm(\021)1651 3037 y Ft(m)1710
3045 y FA(\003)1772 3112 y Fr(\024)22 b Fu(C)6 b(\015)1972
3075 y Fs(\000)p Ft(m)2083 3083 y FA(\003)1967 3136 y
Fp(\001)2136 3112 y Fu(e)2185 3041 y Fp(1)p 2184 3055
34 4 v 2184 3103 a(2)2227 3074 y Ft(\015)2262 3082 y
Fn(\001)2312 3074 y Fp([)p Ft(y)r Fs(\000)p Ft(x)p Fp(])2479
3112 y Fu(:)706 b Fv(\(1.7\))515 3325 y Fo(Pr)l(o)l(of.)43
b Fv(W)-7 b(e)19 b(ma)n(y)f(assume)g(that)i Fr(j)p Fu(y)s
Fr(j)i Fu(>)h Fr(j)p Fu(x)p Fr(j)h Fu(>)e Fv(1)d(since)g(otherwise)e
(the)j(estimate)f(is)f(ob)n(vious.)515 3425 y(Also,)26
b(it)h(is)f(easy)f(to)h(see)g(that)h(the)f(estimate)g(for)g(0)d
Fr(\024)f Fu(m)2304 3437 y Fs(\003)2365 3425 y Fu(<)h
Fv(1)j(follo)n(ws)f(from)h(the)h(one)f(for)515 3524 y
Fu(m)588 3536 y Fs(\003)649 3524 y Fr(\025)c Fv(1.)37
b(Belo)n(w)27 b(w)n(e)g(assume)g(that)1521 3689 y Fr(j)p
Fu(y)s Fr(j)22 b Fu(>)h Fr(j)p Fu(x)p Fr(j)h Fu(>)e Fv(1)p
Fu(;)97 b(m)2160 3701 y Fs(\003)2221 3689 y Fr(\025)23
b Fv(1)p Fu(;)515 3853 y Fv(and)29 b(pro)n(v)n(e)f(the)h(estimate)g
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y FA(\003)1894 3853 y Fu(e)1933 3823 y Ft(m)1992 3831
y FA(\003)2027 3823 y Fp(\(ln)10 b Ft(m)2178 3831 y FA(\003)2213
3823 y Fp(+1\))2327 3853 y Fv(.)42 b(Denoting)29 b Fu(r)g
Fv(=)2919 3820 y Fp(1)p 2919 3834 V 2919 3882 a(2)2962
3853 y Fu(\015)3005 3865 y Fp(\001)3063 3853 y Fu(=m)3178
3865 y Fs(\003)3245 3853 y Fv(and)515 3953 y(taking)e(logarithm)f(of)34
b(\(1.7\),)28 b(w)n(e)f(rewrite)g(it)h(as)1292 4117 y(ln)14
b Fr(j)p Fu(y)s Fr(j)19 b(\000)f Fv(ln)c Fr(j)p Fu(x)p
Fr(j)23 b(\024)g(\000)14 b Fv(ln)f Fu(r)22 b Fv(+)c Fu(r)r
Fv([)p Fu(y)k Fr(\000)c Fu(x)p Fv(])h(+)f(1)p Fu(:)515
4282 y Fv(By)30 b(the)h(triangle)f(inequalit)n(y)i(\(1.4\))e(\(with)i
Fu(y)f Fv(=)d(0)i(and)h Fu(z)j Fv(re-denoted)c(as)g Fu(y)s
Fv(\),)i([)p Fu(y)23 b Fr(\000)d Fu(x)p Fv(])29 b Fr(\025)515
4381 y(j)p Fu(y)s Fr(j)18 b(\000)g(j)p Fu(x)p Fr(j)p
Fv(.)38 b(So)27 b(it)h(su\016ce)f(to)h(c)n(hec)n(k)f(that)1364
4594 y(ln)1459 4538 y Fr(j)p Fu(y)s Fr(j)p 1457 4575
94 4 v 1457 4651 a(j)p Fu(x)p Fr(j)1584 4594 y(\024)c(\000)14
b Fv(ln)f Fu(r)21 b Fv(+)d Fu(r)r Fv(\()p Fr(j)p Fu(y)s
Fr(j)h(\000)g(j)p Fu(x)p Fr(j)p Fv(\))g(+)f(1)p Fu(;)515
4807 y Fv(where)j Fr(j)p Fu(y)s Fr(j)i Fu(>)f Fr(j)p
Fu(x)p Fr(j)i Fu(>)f Fv(1)e(and)h(0)g Fu(<)h(r)j(<)c
Fv(1.)35 b(F)-7 b(or)21 b Fr(j)p Fu(x)p Fr(j)j Fu(<)e
Fr(j)p Fu(y)s Fr(j)h(\024)g Fv(2)p Fr(j)p Fu(x)p Fr(j)f
Fv(the)g(estimate)g(follo)n(ws)e(from)515 4907 y(the)34
b(inequalit)n(y)f(ln)14 b(2)32 b Fu(<)h Fv(1.)54 b(F)-7
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%%Page: 7 7
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%%Page: 8 8
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y Fp(2)1718 2340 y Fr(\032)23 b Fu(X)r(;)96 b(X)2065
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2708 y(2)1202 2688 y Fv(=)c Fr(\000)14 b Fv(id,)27 b(w)n(e)f(in)n(tro)r
(duce)g(in)h Fq(R)2133 2658 y Fp(2)2197 2688 y Fv(a)f(complex)g
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2887 y(linear)h(op)r(erators,)h(while)h Fu(X)1441 2899
y Ft(q)1511 2887 y Fv(is)f(the)h(linear)f(space)f(of)i(complex-an)n
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Fr(!)23 b Fu(X)1733 3279 y Ft(p)1771 3267 y Fu(;)97 b(X)1960
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Ft(q)2225 3267 y Fr(!)23 b Fu(X)2400 3279 y Ft(q)2437
3267 y Fu(;)97 b(X)2626 3279 y Ft(q)2679 3267 y Fr(\002)17
b Fu(X)2830 3279 y Ft(p)2891 3267 y Fr(!)23 b Fu(X)3066
3279 y Ft(q)3102 3267 y Fu(:)42 b Fv(\(1.15\))639 3448
y(W)-7 b(e)34 b(de\014ne)f(the)h(follo)n(wing)e(norms)g
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3448 y Fv(and)33 b Fr(kj)22 b(\001)g(kj)2441 3418 y Fs(\000)2441
3468 y Ft(\015)2530 3448 y Fv(on)33 b(the)g(spaces)f
Fu(M)3149 3418 y Ft(X)3140 3468 y(\015)3245 3448 y Fv(and)515
3558 y Fu(M)605 3528 y Ft(X)5 b(c)596 3578 y(\015)697
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3750 y Ft(s;s)1572 3733 y FA(0)1605 3680 y Fr(fj)p Fu(q)s(A)1772
3692 y Ft(ss)1834 3676 y FA(0)1861 3680 y Fr(j)p Fu(e)1923
3646 y Ft(\015)t Fs(j)p Ft(s)p Fp(+)p Ft(s)2095 3621
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y Fs(\000)2312 4217 y Ft(\015)2390 4196 y Fv(=)22 b Fr(1)28
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%%Page: 9 9
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Fu(M)1871 1789 y Fs(\007)1862 1844 y Ft(\015)1926 1824
y Fu(;)97 b(A)24 b Fr(!)f Fu(A)p Fv(\007)p Fu(:)3167
1758 y Fv(\(1.16\))515 2001 y Fo(Example)48 b Fv(1.7)p
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2001 y Fv(the)d(tensor)e(pro)r(duct)i Fu(y)2967 1970
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b(with)i(en)n(tries)f(in)h Fu(X)7 b Fv(.)36 b(If)28 b
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2267 y Fr(k)p Fu(y)2108 2232 y Fp(2)2145 2267 y Fr(k)2187
2279 y Ft(\015)2229 2267 y Fu(e)2268 2232 y Fs(\000)p
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y Fu(:)515 2433 y Fv(Since)28 b Fr(j)p Fu(s)p Fr(j)18
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2565 y Fp(1)1406 2600 y Fr(\012)18 b Fu(y)1533 2565 y
Fp(2)1570 2600 y Fr(kj)1635 2565 y Fs(\006)1635 2620
y Ft(\015)1714 2600 y Fr(\024)k(k)p Fu(y)1887 2565 y
Fp(1)1924 2600 y Fr(k)1966 2612 y Ft(\015)2008 2600 y
Fr(k)p Fu(y)2094 2565 y Fp(2)2130 2600 y Fr(k)2172 2612
y Ft(\015)2297 2600 y Fv(if)84 b Fu(\015)27 b Fr(\025)c
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y Ft(\015)1405 2776 y Fv(if)28 b Fu(y)1525 2746 y Fp(1)1562
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b Fu(Y)1848 2746 y Ft(X)1829 2797 y(\015)1939 2776 y
Fv(and)k Fu(\015)h Fr(\025)23 b Fv(0.)p 3318 2776 4 57
v 3322 2724 50 4 v 3322 2776 V 3372 2776 4 57 v 639 2925
a(Multiplication)k(of)f(matrices)f(from)g(the)i(spaces)e
Fu(M)2283 2894 y Fs(\006)2274 2945 y Ft(\015)2364 2925
y Fv(agrees)g(with)h(Theorem)f(1.5,)h(as)515 3024 y(sp)r(eci\014es)h
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3443 y Fr(!)23 b Fu(M)1819 3409 y Fp(+)1810 3464 y Ft(\015)1874
3443 y Fu(;)99 b(M)2086 3408 y Fs(\000)2077 3468 y Ft(\015)2116
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3409 y Fp(+)2599 3464 y Ft(\015)2662 3443 y Fu(;)482
b Fv(\(1.18\))1207 3626 y Fu(M)1297 3590 y Fp(+)1288
3650 y Ft(\015)1327 3633 y FA(0)1371 3626 y Fr(\002)18
b Fu(M)1544 3591 y Fs(\000)1535 3646 y Ft(\015)1623 3626
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3626 y Fu(;)99 b(M)2087 3590 y Fs(\000)2078 3650 y Ft(\015)2117
3633 y FA(0)2161 3626 y Fr(\002)18 b Fu(M)2334 3591 y
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3591 y Fs(\000)2599 3646 y Ft(\015)2663 3626 y Fu(:)515
3781 y Fo(Mor)l(e)l(over,)37 b Fr(kj)p Fu(AB)t Fr(kj)1170
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b Fu(C)6 b Fv(\()p Fu(\015)1495 3751 y Fs(0)1540 3781
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y Fs(\000)p Ft(d)1842 3781 y Fr(kj)p Fu(A)p Fr(kj)2034
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Fr(kj)2676 3998 y Fp(+)2676 4058 y Ft(\015)2715 4042
y FA(0)2763 4034 y Fv(=)h(1)p Fu(;)59 b Fr(kj)p Fu(B)t
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y Fv(=)22 b(1)515 4134 y(and)27 b(denote)h Fu(D)d Fv(=)d
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1190 4317 y Fr(j)p Fu(pD)1324 4329 y Ft(ij)1382 4317
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4317 y Fr(j)14 b(j)p Fu(pB)2039 4329 y Ft(sj)2105 4317
y Fr(j)k Fv(+)g Fr(j)p Fu(q)s(A)2354 4329 y Ft(is)2413
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y Fr(j)2666 4249 y Fm(\001)753 4541 y Fr(\024)841 4462
y Fm(X)885 4636 y Ft(s)975 4474 y Fm(\000)1013 4541 y
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4541 y Fr(\024)k Fu(C)6 b Fv(\()p Fu(\015)2497 4507 y
Fs(0)2539 4541 y Fr(\000)18 b(j)p Fu(\015)5 b Fr(j)p
Fv(\))2748 4507 y Fs(\000)p Ft(d)2839 4541 y Fu(e)2878
4507 y Fs(\000)p Ft(\015)t Fs(j)p Ft(i)p Fs(\000)p Ft(j)s
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%%Page: 10 10
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523 y Fv(.)639 623 y(The)23 b(three)g(remaining)f(estimates)g(follo)n
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b(\007)2477 655 y Fm(\020)2526 747 y Fv(\(\007)p Fu(A)p
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b(e)40 b(see)e(that)515 876 y Fu(AB)27 b Fr(2)d Fu(M)836
845 y Fs(\000)827 896 y Ft(\015)919 876 y Fv(and)j(satis\014es)g(the)h
(desired)f(estimate.)p 3318 876 4 57 v 3322 823 50 4
v 3322 876 V 3372 876 4 57 v 515 1150 a Fw(2)134 b(T\177)-67
b(oplitz{Lipsc)l(hitz)46 b(matrices)g(and)f(op)t(erators)515
1349 y Fj(2.1)112 b(De\014nitions)38 b(and)h(examples)515
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Fv(\))g Fr(2)h Fu(M)2413 1472 y Ft(c)2404 1522 y(\015)2480
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y(prop)r(erties)26 b(TL1)h({)h(TL3:)639 1701 y Fl(TL1)p
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1713 y Fp(1)1344 1701 y Fv(,)h Fu(b)1431 1713 y Fp(2)1491
1701 y Fr(2)23 b Fq(Z)1624 1671 y Ft(d)1663 1701 y Fv(,)28
b Fu(a)23 b Fr(6)p Fv(=)g(0,)k(the)h(t)n(w)n(o)f(limits)1283
1884 y Fu(A)1345 1848 y Fs(1\006)1345 1909 y Ft(a;b)1430
1917 y Fn(1)1463 1909 y Ft(;b)1512 1917 y Fn(2)1572 1884
y Fv(=)148 b(lim)1659 1943 y Ft(t)p Fs(!1)p Ft(;)12 b(ta)p
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y Fp(\()p Ft(ta)p Fp(+)p Ft(b)2270 1907 y Fn(1)2303 1899
y Fp(\))p Fs(\006)p Fp(\()p Ft(ta)p Fp(+)p Ft(b)2548
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2112 y(exist)27 b(and)h(are)e(\014nite.)639 2211 y Fl(TL2)p
Fv(.)38 b(There)27 b(exists)g(\003)c Fr(\025)f Fv(6)28
b(suc)n(h)f(that)837 2394 y Fu(A)p Fv(\()p Fu(a;)14 b(b)1048
2406 y Fp(1)1085 2394 y Fu(;)g(b)1158 2406 y Fp(2)1195
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2394 y Fv(;)37 b(\003\))23 b Fu(<)f Fr(1)p Fu(;)515 2576
y Fv(where)981 2676 y Fu(A)1043 2642 y Fs(\006)1099 2676
y Fv(\()p Fu(a;)14 b(b)1248 2688 y Fp(1)1285 2676 y Fu(;)g(b)1358
2688 y Fp(2)1395 2676 y Fv(;)37 b(\003\))23 b(=)g(sup)p
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Fp(+)p Ft(b)2105 2699 y Fn(1)2137 2691 y Fp(\))p Fs(\006)p
Fp(\()p Ft(ta)p Fp(+)p Ft(b)2382 2699 y Fn(2)2415 2691
y Fp(\))2463 2676 y Fr(\000)18 b Fu(A)2608 2641 y Fs(1\006)2608
2701 y Ft(a;b)2693 2709 y Fn(1)2726 2701 y Ft(;b)2775
2709 y Fn(2)2811 2676 y Fr(jg)c Fu(;)515 2826 y Fv(and)27
b(the)h(suprem)n(um)f(is)h(tak)n(en)f(o)n(v)n(er)f(all)h
Fu(t)c(>)g Fv(0)k(suc)n(h)g(that)1201 3008 y Fr(j)p Fu(ta)18
b Fv(+)g Fu(b)1435 3020 y Ft(j)1470 3008 y Fr(j)23 b(\025)f
Fv(\003\(1)c(+)g Fr(j)p Fu(a)p Fr(j)h Fv(+)f Fr(j)p Fu(b)2087
3020 y Ft(j)2121 3008 y Fr(j)p Fv(\))p Fr(j)p Fu(a)p
Fr(j)p Fu(;)98 b(j)28 b Fv(=)22 b(1)p Fu(;)28 b Fv(2)p
Fu(:)514 b Fv(\(2.2\))639 3191 y(Let)28 b(us)g(de\014ne)1265
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Fr(kj)1511 3850 y Ft(\015)t(;)10 b Fp(\003)1652 3838
y Fv(:=)23 b Fr(kj)p Fu(A)p Fr(kj)1955 3850 y Ft(\015)2015
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y Fv(W)-7 b(e)28 b(denote)849 4204 y Fu(M)939 4169 y
Ft(c)930 4224 y(\015)t(;)10 b Fp(\003)1071 4204 y Fv(=)23
b Fr(f)p Fu(A)g Fr(2)g Fu(M)1454 4169 y Ft(c)1445 4224
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y Ft(\015)t(;)10 b Fp(\003)1890 4204 y Fu(<)23 b Fr(1g)14
b Fu(;)179 b(M)2400 4216 y Ft(\015)t(;)p Fp(\003)2531
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4204 y Fu(:)515 4386 y Fv(These)24 b(are)g(a)h(complex)f(and)h(a)g
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Fu(A)p Fr(kj)2815 4398 y Ft(\015)t(;)10 b Fp(\003)2959
4386 y Fv(gro)n(ws)23 b(when)515 4486 y Fu(\015)32 b
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Fn(2)1928 4669 y Fv(if)83 b Fu(\015)2102 4681 y Fp(1)2162
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y Fu(;)28 b Fv(\003)2439 4681 y Fp(1)2499 4669 y Fr(\024)23
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y(the)k(spaces)e Fu(M)1002 4921 y Ft(c)993 4974 y(\015)t(;)p
Fp(\003)1128 4951 y Fv(are)h(de\014ned)g(b)n(y)h(the)g(same)e
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5255 y(10)p eop end
%%Page: 11 11
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546 y(\015)t(;)p Fp(\003)735 523 y Fv(whic)n(h)32 b(w)n(e)f(discuss)g
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(the)515 623 y(same)27 b(pro)r(of.)639 772 y(Clearly)-7
b(,)27 b(if)h Fu(A)23 b Fr(2)h Fu(M)1274 742 y Ft(c)1265
795 y(\015)t(;)10 b Fp(\003)1411 772 y Fv(and)27 b Fu(s)1611
784 y Ft(j)1669 772 y Fv(=)c Fu(ta)18 b Fv(+)g Fu(b)1968
784 y Ft(j)2031 772 y Fv(satis\014es)27 b(\(2.2\))g(for)g
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994 y Ft(;)f(b)1755 1002 y Fn(2)1811 969 y Fv(+)18 b
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Fu(b)2627 1164 y Fp(2)2664 1152 y Fv(,)g(b)r(ounded)g(b)n(y)f(one)h(in)
515 1252 y(mo)r(dulus)h(if)32 b(\(2.2\))26 b(holds.)36
b(The)25 b(\014rst)h(term)g(in)g(the)g(r.h.s.)36 b(ma)n(y)25
b(b)r(e)h(b)r(ounded)g(in)g(terms)f(of)515 1351 y Fr(kj)p
Fu(A)p Fr(kj)707 1363 y Ft(\015)t(;)10 b Fp(\003)825
1351 y Fv(.)36 b(Indeed,)26 b(since)f(for)f Fu(t)f Fr(\035)g
Fv(1)i(w)n(e)f(ha)n(v)n(e)g([\()p Fu(ta)13 b Fv(+)g Fu(b)2290
1363 y Fp(1)2327 1351 y Fv(\))g Fr(\006)g Fv(\()p Fu(ta)g
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Fr(j)p Fu(b)2945 1363 y Fp(1)2995 1351 y Fr(\000)13 b
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y Fr(j)p Fu(A)1231 1559 y Fp(\()p Ft(ta)p Fp(+)p Ft(b)1398
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1518 y Fn(2)2196 1510 y Fs(j)2219 1544 y Fr(kj)p Fu(A)p
Fr(kj)2411 1556 y Ft(\015)2619 1544 y Fr(8)14 b Fv(0)23
b Fr(6)p Fv(=)g Fu(a;)k(b)2962 1556 y Fp(1)2999 1544
y Fu(;)14 b(b)3072 1556 y Fp(2)3132 1544 y Fr(2)23 b
Fq(Z)3265 1510 y Ft(d)3318 1544 y Fu(:)3208 1673 y Fv(\(2.4\))639
1772 y(F)-7 b(or)27 b Fu(a)c Fr(2)h Fq(Z)989 1742 y Ft(d)1046
1772 y Fr(n)18 b(f)p Fv(0)p Fr(g)p Fv(,)26 b(\003)d Fr(\025)g
Fv(6)k(and)g Fu(t)c Fr(\025)g Fv(0)k(w)n(e)g(set)980
1972 y Fu(O)1045 1937 y Ft(a)1043 1992 y(t)1086 1972
y Fv(\(\003\))c(=)1347 1893 y Fm([)1319 2077 y Ft(b)p
Fs(2)p Fg(Z)1432 2060 y Ff(d)1466 1972 y Fr(f)p Fv(\()p
Fu(ta)c Fv(+)f Fu(b)p Fv(\))23 b Fr(j)g(j)p Fu(ta)18
b Fv(+)g Fu(b)p Fr(j)23 b(\025)f Fv(\003\(1)c(+)g Fr(j)p
Fu(a)p Fr(j)h Fv(+)f Fr(j)p Fu(b)p Fr(j)p Fv(\))p Fr(j)p
Fu(a)p Fr(jg)p Fu(:)515 2238 y Fv(Note)27 b(that)h(sets)g
Fu(O)1123 2208 y Ft(a)1121 2259 y(t)1146 2267 y Fn(1)1183
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y Fv(ma)n(y)k(in)n(tersect)g(eac)n(h)g(other.)639 2338
y(W)-7 b(e)28 b(ha)n(v)n(e)1109 2521 y(\()p Fu(ta)18
b Fv(+)g Fu(b)p Fv(\))23 b Fr(2)h Fu(O)1551 2486 y Ft(a)1549
2541 y(t)1592 2521 y Fv(\(\003\))51 b Fr(\))f Fu(t)23
b Fr(\025)g Fv(\(\003)18 b Fr(\000)g Fv(1\)\(1)g(+)g
Fr(j)p Fu(a)p Fr(j)h Fv(+)f Fr(j)p Fu(b)p Fr(j)p Fv(\))423
b(\(2.5\))515 2703 y(and)1357 2803 y Fu(O)1422 2769 y
Ft(a)1420 2823 y(t)1463 2803 y Fv(\(\003)1553 2815 y
Fp(1)1590 2803 y Fv(\))24 b Fr(\032)e Fu(O)1798 2769
y Ft(a)1796 2823 y(t)1839 2803 y Fv(\(\003)1929 2815
y Fp(2)1966 2803 y Fv(\))83 b(if)h(\003)2271 2815 y Fp(1)2331
2803 y Fu(>)23 b Fv(\003)2477 2815 y Fp(2)2513 2803 y
Fu(:)672 b Fv(\(2.6\))515 2966 y(If)28 b Fu(s)23 b Fv(=)f
Fu(ta)d Fv(+)f Fu(b)k Fr(2)i Fu(O)1125 2935 y Ft(a)1123
2986 y(t)1166 2966 y Fv(\(\003\),)k(then)g Fr(j)p Fu(b)p
Fr(j)22 b(\024)1755 2925 y Fs(j)p Ft(s)p Fs(j)p 1730
2947 121 4 v 1730 2994 a Fp(\003)p Fs(j)p Ft(a)p Fs(j)1884
2966 y Fr(\024)1982 2925 y Ft(t)p Fs(j)p Ft(a)p Fs(j)p
Fp(+)p Fs(j)p Ft(b)p Fs(j)p 1982 2947 221 4 v 2031 2994
a Fp(\003)p Fs(j)p Ft(a)p Fs(j)2212 2966 y Fu(:)27 b
Fv(Therefore,)1729 3207 y Fr(j)p Fu(b)p Fr(j)c(\024)2017
3151 y Fu(t)p 1931 3188 201 4 v 1931 3264 a Fv(\003)18
b Fr(\000)g Fv(1)2142 3207 y Fu(:)1043 b Fv(\(2.7\))639
3425 y(Let)28 b Fu(ta)18 b Fv(+)g Fu(b)999 3437 y Ft(j)1057
3425 y Fr(2)23 b Fu(O)1200 3395 y Ft(a)1198 3445 y(t)1241
3425 y Fv(\(\003\),)28 b Fu(j)g Fv(=)23 b(1)p Fu(;)k
Fv(2.)37 b(Then)1133 3607 y Fr(j)p Fu(ta)p Fr(j)23 b(\025)g
Fv(\003\(1)17 b(+)h Fr(j)p Fu(a)p Fr(j)h Fv(+)f Fr(j)p
Fu(b)1847 3619 y Ft(j)1882 3607 y Fr(j)p Fv(\))p Fr(j)p
Fu(a)p Fr(j)g(\000)g(j)p Fu(b)2187 3619 y Ft(j)2222 3607
y Fr(j)23 b Fu(>)g Fr(j)p Fu(b)2415 3619 y Ft(j)2450
3607 y Fr(j)p Fv(\(\003)18 b Fr(\000)g Fv(1\))p Fu(;)515
3790 y Fv(and)27 b(2)p Fr(j)p Fu(ta)p Fr(j)c Fu(>)f Fv(\(\003)d
Fr(\000)f Fv(1\)\()p Fr(j)p Fu(b)1305 3802 y Fp(1)1342
3790 y Fr(j)g Fv(+)g Fr(j)p Fu(b)1525 3802 y Fp(2)1562
3790 y Fr(j)p Fv(.)37 b(Hence,)1290 3973 y Fr(j)p Fv(2)p
Fu(ta)18 b Fv(+)g Fu(b)1566 3985 y Fp(1)1621 3973 y Fv(+)h
Fu(b)1741 3985 y Fp(2)1777 3973 y Fr(j)24 b Fu(>)e Fv(\(\003)d
Fr(\000)f Fv(2\)\()p Fr(j)p Fu(b)2268 3985 y Fp(1)2305
3973 y Fr(j)g Fv(+)g Fr(j)p Fu(b)2488 3985 y Fp(2)2525
3973 y Fr(j)p Fv(\))p Fu(:)605 b Fv(\(2.8\))515 4155
y(Since)24 b Fr(j)p Fu(b)787 4167 y Fp(1)836 4155 y Fr(\000)12
b Fu(b)949 4167 y Fp(2)986 4155 y Fr(j)23 b(\024)f(j)p
Fu(b)1178 4167 y Fp(1)1215 4155 y Fr(j)12 b Fv(+)g Fr(j)p
Fu(b)1386 4167 y Fp(2)1423 4155 y Fr(j)25 b Fv(and)f(\003)e
Fr(\025)h Fv(3,)i(then)g Fr(j)p Fv(2)p Fu(ta)12 b Fv(+)g
Fu(b)2337 4167 y Fp(1)2385 4155 y Fv(+)g Fu(b)2498 4167
y Fp(2)2534 4155 y Fr(j)23 b Fu(>)g Fr(j)p Fu(b)2727
4167 y Fp(1)2776 4155 y Fr(\000)12 b Fu(b)2889 4167 y
Fp(2)2925 4155 y Fr(j)p Fv(.)36 b(Therefore,)801 4338
y([\()p Fu(ta)19 b Fv(+)f Fu(b)1068 4350 y Fp(1)1105
4338 y Fv(\))g Fr(\000)g Fv(\()p Fu(ta)h Fv(+)f Fu(b)1482
4350 y Fp(2)1519 4338 y Fv(\)])23 b(=)g([\()p Fu(ta)c
Fv(+)f Fu(b)1952 4350 y Fp(1)1988 4338 y Fv(\))h Fr(\000)f
Fv(\()p Fr(\000)p Fu(ta)g Fr(\000)g Fu(b)2430 4350 y
Fp(2)2467 4338 y Fv(\)])24 b(=)e Fr(j)p Fu(b)2692 4350
y Fp(1)2748 4338 y Fr(\000)c Fu(b)2867 4350 y Fp(2)2904
4338 y Fr(j)1852 4463 y(8)p Fv(\()p Fu(ta)g Fv(+)g Fu(b)2141
4475 y Fp(1)2178 4463 y Fv(\))p Fu(;)28 b Fv(\()p Fu(ta)19
b Fv(+)f Fu(b)2505 4475 y Fp(2)2542 4463 y Fv(\))23 b
Fr(2)h Fu(O)2741 4428 y Ft(a)2739 4483 y(t)2781 4463
y Fv(\(\003\))p Fu(:)282 b Fv(\(2.9\))515 4645 y(Similar,)1346
4745 y Fr(h)p Fv(\()p Fu(ta)19 b Fv(+)f Fu(b)p Fv(\))p
Fu(;)c(ta)p Fr(i)23 b(\025)g(j)p Fu(ta)p Fr(j)p Fv(\()p
Fr(j)p Fu(ta)p Fr(j)18 b(\000)g(j)p Fu(b)p Fr(j)p Fv(\))23
b Fu(>)g Fv(0)619 b(\(2.10\))515 4894 y(if)28 b Fu(ta)18
b Fv(+)g Fu(b)23 b Fr(2)g Fu(O)968 4864 y Ft(a)966 4915
y(t)1009 4894 y Fv(\(\003\).)1905 5255 y(11)p eop end
%%Page: 12 12
TeXDict begin 12 11 bop 515 523 a Fl(Lemma)33 b(2.1.)41
b Fo(L)l(et)30 b Fv(\003)1248 535 y Fp(1)1304 523 y Fr(\000)19
b Fv(3)25 b Fr(\025)f Fv(\003)1602 535 y Fp(2)1665 523
y Fr(\025)g Fv(6)31 b Fo(and)g Fu(s)2028 535 y Fp(1)2090
523 y Fv(=)25 b Fu(ta)19 b Fv(+)g Fu(b)2393 535 y Fp(1)2455
523 y Fr(2)26 b Fu(O)2601 493 y Ft(a)2599 544 y(t)2641
523 y Fv(\(\003)2731 535 y Fp(1)2769 523 y Fv(\))p Fo(,)32
b Fu(s)2897 535 y Fp(2)2959 523 y Fv(=)25 b Fu(ta)19
b Fv(+)g Fu(b)3262 535 y Fp(2)3324 523 y Fr(62)515 623
y Fu(O)580 593 y Ft(a)578 643 y(t)621 623 y Fv(\(\003)711
635 y Fp(2)748 623 y Fv(\))p Fo(.)39 b(Then)1041 855
y Fr(j)p Fu(s)1103 867 y Fp(1)1159 855 y Fr(\000)18 b
Fu(s)1281 867 y Fp(2)1318 855 y Fr(j)23 b(\025)g Fu(t)1482
738 y Fm(\022)1651 799 y Fv(1)p 1553 836 238 4 v 1553
912 a(\003)1611 924 y Fp(2)1666 912 y Fv(+)18 b(1)1819
855 y Fr(\000)2010 799 y Fv(1)p 1912 836 V 1912 912 a(\003)1970
924 y Fp(1)2025 912 y Fr(\000)g Fv(1)2160 738 y Fm(\023)2244
855 y Fu(>)23 b(t)2385 799 y Fv(\003)2443 811 y Fp(1)2499
799 y Fr(\000)18 b Fv(\003)2640 811 y Fp(2)2695 799 y
Fr(\000)g Fv(2)p 2385 836 435 4 v 2555 912 a(\003)2613
883 y Fp(2)2613 934 y(1)2830 855 y Fu(:)314 b Fv(\(2.11\))515
1088 y Fo(Pr)l(o)l(of.)43 b Fv(Let)33 b(us)g(denote)g(\003)1372
1058 y Fs(0)1372 1109 y Fp(1)1441 1088 y Fv(=)f(\003)1596
1100 y Fp(1)1655 1088 y Fr(\000)22 b Fv(1)32 b(and)h(\003)2041
1058 y Fs(0)2041 1109 y Fp(2)2110 1088 y Fv(=)f(\003)2265
1100 y Fp(2)2324 1088 y Fv(+)22 b(1.)53 b(Since)33 b
Fu(s)2790 1100 y Fp(1)2859 1088 y Fr(2)g Fu(O)3012 1058
y Ft(a)3010 1108 y(t)3053 1088 y Fv(\(\003)3143 1100
y Fp(1)3180 1088 y Fv(\))g(and)515 1188 y(\003)573 1200
y Fp(1)610 1188 y Fr(j)p Fu(a)p Fr(j)18 b(\000)g Fv(1)23
b Fr(\025)f Fv(\003)1011 1157 y Fs(0)1011 1208 y Fp(1)1048
1188 y Fr(j)p Fu(a)p Fr(j)p Fv(,)28 b(then)819 1418 y
Fr(j)p Fu(b)878 1430 y Fp(1)915 1418 y Fr(j)23 b(\024)1059
1362 y Fu(t)p Fr(j)p Fu(a)p Fr(j)18 b(\000)g Fv(\003)1338
1374 y Fp(1)1375 1362 y Fr(j)p Fu(a)p Fr(j)p Fv(\(1)g(+)g
Fr(j)p Fu(a)p Fr(j)p Fv(\))p 1059 1399 704 4 v 1247 1475
a(\003)1305 1487 y Fp(1)1342 1475 y Fr(j)p Fu(a)p Fr(j)g(\000)g
Fv(1)1796 1418 y Fr(\024)1893 1362 y Fu(t)h Fr(\000)f
Fv(\003)2083 1374 y Fp(1)2120 1362 y Fv(\(1)g(+)g Fr(j)p
Fu(a)p Fr(j)p Fv(\))p 1893 1399 524 4 v 2108 1475 a(\003)2166
1446 y Fs(0)2166 1497 y Fp(1)2450 1418 y Fr(\024)2580
1362 y Fu(t)p 2548 1399 95 4 v 2548 1475 a Fv(\003)2606
1446 y Fs(0)2606 1497 y Fp(1)2671 1418 y Fr(\000)g Fv(\(1)g(+)g
Fr(j)p Fu(a)p Fr(j)p Fv(\))p Fu(:)515 1651 y Fv(Since)28
b Fu(s)771 1663 y Fp(2)840 1651 y Fu(=)-51 b Fr(2)23
b Fu(O)974 1621 y Ft(a)972 1671 y(t)1015 1651 y Fv(\(\003)1105
1663 y Fp(2)1142 1651 y Fv(\))28 b(and)g(\003)1422 1663
y Fp(2)1458 1651 y Fr(j)p Fu(a)p Fr(j)19 b Fv(+)f(1)k
Fr(\024)h Fv(\003)1860 1621 y Fs(0)1860 1672 y Fp(2)1897
1651 y Fr(j)p Fu(a)p Fr(j)p Fv(,)28 b(then)819 1882 y
Fr(j)p Fu(b)878 1894 y Fp(2)915 1882 y Fr(j)23 b(\025)1059
1825 y Fu(t)p Fr(j)p Fu(a)p Fr(j)18 b(\000)g Fv(\003)1338
1837 y Fp(2)1375 1825 y Fr(j)p Fu(a)p Fr(j)p Fv(\(1)g(+)g
Fr(j)p Fu(a)p Fr(j)p Fv(\))p 1059 1862 704 4 v 1247 1939
a(\003)1305 1951 y Fp(2)1342 1939 y Fr(j)p Fu(a)p Fr(j)g
Fv(+)g(1)1796 1882 y Fr(\025)1893 1825 y Fu(t)h Fr(\000)f
Fv(\003)2083 1795 y Fs(0)2083 1846 y Fp(2)2120 1825 y
Fv(\(1)g(+)g Fr(j)p Fu(a)p Fr(j)p Fv(\))p 1893 1862 524
4 v 2108 1939 a(\003)2166 1910 y Fs(0)2166 1961 y Fp(2)2450
1882 y Fr(\025)2580 1825 y Fu(t)p 2548 1862 95 4 v 2548
1939 a Fv(\003)2606 1910 y Fs(0)2606 1961 y Fp(2)2671
1882 y Fr(\000)g Fv(\(1)g(+)g Fr(j)p Fu(a)p Fr(j)p Fv(\))p
Fu(:)515 2114 y Fv(As)27 b Fr(j)p Fu(s)699 2126 y Fp(1)755
2114 y Fv(+)18 b Fu(s)877 2126 y Fp(2)914 2114 y Fr(j)23
b(\025)g(j)p Fu(b)1107 2126 y Fp(2)1144 2114 y Fr(j)18
b(\000)h(j)p Fu(b)1328 2126 y Fp(1)1364 2114 y Fr(j)p
Fv(,)28 b(then)g(\(2.11\))f(follo)n(ws.)p 3318 2114 4
57 v 3322 2062 50 4 v 3322 2114 V 3372 2114 4 57 v 515
2280 a Fl(Lemma)k(2.2.)40 b Fo(If)30 b(under)g(the)g(assumptions)g(of)h
(L)l(emma)e(2.1)j Fr(h)p Fu(a;)14 b(s)2660 2292 y Fp(2)2697
2280 y Fr(i)23 b(\025)g Fv(0)p Fo(,)29 b(then)1642 2463
y Fv([)p Fu(s)1704 2475 y Fp(1)1759 2463 y Fr(\000)18
b Fu(s)1881 2475 y Fp(2)1919 2463 y Fv(])23 b Fr(\025)f
Fu(t)p Fv(\003)2140 2427 y Fs(\000)p Fp(2)2140 2485 y(1)2229
2463 y Fu(:)915 b Fv(\(2.12\))515 2646 y Fo(Pr)l(o)l(of.)43
b Fv(Due)32 b(to)f(\(2.11\))o(,)h Fr(j)p Fu(s)1387 2658
y Fp(1)1446 2646 y Fr(\000)20 b Fu(s)1570 2658 y Fp(2)1607
2646 y Fr(j)30 b(\025)g Fv(r.h.s.)48 b(of)38 b(\(2.12\))o(,)32
b(so)f(w)n(e)g(only)g(ha)n(v)n(e)f(to)h(estimate)515
2745 y(from)38 b(b)r(elo)n(w)g Fr(j)p Fu(s)1030 2757
y Fp(1)1093 2745 y Fv(+)25 b Fu(s)1222 2757 y Fp(2)1259
2745 y Fr(j)p Fv(.)70 b(Let)38 b(us)h(set)f Fu(m)j Fv(=)g
Fr(dj)p Fu(b)2108 2757 y Fp(1)2144 2745 y Fr(jj)p Fu(a)p
Fr(j)2257 2715 y Fs(\000)p Fp(1)2347 2745 y Fr(e)p Fu(;)d
Fv(where)g Fr(d)p Fu(x)p Fr(e)h Fv(stands)f(for)g(the)515
2845 y(smallest)27 b(in)n(teger)g Fr(\025)22 b Fu(x)p
Fv(.)38 b(W)-7 b(e)28 b(rewrite)e Fu(s)1767 2857 y Fp(1)1832
2845 y Fv(as)1301 3027 y Fu(s)1340 3039 y Fp(1)1400 3027
y Fv(=)d(\()p Fu(t)c Fr(\000)f Fu(m)p Fv(\))p Fu(a)g
Fv(+)g Fu(b)1938 2993 y Fs(0)1938 3048 y Fp(1)1975 3027
y Fu(;)97 b(b)2131 2993 y Fs(0)2131 3048 y Fp(1)2191
3027 y Fv(=)23 b Fu(b)2315 3039 y Fp(1)2370 3027 y Fv(+)18
b Fu(ma:)515 3210 y Fv(Clearly)-7 b(,)1750 3310 y Fr(h)p
Fu(a;)14 b(b)1899 3275 y Fs(0)1899 3330 y Fp(1)1936 3310
y Fr(i)23 b(\025)g Fv(0)p Fu(:)1023 b Fv(\(2.13\))515
3459 y(Due)28 b(to)f(\(2.7\),)h Fr(j)p Fu(b)1071 3471
y Fp(1)1108 3459 y Fr(j)23 b(\024)1304 3427 y Ft(t)p
1251 3441 131 4 v 1251 3488 a Fp(\003)p Fs(\000)p Fp(1)1392
3459 y Fv(.)37 b(Therefore)1295 3696 y Fu(m)23 b Fr(\024)1478
3604 y Fm(l)1695 3640 y Fu(t)p 1532 3677 356 4 v 1532
3753 a Fr(j)p Fu(a)p Fr(j)p Fv(\(\003)18 b Fr(\000)g
Fv(1\))1897 3604 y Fm(m)1964 3696 y Fu(<)2204 3640 y
Fv(2)p Fu(t)p 2062 3677 V 2062 3753 a Fr(j)p Fu(a)p Fr(j)p
Fv(\(\003)g Fr(\000)g Fv(1\))2450 3696 y Fr(\024)2553
3640 y Fu(t)p 2548 3677 42 4 v 2548 3753 a Fv(4)515 3927
y(\(w)n(e)27 b(use)f(\(2.5\))h(to)g(get)f(the)i(second)e(inequalit)n(y)
g(and)h(use)g(that)g(\003)c Fr(\025)f Fv(\003)2750 3897
y Fs(0)2790 3927 y Fv(+)17 b(\003)2930 3939 y Fp(\001)3011
3927 y Fr(\025)23 b Fv(9)k(to)f(get)515 4027 y(the)i(third)g(one\).)36
b(Hence,)28 b Fu(t)19 b Fr(\000)f Fu(m)23 b Fr(\025)1677
3994 y Fp(3)p 1677 4008 34 4 v 1677 4055 a(4)1720 4027
y Fu(t)p Fv(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)644 4219
y Fr(j)p Fu(s)706 4231 y Fp(1)761 4219 y Fv(+)18 b Fu(s)883
4231 y Fp(2)920 4219 y Fr(j)943 4184 y Fp(2)1004 4219
y Fv(=)k Fr(j)p Fu(s)1153 4231 y Fp(2)1191 4219 y Fr(j)1214
4184 y Fp(2)1269 4219 y Fv(+)c Fr(j)p Fv(\()p Fu(t)h
Fr(\000)f Fu(m)p Fv(\))p Fu(a)p Fr(j)1711 4184 y Fp(2)1767
4219 y Fv(+)g Fr(j)p Fu(b)1909 4184 y Fs(0)1909 4239
y Fp(1)1946 4219 y Fr(j)1969 4184 y Fp(2)2025 4219 y
Fv(+)g(2)p Fu(t)p Fr(h)p Fu(a;)27 b(s)2345 4231 y Fp(2)2382
4219 y Fr(i)19 b Fv(+)f(2)p Fr(h)p Fu(b)2626 4184 y Fs(0)2626
4239 y Fp(1)2663 4219 y Fu(;)28 b(s)2753 4231 y Fp(2)2790
4219 y Fr(i)18 b Fv(+)g(2)p Fu(t)p Fr(h)p Fu(b)3063 4184
y Fs(0)3063 4239 y Fp(1)3100 4219 y Fu(;)28 b(a)p Fr(i)p
Fu(:)515 4401 y Fv(Since)g Fr(h)p Fu(a;)14 b(s)884 4413
y Fp(2)921 4401 y Fr(i)23 b(\025)g Fv(0)k(b)n(y)g(assumption,)g(then)h
(using)h(\(2.13\))e(w)n(e)g(get)g(that)561 4584 y Fr(j)p
Fu(s)623 4596 y Fp(1)678 4584 y Fv(+)18 b Fu(s)800 4596
y Fp(2)837 4584 y Fr(j)860 4550 y Fp(2)921 4584 y Fr(\025)k(j)p
Fu(s)1070 4596 y Fp(2)1108 4584 y Fr(j)1131 4550 y Fp(2)1186
4584 y Fv(+)c Fr(j)p Fv(\()p Fu(t)h Fr(\000)f Fu(m)p
Fv(\))p Fu(a)p Fr(j)1628 4550 y Fp(2)1684 4584 y Fv(+)g
Fr(j)p Fu(b)1826 4550 y Fs(0)1826 4604 y Fp(1)1863 4584
y Fr(j)1886 4550 y Fp(2)1942 4584 y Fv(+)g(2)p Fr(h)p
Fu(b)2135 4550 y Fs(0)2135 4604 y Fp(1)2171 4584 y Fu(;)28
b(s)2261 4596 y Fp(2)2298 4584 y Fr(i)c Fv(=)e(\()p Fu(t)d
Fr(\000)f Fu(m)p Fv(\))2710 4550 y Fp(2)2747 4584 y Fr(j)p
Fu(a)p Fr(j)2837 4550 y Fp(2)2893 4584 y Fv(+)g Fr(j)p
Fu(s)3038 4596 y Fp(2)3094 4584 y Fv(+)g Fu(b)3213 4550
y Fs(0)3213 4604 y Fp(1)3250 4584 y Fr(j)3273 4550 y
Fp(2)3310 4584 y Fu(:)515 4766 y Fv(Using)27 b(that)h
Fu(t)18 b Fr(\000)g Fu(m)23 b Fr(\025)1253 4734 y Fp(3)p
1253 4748 V 1253 4795 a(4)1296 4766 y Fu(t)28 b Fv(w)n(e)f(get)g
(\(2.12\))o(.)p 3318 4766 4 57 v 3322 4714 50 4 v 3322
4766 V 3372 4766 4 57 v 1905 5255 a(12)p eop end
%%Page: 13 13
TeXDict begin 13 12 bop 639 523 a Fv(Elemen)n(ts)27 b(of)h(the)g(space)
1533 718 y Fu(M)1623 683 y Ft(c)1614 738 y(T)9 b(L)1735
718 y Fv(=)1939 639 y Fm([)1823 821 y Fs(j)p Ft(\015)t
Fs(j\024)p Fp(1)p Ft(;)g Fp(\003)p Fs(\025)p Fp(6)2161
718 y Fu(M)2251 683 y Ft(c)2242 738 y(\015)t(;)h Fp(\003)515
1002 y Fv(are)27 b(called)h Fo(T\177)-42 b(oplitz{Lipschitz)34
b(matric)l(es)p Fv(.)1864 972 y Fp(1)1969 1002 y Fv(The)29
b(linear)f(op)r(erators)e(whic)n(h)i(they)h(de\014ne)515
1102 y(are)d(called)i Fo(T\177)-42 b(oplitz{Lipschitz)33
b(op)l(er)l(ators)p Fv(.)515 1235 y Fo(Example)43 b Fv(2.3)p
Fo(.)j Fv(\(T\177)-42 b(oplitz)34 b(matrices,)i(see)e([HS78].\))59
b(Let)34 b Fr(A)i Fv(:)f Fq(Z)2599 1204 y Ft(d)2672 1235
y Fr(!)g Fq(R)g Fv(b)r(e)g(a)f(function)515 1334 y(with)28
b(compact)f(supp)r(ort,)g(and)1585 1517 y Fu(A)1647 1529
y Ft(s)1678 1537 y Fn(1)1722 1529 y Ft(s)1753 1537 y
Fn(2)1813 1517 y Fv(=)c Fr(A)p Fv(\()p Fu(s)2038 1529
y Fp(1)2094 1517 y Fr(\000)18 b Fu(s)2216 1529 y Fp(2)2253
1517 y Fv(\))p Fu(:)515 1699 y Fv(Then)i Fu(A)786 1714
y Fp(\()p Ft(ta)p Fp(+)p Ft(b)953 1722 y Fn(1)986 1714
y Fp(\))p Fs(\000)p Fp(\()p Ft(ta)p Fp(+)p Ft(b)1231
1722 y Fn(2)1264 1714 y Fp(\))1317 1699 y Fv(=)j Fr(A)p
Fv(\(2)p Fu(ta)t Fv(+)t Fu(b)1728 1711 y Fp(1)1769 1699
y Fv(+)t Fu(b)1874 1711 y Fp(2)1910 1699 y Fv(\))e(v)-5
b(anishes)20 b(if)h Fu(t)i Fr(\035)g Fv(1,)f(and)e Fu(A)2813
1714 y Fp(\()p Ft(ta)p Fp(+)p Ft(b)2980 1722 y Fn(1)3013
1714 y Fp(\))11 b(\()p Ft(t)h(a)p Fp(+)p Ft(b)3229 1722
y Fn(2)3261 1714 y Fp(\))3314 1699 y Fv(=)515 1799 y
Fr(A)p Fv(\()p Fu(b)649 1811 y Fp(1)705 1799 y Fr(\000)18
b Fu(b)824 1811 y Fp(2)861 1799 y Fv(\).)37 b(So)27 b(TL1)g(holds)g
(and)1290 1981 y Fu(A)1352 1946 y Fs(1\000)1352 2006
y Ft(a;b)1437 2014 y Fn(1)1469 2006 y Ft(;b)1518 2014
y Fn(2)1578 1981 y Fv(=)c(0)p Fu(;)96 b(A)1889 1946 y
Fs(1)p Fp(+)1889 2006 y Ft(a;b)1974 2014 y Fn(1)2007
2006 y Ft(;b)2056 2014 y Fn(2)2115 1981 y Fv(=)23 b Fr(A)p
Fv(\()p Fu(b)2337 1993 y Fp(1)2393 1981 y Fr(\000)18
b Fu(b)2512 1993 y Fp(2)2549 1981 y Fv(\))p Fu(:)515
2164 y Fv(The)29 b(prop)r(erties)g(TL2)g(and)h(TL3)f(ob)n(viously)f
(hold,)i(so)f(T\177)-42 b(oplitz)30 b(matrices)f(b)r(elong)g(to)g(all)
515 2264 y(spaces)d Fu(M)850 2276 y Ft(\015)t(;)p Fp(\003)958
2264 y Fv(.)639 2363 y(In)36 b(particular,)h(the)g(iden)n(tit)n(y)f
(matrix)f(is)h(T\177)-42 b(oplitz)36 b(with)g Fr(A)h
Fv(=)g Fl(1)2754 2378 y Fs(f)p Fd(0)p Fs(g)2864 2363
y Fv(.)62 b(In)36 b(this)g(case)515 2472 y Fu(A)577 2487
y Fp(\()p Ft(ta)p Fp(+)p Ft(b)744 2495 y Fn(1)777 2487
y Fp(\))11 b(\()p Ft(ta)p Fp(+)p Ft(b)981 2495 y Fn(2)1013
2487 y Fp(\))1062 2472 y Fr(\000)18 b Fu(A)1207 2437
y Fs(1)p Fp(+)1207 2497 y Ft(a;b)1292 2505 y Fn(1)1325
2497 y Ft(;b)1374 2505 y Fn(2)1433 2472 y Fv(=)23 b Fl(1)1569
2487 y Fs(f)p Fp(0)p Fs(g)1674 2472 y Fv(\(2)p Fu(ta)18
b Fv(+)h Fu(b)1960 2484 y Fp(1)2015 2472 y Fv(+)f Fu(b)2134
2484 y Fp(2)2171 2472 y Fv(\))23 b(=)g(0)28 b(since)f
Fr(j)p Fv(2)p Fu(ta)18 b Fv(+)g Fu(b)g Fr(\000)g Fv(1)g(+)g
Fu(b)3143 2484 y Fp(2)3180 2472 y Fr(j)24 b Fu(>)e Fv(0.)515
2572 y(This)33 b(inequalit)n(y)f(is)h(ob)n(vious)f(if)h
Fu(b)1612 2584 y Fp(1)1681 2572 y Fv(=)f Fu(b)1814 2584
y Fp(2)1883 2572 y Fv(=)f(0,)j(otherwise)e(it)i(follo)n(ws)e(from)g
(\(2.8\).)53 b(So,)515 2672 y Fr(h)p Fv(id)p Fr(i)648
2684 y Ft(\015)t(;)p Fp(\003)779 2672 y Fv(=)23 b(1)55
b(for)27 b(an)n(y)g Fr(j)p Fu(\015)5 b Fr(j)22 b(\024)h
Fv(1)p Fu(;)14 b Fv(\003)22 b Fr(\025)h Fv(6.)p 3318
2672 4 57 v 3322 2619 50 4 v 3322 2672 V 3372 2672 4
57 v 515 2804 a Fo(Example)38 b Fv(2.4)p Fo(.)k Fv(\(Hank)n(el)29
b(matrices,)f(see)h([HS78)o(].\))42 b(Let)29 b Fu(A)2385
2816 y Ft(s)2416 2824 y Fn(1)2449 2816 y Ft(s)2480 2824
y Fn(2)2542 2804 y Fv(=)c Fr(A)p Fv(\()p Fu(s)2769 2816
y Fp(1)2826 2804 y Fv(+)19 b Fu(s)2949 2816 y Fp(2)2986
2804 y Fv(\),)30 b(where)f Fr(A)515 2904 y Fv(is)e(a)g(function)i(on)e
Fq(Z)1163 2874 y Ft(d)1230 2904 y Fv(with)h(compact)f(supp)r(ort.)37
b(No)n(w)1290 3087 y Fu(A)1352 3051 y Fs(1)p Fp(+)1352
3112 y Ft(a;b)1437 3120 y Fn(1)1469 3112 y Ft(;b)1518
3120 y Fn(2)1578 3087 y Fr(\021)23 b Fv(0)p Fu(;)96 b(A)1889
3051 y Fs(1\000)1889 3112 y Ft(a;b)1974 3120 y Fn(1)2007
3112 y Ft(;b)2056 3120 y Fn(2)2115 3087 y Fv(=)23 b Fr(A)p
Fv(\()p Fu(b)2337 3099 y Fp(1)2393 3087 y Fr(\000)18
b Fu(b)2512 3099 y Fp(2)2549 3087 y Fv(\))p Fu(:)515
3269 y Fv(Again,)43 b(relations)d(TL1-TL3)f(hold)i(for)f(all)h
(\(admissible\))f Fu(\015)46 b Fv(and)41 b(\003,)i(so)d(the)i(Hank)n
(el)515 3369 y(matrices)27 b(also)f(b)r(elong)h(to)h(all)f(spaces)g
Fu(M)1832 3381 y Ft(\015)t(;)p Fp(\003)1939 3369 y Fv(.)p
3318 3369 V 3322 3316 50 4 v 3322 3369 V 3372 3369 4
57 v 515 3502 a Fo(Example)46 b Fv(2.5)p Fo(.)h Fv(\(Compactly)37
b(supp)r(orted)g(matrices\).)66 b(Let)37 b Fu(A)2557
3514 y Ft(s)2588 3522 y Fn(1)2632 3514 y Ft(s)2663 3522
y Fn(2)2738 3502 y Fv(b)r(e)h(a)f(matrix)f(suc)n(h)515
3601 y(that)25 b Fu(A)754 3613 y Ft(s)785 3621 y Fn(1)830
3613 y Ft(s)861 3621 y Fn(2)921 3601 y Fv(=)d(0)j(if)h
Fr(j)p Fu(s)1211 3613 y Fp(1)1248 3601 y Fr(j)14 b Fv(+)f
Fr(j)p Fu(s)1425 3613 y Fp(2)1463 3601 y Fr(j)23 b(\025)f
Fu(C)1655 3613 y Ft(A)1735 3601 y Fv(for)j(a)g(suitable)g(constan)n(t)g
Fu(C)2626 3613 y Ft(A)2680 3601 y Fv(.)36 b(This)26 b(is)f(a)g(T\177)
-42 b(oplitz{)515 3701 y(Lipsc)n(hitz)27 b(matrix)g(suc)n(h)h(that)f
Fr(h)p Fu(A)p Fr(i)1632 3713 y Ft(\015)t(;)12 b Fp(\003)1775
3701 y Fv(=)23 b(0,)k(if)h(\003)g(is)f(su\016cien)n(tly)h(large.)p
3318 3701 V 3322 3648 50 4 v 3322 3701 V 3372 3701 4
57 v 515 3834 a Fo(Example)j Fv(2.6)p Fo(.)k Fv(\(T)-7
b(ensor)20 b(pro)r(duct)h(of)g(v)n(ectors\).)33 b(Let)21
b Fu(y)2238 3803 y Fp(1)2275 3834 y Fu(;)14 b(y)2356
3803 y Fp(2)2416 3834 y Fr(2)24 b Fu(Y)2543 3846 y Ft(\015)2608
3834 y Fv(=)f Fu(Y)2744 3846 y Ft(\015)t(;m)2862 3854
y FA(\003)2900 3834 y Fv(,)g(where)d Fu(\015)28 b Fr(\025)22
b Fv(0)515 3933 y(and)27 b Fu(m)749 3945 y Fs(\003)810
3933 y Fr(\025)c Fv(1.)36 b(W)-7 b(e)28 b(de\014ne)g
Fu(A)c Fv(=)e Fu(y)1599 3903 y Fp(1)1655 3933 y Fr(\012)c
Fu(y)1782 3903 y Fp(2)1819 3933 y Fv(,)27 b(i.e.)37 b
Fu(A)2074 3945 y Ft(ss)2136 3929 y FA(0)2187 3933 y Fv(=)22
b Fu(y)2318 3903 y Fp(1)2315 3954 y Ft(s)2355 3933 y
Fu(y)2399 3903 y Fp(2)2396 3956 y Ft(s)2427 3939 y FA(0)2454
3933 y Fv(.)37 b(Then)1110 4130 y Fr(j)p Fu(A)1195 4142
y Ft(ss)1257 4125 y FA(0)1284 4130 y Fr(j)24 b(\024)e(k)p
Fu(y)1504 4095 y Fp(1)1541 4130 y Fr(k)1583 4142 y Ft(\015)1625
4130 y Fr(k)p Fu(y)1711 4095 y Fp(2)1747 4130 y Fr(k)1789
4142 y Ft(\015)1831 4130 y Fu(e)1870 4095 y Fs(\000)p
Ft(\015)t Fp(\()p Fs(j)p Ft(s)p Fs(j)p Fp(+)p Fs(j)p
Ft(s)2160 4070 y FA(0)2181 4095 y Fs(j)p Fp(\))2231 4130
y Fr(h)p Fu(s)p Fr(i)2334 4095 y Fs(\000)p Ft(m)2445
4103 y FA(\003)2484 4130 y Fr(h)p Fu(s)2555 4095 y Fs(0)2579
4130 y Fr(i)2611 4095 y Fs(\000)p Ft(m)2722 4103 y FA(\003)2761
4130 y Fu(:)515 4312 y Fv(So)k Fu(A)691 4277 y Fs(1\006)691
4337 y Ft(a;b)776 4345 y Fn(1)809 4337 y Ft(;b)858 4345
y Fn(2)918 4312 y Fv(=)c(0.)36 b(Due)28 b(to)f(\(2.2\))o(,)h(\(2.5\))e
(and)h(\(2.9\),)g(for)f(an)n(y)g Fu(ta)17 b Fv(+)g Fu(b)2672
4324 y Ft(j)2730 4312 y Fr(2)23 b Fu(O)2873 4282 y Ft(a)2871
4333 y(t)2914 4312 y Fv(\(\003\),)k Fu(j)h Fv(=)23 b(1)p
Fu(;)14 b Fv(2,)515 4412 y(\003)22 b Fr(\025)h Fv(6,)k(w)n(e)h(ha)n(v)n
(e)515 4639 y Fr(j)p Fu(ta)14 b Fv(+)f Fu(b)740 4651
y Ft(j)775 4639 y Fr(j)23 b(\025)g Fu(t)939 4522 y Fm(\022)1000
4639 y Fv(1)14 b Fr(\000)1224 4583 y Fv(1)p 1144 4620
201 4 v 1144 4696 a(\003)k Fr(\000)g Fv(1)1355 4522 y
Fm(\023)1439 4639 y Fr(\025)1537 4583 y Fv(1)p 1537 4620
42 4 v 1537 4696 a(2)1588 4639 y Fu(t;)97 b(t)23 b Fr(\025)g
Fv(2\(\003)13 b Fr(\000)h Fv(1\))p Fu(;)97 b Fr(j)p Fu(ta)14
b Fv(+)f Fu(b)2522 4651 y Fp(1)2559 4639 y Fr(j)h Fv(+)g
Fr(j)p Fu(ta)g Fv(+)g Fu(b)2901 4651 y Fp(2)2938 4639
y Fr(j)23 b(\025)f(j)p Fu(b)3130 4651 y Fp(1)3181 4639
y Fr(\000)14 b Fu(b)3296 4651 y Fp(2)3333 4639 y Fr(j)p
Fu(:)p 515 4771 1146 4 v 607 4825 a Fc(1)642 4848 y Fh(It)29
b(w)n(ould)f(b)r(e)g(more)g(consisten)n(t)i(to)e(call)g(these)h
(matrices)f(Hank)n(el{T\177)-35 b(oplitz{Lipsc)n(hitz)31
b(\(see)e(b)r(elo)n(w)515 4927 y(Examples)e(2.3,)h(2.4\),)g(reserving)f
(the)h(name)g(T\177)-35 b(oplitz{Lipsc)n(hitz)29 b(for)d(the)i
(matrices,)g(de\014ned)h(b)r(elo)n(w)e(in)515 5006 y(Section)e(2.3.)31
b(Unfortunately)-6 b(,)24 b(the)h(former)d(name)j(is)e(hardly)g
(acceptable.)1905 5255 y Fv(13)p eop end
%%Page: 14 14
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(estimate)966 695 y Fr(j)p Fu(A)1051 710 y Fp(\()p Ft(ta)p
Fp(+)p Ft(b)1218 718 y Fn(1)1251 710 y Fp(\))p Fs(\006)p
Fp(\()p Ft(ta)p Fp(+)p Ft(b)1496 718 y Fn(2)1528 710
y Fp(\))1558 695 y Fr(j)23 b(\024)g(k)p Fu(y)1778 661
y Fp(1)1814 695 y Fr(k)1856 707 y Ft(\015)1899 695 y
Fr(k)p Fu(y)1985 661 y Fp(2)2021 695 y Fr(k)2063 707
y Ft(\015)2105 695 y Fv(4)p Fu(t)2177 661 y Fs(\000)p
Fp(2)2266 695 y Fu(e)2305 661 y Fs(\000)p Ft(\015)t Fp(\()p
Fs(j)p Ft(ta)p Fp(+)p Ft(b)2583 669 y Fn(1)2614 661 y
Fs(j)p Fp(+)p Fs(j)p Ft(ta)p Fp(+)p Ft(b)2846 669 y Fn(2)2878
661 y Fs(j)p Fp(\))1254 836 y Fr(\024)f(k)p Fu(y)1427
802 y Fp(1)1464 836 y Fr(k)1506 848 y Ft(\015)1548 836
y Fr(k)p Fu(y)1634 802 y Fp(2)1670 836 y Fr(k)1712 848
y Ft(\015)1754 836 y Fu(t)1784 802 y Fs(\000)p Fp(1)1873
836 y Fv(2\(\003)c Fr(\000)g Fv(1\))2180 802 y Fs(\000)p
Fp(1)2269 836 y Fu(e)2308 802 y Fs(\000)p Ft(\015)t Fs(j)p
Ft(b)2448 810 y Fn(1)2480 802 y Fs(\000)p Ft(b)2561 810
y Fn(2)2593 802 y Fs(j)2617 836 y Fu(:)515 1009 y Fv(That)27
b(is,)1284 1108 y Fr(h)p Fu(A)p Fr(i)1410 1120 y Ft(\015)t(;)p
Fp(\003)1542 1108 y Fr(\024)c Fv(2\(\003)18 b Fr(\000)g
Fv(1\))1937 1074 y Fs(\000)p Fp(1)2026 1108 y Fr(k)p
Fu(y)2112 1074 y Fp(1)2148 1108 y Fr(k)2190 1120 y Ft(\015)2232
1108 y Fr(k)p Fu(y)2318 1074 y Fp(2)2355 1108 y Fr(k)2397
1120 y Ft(\015)2439 1108 y Fu(:)p 2549 1108 4 57 v 2553
1056 50 4 v 2553 1108 V 2602 1108 4 57 v 705 w Fv(\(2.14\))639
1297 y(F)-7 b(or)19 b(matrices,)i(formed)e(b)n(y)h(2)s
Fr(\002)s Fv(2{blo)r(c)n(ks,)e(the)i(righ)n(t)f(de\014nition)h(of)g
(T\177)-42 b(oplitz{Lipsc)n(hitz)515 1397 y(matrices)23
b(in)h(the)h(spaces)e Fu(M)1418 1367 y Fs(\006)1409 1418
y Ft(\015)1497 1397 y Fv(\(see)h(section)g(1.4\))f(is)h(the)h(follo)n
(wing.)34 b(T)-7 b(ak)n(e)23 b(an)n(y)g Fu(A)h Fr(2)f
Fu(M)3301 1367 y Fp(+)3292 1418 y Ft(\015)3356 1397 y
Fv(.)515 1497 y(Then)1339 1596 y(\()p Fu(pA)p Fv(\))1507
1561 y Fs(1\000)1507 1621 y Ft(a;b)1592 1629 y Fn(1)1625
1621 y Ft(;b)1674 1629 y Fn(2)1733 1596 y Fr(\021)g Fv(0)14
b Fu(;)96 b Fv(\()p Fu(q)s(A)p Fv(\))2162 1561 y Fs(1)p
Fp(+)2162 1621 y Ft(a;b)2247 1629 y Fn(1)2280 1621 y
Ft(;b)2329 1629 y Fn(2)2389 1596 y Fr(\021)23 b Fv(0)14
b Fu(:)515 1751 y Fv(Accordingly)-7 b(,)26 b(for)h Fu(A)d
Fr(2)f Fu(M)1373 1721 y Fp(+)1364 1771 y Ft(\015)1456
1751 y Fv(w)n(e)k(de\014ne)555 1941 y Fr(h)p Fu(A)p Fr(i)681
1906 y Fp(+)681 1966 y Ft(\015)t(;)12 b Fp(\003)824 1941
y Fv(=)23 b(max)1080 1874 y Fm(\000)1270 1941 y Fv(sup)1132
2017 y Ft(a)p Fs(6)p Fp(=0)p Ft(;b)1301 2025 y Fn(1)1334
2017 y Ft(;b)1383 2025 y Fn(2)1415 2017 y Fs(2)p Fg(Z)1499
2000 y Ff(d)1547 1941 y Fu(e)1586 1907 y Ft(\015)t Fs(j)p
Ft(b)1674 1915 y Fn(1)1705 1907 y Fs(\000)p Ft(b)1786
1915 y Fn(2)1819 1907 y Fs(j)1843 1941 y Fv(\()p Fu(pA)p
Fv(\))2011 1907 y Fp(+)2067 1941 y Fv(\()p Fu(a;)k(b)2229
1953 y Fp(1)2266 1941 y Fu(;)h(b)2353 1953 y Fp(2)2390
1941 y Fv(;)37 b(\003\))14 b Fu(;)1995 2148 y Fv(sup)1857
2223 y Ft(a)p Fs(6)p Fp(=0)p Ft(;b)2026 2231 y Fn(1)2058
2223 y Ft(;b)2107 2231 y Fn(2)2139 2223 y Fs(2)p Fg(Z)2223
2206 y Ff(d)2272 2148 y Fu(e)2311 2113 y Ft(\015)t Fs(j)p
Ft(b)2399 2121 y Fn(1)2430 2113 y Fs(\000)p Ft(b)2511
2121 y Fn(2)2543 2113 y Fs(j)2567 2148 y Fv(\()p Fu(q)s(A)p
Fv(\))2733 2113 y Fs(\000)2790 2148 y Fv(\()p Fu(a;)28
b(b)2953 2160 y Fp(1)2990 2148 y Fu(;)g(b)3077 2160 y
Fp(2)3114 2148 y Fv(;)37 b(\003\))3264 2080 y Fm(\001)3315
2148 y Fu(;)515 2381 y Fv(and)1421 2481 y Fr(kj)p Fu(A)p
Fr(kj)1613 2445 y Fp(+)1613 2505 y Ft(\015)t(;)10 b Fp(\003)1754
2481 y Fv(=)23 b Fr(kj)p Fu(A)p Fr(kj)2034 2447 y Fp(+)2034
2501 y Ft(\015)2107 2481 y Fv(+)18 b Fr(h)p Fu(A)p Fr(i)2316
2445 y Fp(+)2316 2505 y Ft(\015)t(;)11 b Fp(\003)2450
2481 y Fu(:)515 2624 y Fv(Finally)-7 b(,)27 b(w)n(e)h(set)846
2797 y Fu(M)936 2761 y Fp(+)p Ft(c)927 2821 y(\015)t(;)10
b Fp(\003)1069 2797 y Fv(=)23 b Fr(f)p Fu(A)g Fr(2)g
Fu(M)1452 2762 y Fp(+)p Ft(c)1443 2817 y(\015)1559 2797
y Fr(j)h(kj)p Fu(A)p Fr(kj)1798 2761 y Fp(+)1798 2821
y Ft(\015)t(;)9 b Fp(\003)1939 2797 y Fu(<)23 b Fr(1g)14
b Fu(;)96 b(M)2375 2761 y Fp(+)2366 2821 y Ft(\015)t(;)10
b Fp(\003)2508 2797 y Fv(=)22 b Fu(M)2685 2761 y Fp(+)p
Ft(c)2676 2821 y(\015)t(;)10 b Fp(\003)2813 2797 y Fr(\\)19
b Fu(M)2968 2809 y Ft(\015)3024 2797 y Fu(:)515 2985
y Fv(More)25 b(on)g(the)i(spaces)d Fu(M)1325 2949 y Fp(+)p
Ft(c)1316 3009 y(\015)t(;)11 b Fp(\003)1461 2985 y Fv(see)26
b(in)g([EK05)n(],)g(section)g(on)f(the)i(T\177)-42 b(oplitz{Lipsc)n
(hitz)25 b(natri-)515 3085 y(ces.)515 3315 y Fj(2.2)112
b(Multiplication)39 b(of)e(the)h(T\177)-56 b(oplitz{Lipsc)m(hitz)37
b(matrices)515 3468 y Fv(The)28 b(space)g Fu(M)999 3438
y Ft(c)990 3491 y(T)9 b(L)1115 3468 y Fv(is)28 b(linear.)38
b(In)29 b(this)f(section)g(w)n(e)g(sho)n(w)f(that)i Fu(M)2609
3438 y Ft(c)2600 3491 y(T)9 b(L)2726 3468 y Fv(also)27
b(is)h(an)g(algebra.)515 3568 y(The)f(corresp)r(onding)f(result)h
(follo)n(ws)g(from)515 3726 y Fl(Theorem)g(2.7.)36 b
Fo(L)l(et)26 b Fu(A)d Fr(2)h Fu(M)1496 3696 y Ft(c)1487
3749 y(\015)1526 3732 y FA(0)1547 3749 y Ft(;)p Fp(\003)1612
3732 y FA(0)1665 3726 y Fo(and)j Fu(B)g Fr(2)d Fu(M)2082
3696 y Ft(c)2073 3749 y(\015)2112 3732 y FA(0)2133 3749
y Ft(;)p Fp(\003)2228 3726 y Fo(for)k(some)e Fu(\015)2614
3696 y Fs(0)2660 3726 y Fu(>)d Fv(0)p Fo(,)k Fu(\015)g
Fr(2)d Fv([)p Fr(\000)p Fv(1)p Fu(;)14 b Fv(1])24 b Fo(and)515
3826 y Fv(\003)573 3795 y Fs(0)619 3826 y Fr(\025)e Fv(6)p
Fo(.)39 b(Then)1157 3998 y Fr(kj)p Fu(AB)t Fr(kj)1416
4010 y Ft(\015)t(;)10 b Fp(\003)1558 3998 y Fr(\024)23
b Fu(C)6 b Fr(kj)p Fu(A)p Fr(kj)1903 4010 y Ft(\015)1942
3994 y FA(0)1963 4010 y Ft(;)p Fp(\003)2028 3994 y FA(0)2069
3998 y Fr(kj)p Fu(B)t Fr(kj)2266 4010 y Ft(\015)t(;)p
Fp(\003)2370 3994 y FA(0)2395 3998 y Fu(\015)2443 3962
y Fs(\000)p Ft(d)p Fs(\000)p Fp(1)2438 4022 y(\001)2618
3998 y Fv(\003)2676 3964 y Fp(2)2713 3998 y Fu(;)431
b Fv(\(2.15\))1157 4179 y Fr(kj)p Fu(B)t(A)p Fr(kj)1416
4191 y Ft(\015)t(;)10 b Fp(\003)1558 4179 y Fr(\024)23
b Fu(C)6 b Fr(kj)p Fu(A)p Fr(kj)1903 4191 y Ft(\015)1942
4175 y FA(0)1963 4191 y Ft(;)p Fp(\003)2028 4175 y FA(0)2069
4179 y Fr(kj)p Fu(B)t Fr(kj)2266 4191 y Ft(\015)t(;)p
Fp(\003)2370 4175 y FA(0)2395 4179 y Fu(\015)2443 4144
y Fs(\000)p Ft(d)p Fs(\000)p Fp(1)2438 4204 y(\001)2618
4179 y Fv(\003)2676 4145 y Fp(2)2713 4179 y Fu(;)431
b Fv(\(2.16\))515 4323 y Fo(pr)l(ovide)l(d)32 b(that)1432
4422 y Fu(\015)1480 4388 y Fs(0)1526 4422 y Fr(\025)23
b(j)p Fu(\015)5 b Fr(j)18 b Fv(+)g(2)p Fu(\015)1894 4434
y Fp(\001)1952 4422 y Fu(;)57 b Fv(\003)23 b Fr(\025)g
Fv(\003)2259 4388 y Fs(0)2300 4422 y Fv(+)18 b(3)c Fu(;)705
b Fv(\(2.17\))515 4565 y Fo(wher)l(e)30 b Fu(\015)792
4577 y Fp(\001)874 4565 y Fu(>)23 b Fv(0)p Fo(.)38 b(Mor)l(e)l(over,)
953 4750 y Fv(\()p Fu(AB)t Fv(\))1146 4714 y Fs(1)p Fp(+)1146
4775 y Ft(a;)12 b(b)1243 4783 y Fn(1)1275 4775 y Ft(b)1304
4783 y Fn(2)1364 4750 y Fv(=)1452 4671 y Fm(X)1497 4845
y Ft(c)1572 4750 y Fv(\()p Fu(A)1666 4714 y Fs(1)p Fp(+)1666
4775 y Ft(a;)f(b)1762 4783 y Fn(1)1795 4775 y Ft(;)g(c)1860
4750 y Fu(B)1927 4714 y Fs(1)p Fp(+)1923 4775 y Ft(a;)g(c;)g(b)2080
4783 y Fn(2)2135 4750 y Fv(+)18 b Fu(A)2280 4714 y Fs(1\000)2280
4775 y Ft(a;)11 b(b)2376 4783 y Fn(1)2409 4775 y Ft(;)g(c)2474
4750 y Fu(B)2541 4714 y Fs(1\000)2537 4775 y(\000)p Ft(a;)f
Fs(\000)p Ft(c;)h Fs(\000)p Ft(b)2849 4783 y Fn(1)2886
4750 y Fv(\))p Fu(;)226 b Fv(\(2.18\))515 5006 y Fo(and)30
b(the)g(elements)f Fv(\()p Fu(AB)t Fv(\))1348 4971 y
Fs(1\000)1348 5031 y Ft(a;)13 b(b)1446 5039 y Fn(1)1478
5031 y Ft(;)e(b)1538 5039 y Fn(2)1605 5006 y Fo(ar)l(e)30
b(given)g(by)g(similar)h(formulas.)1905 5255 y Fv(14)p
eop end
%%Page: 15 15
TeXDict begin 15 14 bop 515 523 a Fl(Remark.)77 b Fv(Note)42
b(that)f(in)h(di\013erence)f(with)h(Theorem)e(1.5,)k(no)n(w)d(w)n(e)f
(estimate)i(the)515 623 y(pro)r(duct)31 b Fu(AB)k Fv(in)c(a)g(norm,)g
(whic)n(h)g(is)f(strictly)h(w)n(eak)n(er)e(than)i(b)r(oth)g(the)h
(norm,)f(used)g(for)515 722 y Fu(A)d Fv(and)f(the)h(norm,)f(used)h(for)
f Fu(B)t Fv(.)515 933 y Fo(Pr)l(o)l(of.)43 b Fv(W)-7
b(e)28 b(ma)n(y)f(assume)g(that)1462 1107 y Fr(kj)p Fu(A)p
Fr(kj)1654 1119 y Ft(\015)1693 1103 y FA(0)1714 1119
y Ft(;)p Fp(\003)1779 1103 y FA(0)1829 1107 y Fv(=)22
b Fr(kj)p Fu(B)t Fr(kj)2113 1119 y Ft(\015)t(;)p Fp(\003)2217
1103 y FA(0)2266 1107 y Fv(=)h(1)14 b Fu(;)734 b Fv(\(2.19\))639
1281 y(W)-7 b(e)28 b(ha)n(v)n(e)f(to)g(estimate)h(the)g(matrix)f
(elemen)n(ts)g(\()p Fu(AB)t Fv(\))2353 1293 y Ft(s)2384
1301 y Fn(1)2418 1293 y Fs(\006)p Ft(s)2501 1301 y Fn(2)2538
1281 y Fv(,)g(where)1354 1455 y Fu(s)1393 1467 y Ft(j)1451
1455 y Fv(=)22 b Fu(ta)d Fv(+)f Fu(b)1750 1467 y Ft(j)1807
1455 y Fr(2)24 b Fu(O)1951 1421 y Ft(a)1949 1475 y(t)1991
1455 y Fv(\(\003\))p Fu(;)98 b(j)28 b Fv(=)22 b(1)p Fu(;)27
b Fv(2)p Fu(:)627 b Fv(\(2.20\))515 1629 y(Since)27 b(the)g(cases)f
Fu(s)1120 1641 y Fp(1)1175 1629 y Fv(+)17 b Fu(s)1296
1641 y Fp(2)1360 1629 y Fv(and)27 b Fu(s)1560 1641 y
Fp(1)1614 1629 y Fr(\000)17 b Fu(s)1735 1641 y Fp(2)1800
1629 y Fv(are)26 b(similar,)g(w)n(e)h(restrict)f(ourselv)n(es)f(to)i
(the)h(sign)515 1729 y(plus.)639 1828 y(W)-7 b(e)28 b(ha)n(v)n(e)1254
1928 y(\()p Fu(AB)t Fv(\))1447 1940 y Ft(s)1478 1948
y Fn(1)1512 1940 y Ft(s)1543 1948 y Fn(2)1603 1928 y
Fv(=)1705 1849 y Fm(X)1690 2033 y Ft(s)p Fs(2)p Fg(Z)1805
2016 y Ff(d)1854 1928 y Fu(D)1923 1940 y Ft(s)1958 1928
y Fu(;)97 b(D)2147 1940 y Ft(s)2205 1928 y Fv(=)23 b
Fu(A)2355 1940 y Ft(s)2386 1948 y Fn(1)2419 1940 y Ft(s)2454
1928 y Fu(B)2517 1940 y Ft(ss)2579 1948 y Fn(2)2617 1928
y Fu(:)515 2164 y Fv(Let)k(us)h(denote)g(b)n(y)f(\006)1213
2134 y Fp(+)1213 2185 y Ft(a)1296 2164 y Fv(and)g(\006)1517
2134 y Fs(\000)1517 2185 y Ft(a)1601 2164 y Fv(the)h(half-spaces)1092
2338 y(\006)1152 2304 y Fp(+)1152 2359 y Ft(a)1230 2338
y Fv(=)23 b Fr(f)p Fu(s)f Fr(j)h(j)p Fu(s)c Fr(\000)f
Fu(ta)p Fr(j)23 b(\024)f(j)p Fu(s)d Fv(+)f Fu(ta)p Fr(jg)k
Fv(=)h Fr(f)p Fu(s)g Fr(j)g(h)p Fu(s;)14 b(a)p Fr(i)23
b(\025)g Fv(0)p Fr(g)p Fu(;)364 b Fv(\(2.21\))1092 2512
y(\006)1152 2478 y Fs(\000)1152 2533 y Ft(a)1231 2512
y Fv(=)22 b Fr(f)p Fu(s)h Fr(j)g(j)p Fu(s)18 b Fr(\000)g
Fu(ta)p Fr(j)23 b Fu(>)g Fr(j)p Fu(s)18 b Fv(+)g Fu(ta)p
Fr(jg)23 b Fv(=)g Fr(f)p Fu(s)f Fr(j)h(h)p Fu(s;)14 b(a)p
Fr(i)23 b Fu(<)g Fv(0)p Fr(g)p Fu(;)364 b Fv(\(2.22\))515
2657 y(By)28 b(\(2.10\))1106 2756 y Fu(O)1171 2722 y
Ft(a)1169 2777 y(t)1235 2756 y Fv(:=)23 b Fu(O)1411 2722
y Ft(a)1409 2777 y(t)1452 2756 y Fv(\(\003)1542 2722
y Fs(0)1565 2756 y Fv(\))g Fr(\032)g Fv(\006)1768 2722
y Fp(+)1768 2777 y Ft(a)1823 2756 y Fu(;)97 b(O)2008
2721 y Fs(\000)p Ft(a)2006 2777 y(t)2124 2756 y Fv(:=)22
b Fu(O)2299 2721 y Fs(\000)p Ft(a)2297 2777 y(t)2392
2756 y Fv(\(\003)2482 2722 y Fs(0)2505 2756 y Fv(\))i
Fr(\032)e Fv(\006)2708 2722 y Fs(\000)2708 2777 y Ft(a)2764
2756 y Fu(:)515 2901 y Fv(Then)1176 3000 y(\()p Fu(AB)t
Fv(\))1369 3012 y Ft(s)1400 3020 y Fn(1)1434 3012 y Ft(s)1465
3020 y Fn(2)1525 3000 y Fv(=)1635 2921 y Fm(X)1612 3099
y Ft(s)p Fs(2)p Ft(O)1740 3079 y Ff(a)1738 3116 y(t)1791
3000 y Fu(D)1860 3012 y Ft(s)1913 3000 y Fv(+)2041 2921
y Fm(X)1996 3114 y Ft(s)p Fs(2)p Ft(O)2124 3087 y FA(\000)p
Ff(a)2122 3131 y(t)2220 3000 y Fu(D)2289 3012 y Ft(s)2343
3000 y Fv(+)2441 2921 y Fm(X)2426 3099 y Ft(s)p Fs(2)p
Fp(\006)2549 3107 y Ff(t)2590 3000 y Fu(D)2659 3012 y
Ft(s)2695 3000 y Fu(;)449 b Fv(\(2.23\))515 3264 y(where)33
b(\006)821 3276 y Ft(t)883 3264 y Fv(=)g Fq(Z)1036 3234
y Ft(d)1097 3264 y Fr(n)22 b Fv(\()p Fu(O)1258 3234 y
Ft(a)1256 3284 y(t)1322 3264 y Fr([)g Fu(O)1464 3228
y Fs(\000)p Ft(a)1462 3284 y(t)1557 3264 y Fv(\))33 b(=)g(\(\006)1812
3234 y Fp(+)1812 3284 y Ft(a)1890 3264 y Fr(n)22 b Fu(O)2019
3234 y Ft(a)2017 3284 y(t)2059 3264 y Fv(\))h Fr([)g
Fv(\(\006)2284 3234 y Fs(\000)2284 3284 y Ft(a)2363 3264
y Fr(n)f Fu(O)2492 3228 y Fs(\000)p Ft(a)2490 3284 y(t)2584
3264 y Fv(\).)55 b(W)-7 b(riting)34 b Fu(s)f Fr(2)g Fu(O)3230
3234 y Ft(a)3228 3284 y(t)3305 3264 y Fv(as)515 3363
y Fu(s)23 b Fv(=)f Fu(at)d Fv(+)f Fu(c)27 b Fv(and)h(using)g(\(2.3\),)g
(\(2.19\))e(w)n(e)i(get)1059 3549 y Fu(D)1128 3561 y
Ft(s)1187 3549 y Fv(=)22 b Fu(A)1336 3514 y Fs(1)p Fp(+)1336
3574 y Ft(a;)12 b(b)1433 3582 y Fn(1)1465 3574 y Ft(;)f(c)1530
3549 y Fu(B)1597 3514 y Fs(1)p Fp(+)1593 3574 y Ft(a;)g(c;)g(b)1750
3582 y Fn(2)1805 3549 y Fv(+)18 b(3)p Fu(t)1960 3515
y Fs(\000)p Fp(1)2049 3549 y Fr(O)r Fv(\(1\))p Fu(e)2262
3515 y Fs(\000)p Ft(\015)2353 3490 y FA(0)2375 3515 y
Fs(j)p Ft(b)2424 3523 y Fn(1)2456 3515 y Fs(\000)p Ft(c)p
Fs(j\000)p Ft(\015)t Fs(j)p Ft(c)p Fs(\000)p Ft(b)2780
3523 y Fn(2)2810 3515 y Fs(j)515 3723 y Fv(\(note)28
b(that)f Fu(t)c(>)g Fv(1)k(due)h(to)g(\(2.5\))o(\).)37
b(Hence,)722 3835 y Fm(X)700 4013 y Ft(s)p Fs(2)p Ft(O)828
3993 y Ff(a)826 4030 y(t)879 3914 y Fu(D)948 3926 y Ft(s)1006
3914 y Fv(=)1230 3835 y Fm(X)1094 4017 y Fs(f)p Ft(c)p
Fs(j)p Ft(at)p Fp(+)p Ft(c)p Fs(2)p Ft(O)1417 3997 y
Ff(a)1415 4034 y(t)1452 4017 y Fs(g)1500 3914 y Fu(A)1562
3878 y Fs(1)p Fp(+)1562 3939 y Ft(a;)11 b(b)1658 3947
y Fn(1)1691 3939 y Ft(;)g(c)1756 3914 y Fu(B)1823 3878
y Fs(1)p Fp(+)1819 3939 y Ft(a;)g(c;)f(b)1975 3947 y
Fn(2)2031 3914 y Fv(+)18 b Fu(C)6 b(\015)2227 3878 y
Fs(\000)p Ft(d)2222 3938 y Fp(\001)2317 3914 y Fu(t)2347
3880 y Fs(\000)p Fp(1)2436 3914 y Fr(O)r Fv(\(1\))p Fu(e)2649
3880 y Fs(\000)p Ft(\015)t Fs(j)p Ft(b)2789 3888 y Fn(1)2821
3880 y Fs(\000)p Ft(b)2902 3888 y Fn(2)2935 3880 y Fs(j)2958
3914 y Fu(:)186 b Fv(\(2.24\))515 4195 y(Let)27 b Fu(c)c
Fr(2)h Fq(Z)856 4165 y Ft(d)923 4195 y Fv(b)r(e)k(suc)n(h)f(that)h
Fu(at)18 b Fv(+)g Fu(c)23 b Fr(62)g Fu(O)1780 4165 y
Ft(a)1778 4216 y(t)1821 4195 y Fv(.)37 b(Then,)28 b(b)n(y)f(Lemma)h
(2.1,)1493 4369 y Fr(j)p Fu(c)18 b Fr(\000)g Fu(b)1689
4381 y Ft(j)1724 4369 y Fr(j)23 b(\025)g Fu(t)p Fv(\003)1946
4335 y Fs(\000)p Fp(2)2034 4369 y Fu(;)37 b(j)28 b Fv(=)23
b(1)p Fu(;)k Fv(2)p Fu(:)766 b Fv(\(2.25\))515 4543 y(Therefore,)26
b(using)j(\(2.4\))e(and)g(Lemma)h(1.4)e(w)n(e)i(get)669
4647 y Fm(\014)669 4696 y(\014)669 4746 y(\014)669 4796
y(\014)847 4688 y(X)711 4870 y Fs(f)p Ft(c)p Fs(j)p Ft(at)p
Fp(+)p Ft(c)p Fs(62)p Ft(O)1034 4850 y Ff(a)1032 4887
y(t)1069 4870 y Fs(g)1117 4767 y Fu(A)1179 4731 y Fs(1)p
Fp(+)1179 4792 y Ft(a;)12 b(b)1276 4800 y Fn(1)1308 4792
y Ft(;)f(c)1373 4767 y Fu(B)1440 4731 y Fs(1)p Fp(+)1436
4792 y Ft(a;)g(c;)g(b)1593 4800 y Fn(2)1630 4647 y Fm(\014)1630
4696 y(\014)1630 4746 y(\014)1630 4796 y(\014)1680 4767
y Fr(\024)23 b Fu(e)1807 4733 y Fs(\000)p Ft(\015)1894
4741 y Fn(\001)1943 4733 y Ft(t)p Fp(\003)2013 4708 y
FA(\000)p Fn(2)2109 4688 y Fm(X)2154 4862 y Ft(c)2243
4767 y Fu(e)2282 4733 y Fs(\000)p Fp(\()p Fs(j)p Ft(\015)t
Fs(j)p Fp(+)p Ft(\015)2525 4741 y Fn(\001)2573 4733 y
Fp(\))p Fs(j)p Ft(b)2648 4741 y Fn(1)2680 4733 y Fs(\000)p
Ft(c)p Fs(j\000)p Ft(\015)t Fs(j)p Ft(c)p Fs(\000)p Ft(b)3004
4741 y Fn(2)3034 4733 y Fs(j)2032 5006 y Fr(\024)f Fu(C)6
b(t)2214 4972 y Fs(\000)p Fp(1)2304 5006 y Fu(\015)2352
4971 y Fs(\000)p Fp(1)2347 5031 y(\001)2441 5006 y Fv(\003)2499
4972 y Fp(2)2535 5006 y Fu(\015)2583 4971 y Fs(\000)p
Ft(d)2578 5031 y Fp(\001)2674 5006 y Fu(e)2713 4972 y
Fs(\000)p Ft(\015)t Fs(j)p Ft(b)2853 4980 y Fn(1)2884
4972 y Fs(\000)p Ft(b)2965 4980 y Fn(2)2997 4972 y Fs(j)3035
5006 y Fu(:)109 b Fv(\(2.26\))1905 5255 y(15)p eop end
%%Page: 16 16
TeXDict begin 16 15 bop 515 523 a Fv(So)40 b(w)n(e)g(can)g(replace)f
(in)h(\(2.24\))g(the)h(summation)f(o)n(v)n(er)e Fr(f)p
Fu(c)44 b Fr(j)g Fu(at)27 b Fv(+)f Fu(c)44 b Fr(2)h Fu(O)3013
493 y Ft(a)3011 544 y(t)3054 523 y Fr(g)40 b Fv(b)n(y)g(the)515
623 y(summation)27 b(o)n(v)n(er)f Fq(Z)1177 593 y Ft(d)1216
623 y Fv(:)739 726 y Fm(X)717 905 y Ft(s)p Fs(2)p Ft(O)845
885 y Ff(a)843 921 y(t)895 805 y Fu(D)964 817 y Ft(s)1023
805 y Fv(=)1124 726 y Fm(X)1110 910 y Ft(c)p Fs(2)p Fg(Z)1224
894 y Ff(d)1272 805 y Fu(A)1334 770 y Fs(1)p Fp(+)1334
830 y Ft(a;)11 b(b)1430 838 y Fn(1)1463 830 y Ft(;)g(c)1528
805 y Fu(B)1595 770 y Fs(1)p Fp(+)1591 830 y Ft(a;)g(c;)g(b)1748
838 y Fn(2)1803 805 y Fv(+)18 b Fu(C)1945 817 y Fp(1)1982
805 y Fu(t)2012 771 y Fs(\000)p Fp(1)2102 805 y Fr(O)r
Fv(\(1\))p Fu(\015)2324 770 y Fs(\000)p Ft(d)p Fs(\000)p
Fp(1)2319 830 y(\001)2499 805 y Fv(\003)2557 771 y Fp(2)2594
805 y Fu(e)2633 771 y Fs(\000)p Ft(\015)t Fs(j)p Ft(b)2773
779 y Fn(1)2804 771 y Fs(\000)p Ft(b)2885 779 y Fn(2)2918
771 y Fs(j)2942 805 y Fu(:)202 b Fv(\(2.27\))639 1079
y(Similar,)661 1194 y Fm(X)616 1387 y Ft(s)p Fs(2)p Ft(O)744
1360 y FA(\000)p Ff(a)742 1404 y(t)840 1273 y Fu(D)909
1285 y Ft(s)967 1273 y Fv(=)1069 1194 y Fm(X)1055 1378
y Ft(c)p Fs(2)p Fg(Z)1169 1362 y Ff(d)1217 1273 y Fu(A)1279
1238 y Fs(1\000)1279 1298 y Ft(a;)11 b(b)1375 1306 y
Fn(1)1408 1298 y Ft(;)g(c)1472 1273 y Fu(B)1539 1238
y Fs(1\000)1535 1298 y(\000)p Ft(a;)g Fs(\000)p Ft(c;)g
Fs(\000)p Ft(b)1848 1306 y Fn(2)1903 1273 y Fv(+)18 b
Fu(C)2045 1285 y Fp(1)2083 1273 y Fu(t)2113 1239 y Fs(\000)p
Fp(1)2202 1273 y Fr(O)r Fv(\(1\))p Fu(\015)2424 1238
y Fs(\000)p Ft(d)p Fs(\000)p Fp(1)2419 1298 y(\001)2599
1273 y Fv(\003)2657 1239 y Fp(2)2694 1273 y Fu(e)2733
1239 y Fs(\000)p Ft(\015)t Fs(j)p Ft(b)2873 1247 y Fn(1)2905
1239 y Fs(\000)p Ft(b)2986 1247 y Fn(2)3018 1239 y Fs(j)3042
1273 y Fu(:)102 b Fv(\(2.28\))639 1566 y(It)32 b(remains)e(to)i
(estimate)f(the)h(third)f(sum)h(in)f(\(2.23\))o(.)49
b(Let)31 b(us)g(consider)g(a)f(term)i Fu(D)3344 1578
y Ft(s)515 1666 y Fv(with)e Fu(s)c Fr(2)h Fv(\006)913
1678 y Ft(t)942 1666 y Fv(.)42 b(W)-7 b(e)30 b(assume)f(\014rst)g(that)
h Fu(s)c Fr(2)h Fv(\006)2003 1636 y Fp(+)2003 1686 y
Ft(a)2078 1666 y Fr(n)19 b Fu(O)2204 1636 y Ft(a)2202
1686 y(t)2274 1666 y Fv(and)30 b(write)f(it)h(as)f Fu(s)d
Fv(=)g Fu(at)19 b Fv(+)h Fu(c)p Fv(.)42 b(By)515 1765
y(Lemma)27 b(2.2,)g([)p Fu(s)18 b Fr(\000)g Fu(s)1170
1777 y Fp(1)1208 1765 y Fv(])23 b Fr(\025)f Fu(t)p Fv(\003)1429
1735 y Fp(2)1466 1765 y Fv(.)37 b(This)28 b(estimate)f(and)h(\(2.19\))o
(,)g(\(2.17\))e(imply)i(that)869 1964 y Fr(j)p Fu(D)961
1976 y Ft(s)996 1964 y Fr(j)23 b(\024)g Fu(e)1169 1930
y Fs(\000)p Ft(\015)1260 1905 y FA(0)1281 1930 y Fp([)p
Ft(s)1331 1938 y Fn(1)1364 1930 y Fs(\000)p Ft(s)p Fp(])p
Fs(\000)p Ft(\015)t Fp([)p Ft(s)p Fs(\000)p Ft(s)1690
1938 y Fn(2)1722 1930 y Fp(])1768 1964 y Fr(\024)f Fu(e)1894
1930 y Fs(\000)p Ft(\015)1981 1938 y Fn(\001)2031 1930
y Ft(t)p Fp(\003)2101 1905 y FA(\000)p Fn(2)2183 1964
y Fu(e)2222 1930 y Fs(\000)p Fp(\()p Fs(j)p Ft(\015)t
Fs(j)p Fp(+)p Ft(\015)2465 1938 y Fn(\001)2512 1930 y
Fp(\)[)p Ft(s)2588 1938 y Fn(1)2621 1930 y Fs(\000)p
Ft(s)p Fp(])p Fs(\000)p Ft(\015)t Fp([)p Ft(s)p Fs(\000)p
Ft(s)2947 1938 y Fn(2)2979 1930 y Fp(])3002 1964 y Fu(:)515
2147 y Fv(Estimating)30 b(similar)f(the)i(terms)g Fu(D)1669
2159 y Ft(s)1735 2147 y Fv(with)g Fu(s)d Fr(2)g Fv(\006)2137
2117 y Fs(\000)2137 2168 y Ft(a)2213 2147 y Fr(n)20 b
Fu(O)2340 2112 y Fs(\000)p Ft(a)2338 2167 y(t)2464 2147
y Fv(and)30 b(using)g(Lemma)h(1.6)e(w)n(e)515 2247 y(get)e(that)1060
2371 y Fm(\014)1060 2421 y(\014)1117 2362 y(X)1102 2541
y Ft(s)p Fs(2)p Fp(\006)1225 2549 y Ff(t)1267 2441 y
Fu(D)1336 2453 y Ft(s)1371 2371 y Fm(\014)1371 2421 y(\014)1422
2441 y Fr(\024)22 b Fu(e)1548 2407 y Fs(\000)p Ft(\015)1635
2415 y Fn(\001)1684 2407 y Ft(t)p Fp(\003)1754 2382 y
FA(\000)p Fn(2)1865 2362 y Fm(X)1850 2546 y Ft(s)p Fs(2)p
Fg(Z)1965 2530 y Ff(d)2014 2441 y Fu(e)2053 2407 y Fs(\000)p
Fp(\()p Fs(j)p Ft(\015)t Fs(j)p Fp(+)p Ft(\015)2296 2415
y Fn(\001)2344 2407 y Fp(\)[)p Ft(s)2420 2415 y Fn(1)2452
2407 y Fs(\000)p Ft(s)p Fp(])p Fs(\000)p Ft(\015)t Fp([)p
Ft(s)p Fs(\000)p Ft(s)2778 2415 y Fn(2)2810 2407 y Fp(])3167
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%%Page: 17 17
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%%Page: 18 18
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1436 y Fs(\000)p Ft(d)p Fs(\000)p Fp(1)1848 1496 y(\001)2028
1472 y Fv(\003)2086 1437 y Fp(2)2123 1401 y Fm(\014)2123
1451 y(\014)2164 1472 y Fr(kj)p Fu(A)p Fr(kj)2356 1484
y Ft(\015)2395 1467 y FA(0)2417 1484 y Ft(;)p Fp(\003)2482
1467 y FA(0)2509 1401 y Fm(\014)2509 1451 y(\014)2555
1472 y Fr(\002)2638 1401 y Fm(\014)2638 1451 y(\014)2679
1472 y Fr(kj)p Fu(B)t Fr(kj)2876 1484 y Ft(\015)t(;)p
Fp(\003)2980 1467 y FA(0)3006 1401 y Fm(\014)3006 1451
y(\014)3033 1472 y Fu(:)111 b Fv(\(2.32\))515 1654 y
Fo(Pr)l(o)l(of.)43 b Fv(The)28 b(assertion)e(follo)n(ws)g(from)h
(Theorem)g(2.7)g(and)g(\(2.18\),)g(\(2.30\))o(.)p 3318
1654 V 3322 1602 50 4 v 3322 1654 V 3372 1654 4 57 v
639 1870 a(Finally)-7 b(,)34 b(w)n(e)e(note)g(that)h(w)n(e)f(can)g
(de\014ne)h(in)g(a)f(similar)f(w)n(a)n(y)h(the)g(spaces)g(of)g
Fu(n)p Fv(-times)515 1970 y(T\177)-42 b(oplitz{Lipsc)n(hitz)35
b(matrices.)61 b(As)35 b(for)h(the)g(cases)f Fu(n)h Fv(=)h(1)e(and)h
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b(arian)n(t)29 b(with)h(resp)r(ect)f(to)h(m)n(ultiplication)g(of)f
(matrices)g(as)g(w)n(ell)g(as)g(with)h(resp)r(ect)515
2169 y(to)f(taking)f(the)i(\015o)n(w-maps)e(of)h(the)g(nonautonomous)f
(linear)g(systems)h(with)g(co)r(e\016cien)n(ts)515 2269
y(from)e(these)h(spaces.)515 2543 y Fw(3)134 b(Di\013eren)l(tial)48
b(equations)e(with)716 2693 y(the)g(T\177)-67 b(oplitz{Lipsc)l(hitz)45
b(co)t(e\016cien)l(ts)515 2891 y Fj(3.1)112 b(Equations)38
b(with)f(co)s(e\016cien)m(ts)h(in)g Fi(M)2354 2855 y
Fb(c)2344 2916 y(\015)2390 2891 y Fj(.)515 3053 y Fv(F)-7
b(or)40 b(some)g(0)k Fu(<)h(\015)1142 3023 y Fs(0)1210
3053 y Fr(\024)f Fv(1,)g(let)d Fu(A)p Fv(\()p Fu(t)p
Fv(\))46 b Fr(2)f Fu(M)1953 3023 y Ft(c)1944 3076 y(\015)1983
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(linear)g(op)r(erator)f(whic)n(h)515 3153 y(con)n(tin)n(uously)26
b(dep)r(ends)i(on)g Fu(t)p Fv(,)f(and)1593 3335 y Fr(kj)p
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3331 y FA(0)1967 3335 y Fr(\024)c Fu(\026)83 b Fr(8)14
b Fu(t;)907 b Fv(\(3.1\))515 3518 y(where)27 b(0)22 b
Fu(<)h(\026)g Fr(\024)g Fv(1.)36 b(W)-7 b(e)28 b(consider)f(the)h
(linear)f(equation)1678 3700 y(_)-38 b Fu(y)s Fv(\()p
Fu(t)p Fv(\))24 b(=)e Fu(A)p Fv(\()p Fu(t)p Fv(\))p Fu(y)s
Fv(\()p Fu(t)p Fv(\))p Fu(;)979 b Fv(\(3.2\))515 3883
y(and)27 b(denote)h(b)n(y)f Fu(S)1115 3846 y Ft(t)1140
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y Fv(the)h(corresp)r(onding)e(\015o)n(w-maps:)1646 4076
y Fu(S)1702 4039 y Ft(t)1727 4047 y Fn(2)1697 4096 y
Ft(t)1722 4104 y Fn(1)1763 4076 y Fu(y)s Fv(\()p Fu(t)1869
4088 y Fp(1)1906 4076 y Fv(\))e(=)e Fu(y)s Fv(\()p Fu(t)2155
4088 y Fp(2)2193 4076 y Fv(\))p Fu(;)515 4258 y Fv(where)27
b Fu(y)s Fv(\()p Fu(t)p Fv(\))h(is)f(a)g(solution)g(of)34
b(\(3.2\).)j(If)28 b Fu(t)1832 4270 y Fp(1)1892 4258
y Fv(=)23 b(0,)k(w)n(e)h(abbreviate)e Fu(S)2659 4228
y Ft(t)2654 4279 y Fp(0)2714 4258 y Fv(=)d Fu(S)2858
4228 y Ft(t)2887 4258 y Fv(.)639 4358 y(By)34 b(Theorem)g(1.2,)h
Fu(A)p Fv(\()p Fu(t)p Fv(\))h(is)e(a)g(b)r(ounded)h(linear)f(op)r
(erator)f(in)h(an)n(y)g(space)g Fu(Y)3145 4328 y Ft(c)3127
4379 y(\015)3179 4358 y Fv(,)i(0)e Fu(<)515 4458 y(\015)27
b(<)c(\015)721 4428 y Fs(0)744 4458 y Fv(,)i(con)n(tin)n(uous)f(in)h
Fu(t)p Fv(.)36 b(So)25 b Fu(S)1557 4428 y Ft(t)1552 4478
y(t)1577 4486 y Fn(1)1638 4458 y Fv(is)g(a)g(b)r(ounded)g(linear)f(op)r
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4478 y(\015)3305 4458 y Fv(as)515 4574 y(ab)r(o)n(v)n(e,)k
Fu(C)840 4544 y Fp(1)877 4574 y Fv(-smo)r(oth)h(in)g
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4574 y Fu(S)t Fv(\()p Fu(t)p Fv(\))d(=)f Fu(S)2240 4544
y Ft(t)2235 4595 y(t)2260 4603 y Fn(1)2296 4574 y Fr(\000)14
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(equation)1282 4839 y Fu(d)p 1267 4876 74 4 v 1267 4952
a(dt)1364 4874 y Fv(~)1350 4895 y Fu(S)5 b Fv(\()p Fu(t)p
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y(~)1799 4895 y Fu(S)6 b Fv(\()p Fu(t)p Fv(\))19 b(+)f(id\))p
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4907 y Fp(1)2429 4895 y Fv(\))23 b(=)g(0)p Fu(:)571 b
Fv(\(3.3\))1905 5255 y(18)p eop end
%%Page: 19 19
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(hitz)h(as)f(a)g(comp)r(osition)g(of)g(t)n(w)n(o)g(Lipsc)n(hitz)515
922 y(maps.)51 b(So)32 b(it)h(is)f(di\013eren)n(tiable)g(at)g(almost)g
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Fu(t)p Fv(,)h(b)n(y)e(the)515 1021 y(triangle)26 b(inequalit)n(y)i(w)n
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1192 y Fu(:)515 1362 y Fv(Dividing)28 b(this)f(relation)g(b)n(y)g
Fu(")h Fv(and)f(sending)h Fu(")f Fv(to)g(zero)g(w)n(e)g(get)g(that)1345
1501 y Fm(\014)1345 1551 y(\014)1398 1516 y Fu(d)p 1383
1553 74 4 v 1383 1629 a(dt)1466 1572 y Fr(kj)1544 1551
y Fv(~)1531 1572 y Fu(S)t Fv(\()p Fu(t)p Fv(\))p Fr(kj)1745
1584 y Ft(\015)1788 1501 y Fm(\014)1788 1551 y(\014)1838
1572 y Fr(\024)c(kj)2016 1516 y Fu(d)p 2001 1553 V 2001
1629 a(dt)2098 1551 y Fv(~)2084 1572 y Fu(S)t Fv(\()p
Fu(t)p Fv(\))p Fr(kj)2298 1584 y Ft(\015)2424 1572 y
Fv(a.e.)515 1770 y(Due)28 b(to)f(\(3.1\),)h(\(3.3\))o(,)g(the)g(last)f
(inequalit)n(y)h(and)f(Theorem)g(1.5)f(w)n(e)i(ha)n(v)n(e)587
1914 y Fm(\014)587 1963 y(\014)640 1928 y Fu(d)p 625
1965 V 625 2041 a(dt)708 1984 y Fr(kj)786 1963 y Fv(~)773
1984 y Fu(S)t Fv(\()p Fu(t)p Fv(\))p Fr(kj)987 1996 y
Ft(\015)1030 1914 y Fm(\014)1030 1963 y(\014)1081 1984
y Fr(\024)22 b(kj)p Fu(A)p Fv(\()p Fu(t)p Fv(\))1389
1917 y Fm(\000)1442 1963 y Fv(~)1428 1984 y Fu(S)t Fv(\()p
Fu(t)p Fv(\))d(+)f(id)1749 1917 y Fm(\001)1787 1984 y
Fr(kj)1852 1996 y Ft(\015)1917 1984 y Fr(\024)k Fu(\026)d
Fv(+)f Fu(C)6 b(\026\015)2319 1949 y Fs(\000)p Ft(d)2314
2009 y Fp(\001)2409 1984 y Fr(kj)2488 1963 y Fv(~)2474
1984 y Fu(S)f Fv(\()p Fu(t)p Fv(\))p Fr(kj)2689 1996
y Ft(\015)2731 1984 y Fu(;)97 b(\015)2894 1996 y Fp(\001)2976
1984 y Fv(=)23 b Fu(\015)3112 1950 y Fs(0)3153 1984 y
Fr(\000)18 b Fu(\015)5 b(:)515 2197 y Fv(Since)746 2176
y(~)732 2197 y Fu(S)t Fv(\()p Fu(t)849 2209 y Fp(1)887
2197 y Fv(\))23 b(=)g(0,)k(then)h(the)g(Gran)n(w)n(all)e(inequalit)n(y)
h(implies)h(the)g(estimate)1105 2392 y Fr(kj)1184 2371
y Fv(~)1170 2392 y Fu(S)5 b Fv(\()p Fu(t)p Fv(\))p Fr(kj)1385
2404 y Ft(\015)1450 2392 y Fr(\024)23 b Fu(C)1603 2358
y Fs(\000)p Fp(1)1693 2392 y Fu(\015)1741 2358 y Ft(d)1736
2413 y Fp(\001)1794 2300 y Fm(\020)1858 2392 y Fv(exp)1998
2325 y Fm(\000)2036 2392 y Fu(C)6 b(\026\015)2199 2357
y Fs(\000)p Ft(d)2194 2417 y Fp(\001)2290 2392 y Fr(j)p
Fu(t)18 b Fr(\000)g Fu(t)2474 2404 y Fp(1)2512 2392 y
Fr(j)2535 2325 y Fm(\001)2591 2392 y Fr(\000)g Fv(1)2716
2300 y Fm(\021)2765 2392 y Fu(:)515 2602 y Fv(Assuming)27
b(that)h Fu(C)6 b(\026\015)1239 2567 y Fs(\000)p Ft(d)1234
2626 y Fp(\001)1353 2602 y Fr(\024)22 b Fv(1)27 b(and)h
Fr(j)p Fu(t)18 b Fr(\000)g Fu(t)1855 2614 y Fp(1)1893
2602 y Fr(j)23 b(\024)f Fv(1,)28 b(w)n(e)f(ha)n(v)n(e)f
Fr(kj)2511 2581 y Fv(~)2497 2602 y Fu(S)5 b Fv(\()p Fu(t)p
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b Fu(C)2924 2614 y Fp(1)2962 2602 y Fu(\026)p Fr(j)p
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Fr(j)p Fv(.)639 2702 y(W)-7 b(e)28 b(ha)n(v)n(e)f(got)g(the)h(follo)n
(wing)e(result:)515 2858 y Fl(Theorem)e(3.1.)35 b Fo(L)l(et)24
b(a)h(line)l(ar)g(op)l(er)l(ator)g Fu(A)p Fv(\()p Fu(t)p
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2864 y FA(0)2281 2858 y Fo(b)l(e)i(c)l(ontinuous)e(in)i
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2928 y Fs(0)1340 2958 y Fv(\))30 b Fo(satis\014es)1614
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Ft(d)2128 3128 y Fr(\024)22 b Fv(1)p Fu(;)928 b Fv(\(3.4\))515
3299 y Fo(wher)l(e)41 b Fu(C)48 b Fo(is)41 b(a)h(su\016ciently)f(lar)l
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3311 y Fp(1)2701 3299 y Fu(;)14 b(t)2768 3311 y Fp(2)2848
3299 y Fr(\025)43 b Fv(0)e Fo(such)g(that)515 3399 y
Fr(j)p Fu(t)568 3411 y Fp(2)623 3399 y Fr(\000)18 b Fu(t)736
3411 y Fp(1)774 3399 y Fr(j)23 b(\024)f Fv(1)30 b Fo(we)g(have)1445
3498 y Fr(kj)p Fu(S)1566 3461 y Ft(t)1591 3469 y Fn(2)1561
3519 y Ft(t)1586 3527 y Fn(1)1645 3498 y Fr(\000)18 b
Fv(id)c Fr(kj)1876 3510 y Ft(\015)1941 3498 y Fr(\024)23
b Fu(C)6 b(\026)p Fr(j)p Fu(t)2197 3510 y Fp(2)2253 3498
y Fr(\000)18 b Fu(t)2366 3510 y Fp(1)2403 3498 y Fr(j)p
Fu(:)639 3700 y Fv(Rep)r(eating)32 b(the)h(argumen)n(ts)e(ab)r(o)n(v)n
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3799 y(get)27 b(the)h(follo)n(wing)f(result)g(whic)n(h)g(w)n(e)h(will)g
(need)f(later:)515 3956 y Fl(Theorem)e(3.2.)35 b Fo(L)l(et)24
b(us)g(assume)h(that)f Fr(j)p Fu(A)1875 3968 y Ft(ss)1937
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4055 y Fu(\015)563 4025 y Fs(0)609 4055 y Fu(>)c(\015)28
b(>)23 b Fv(0)29 b Fo(and)39 b Fv(\(3.4\))29 b Fo(holds.)40
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y Fr(\000)18 b Fu(t)2113 4067 y Fp(1)2150 4055 y Fr(j)23
b(\024)g Fv(1)29 b Fo(we)h(have)1103 4238 y Fr(j)p Fv(\()p
Fu(S)1214 4201 y Ft(t)1239 4209 y Fn(2)1209 4258 y Ft(t)1234
4266 y Fn(1)1275 4238 y Fv(\))1307 4250 y Ft(ss)1369
4234 y FA(0)1415 4238 y Fr(\000)18 b Fu(\016)1535 4250
y Ft(s;s)1617 4234 y FA(0)1644 4238 y Fr(j)23 b(\024)g
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y Fr(\000)18 b Fu(t)2115 4250 y Fp(2)2152 4238 y Fr(j)p
Fu(e)2214 4204 y Fs(\000)p Ft(\015)t Fs(j)p Ft(s)p Fs(\000)p
Ft(s)2439 4179 y FA(0)2461 4204 y Fs(j)2570 4238 y Fr(8)c
Fu(s;)g(s)2745 4204 y Fs(0)2768 4238 y Fu(:)639 4408
y Fv(Let)23 b Fu(X)30 b Fv(b)r(e)23 b(the)h(algebra)d(of)i
Fu(m)9 b Fr(\002)g Fu(m)24 b Fv(real)e(matrices)g(\()p
Fu(m)h Fr(\025)g Fv(2\),)h(and)f Fu(X)2813 4378 y Ft(c)2869
4408 y Fv(b)r(e)g(the)h(algebra)515 4508 y(of)29 b(complex)f(matrices.)
41 b(Let)29 b Fu(A)p Fv(\()p Fu(t)p Fv(\))d Fr(2)g Fu(M)1810
4478 y Ft(X)5 b(c)1801 4529 y(\015)1932 4508 y Fv(b)r(e)29
b(a)g(linear)f(op)r(erator)f(whic)n(h)i(con)n(tin)n(uously)515
4608 y(dep)r(ends)h(on)g Fu(t)p Fv(.)45 b(Then)31 b(it)f(de\014nes)h
(the)f(linear)g(di\013eren)n(tial)g(equation)f(\(3.2\),)i(where)f
Fu(y)s Fv(\()p Fu(t)p Fv(\))515 4707 y(is)e(an)h(elemen)n(t)g(of)g(the)
g(space)f(of)h(sequences)f(\()p Fu(y)2034 4719 y Ft(s)2095
4707 y Fr(2)d Fq(C)2235 4677 y Ft(m)2298 4707 y Fv(,)30
b Fu(s)25 b Fr(2)g Fq(Z)2550 4677 y Ft(d)2589 4707 y
Fv(\).)41 b(Again,)29 b(it)h(is)e(easy)g(to)515 4807
y(see)23 b(that)i(the)f(argumen)n(ts)f(ab)r(o)n(v)n(e)f(apply)i(to)g
(pro)n(v)n(e)e(that)i Fo(the)j(assertions)g(of)g(The)l(or)l(em)h(3.1)
515 4907 y(r)l(emain)f(true)f(if)i Fu(A)p Fv(\()p Fu(t)p
Fv(\))23 b Fr(2)h Fu(M)1391 4876 y Ft(X)5 b(c)1382 4929
y(\015)1421 4913 y FA(0)1483 4907 y Fo(,)28 b(and)f Fu(S)1750
4870 y Ft(t)1775 4878 y Fn(2)1745 4927 y Ft(t)1770 4935
y Fn(1)1838 4907 y Fo(is)g(an)g(op)l(er)l(ator)h(in)f
Fu(M)2553 4876 y Ft(X)5 b(c)2544 4927 y(\015)2645 4907
y Fo(.)36 b Fv(If)25 b Fu(A)p Fv(\()p Fu(t)p Fv(\))g(is)g(a)f(matrix,)
515 5006 y(formed)j(b)n(y)g(2)18 b Fr(\002)g Fv(2-blo)r(c)n(ks)26
b(\(see)i(section)f(1.4\),)g(w)n(e)g(can)h(sp)r(ecify)f(the)h(result:)
1905 5255 y(19)p eop end
%%Page: 20 20
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b Fv(\))2915 607 y Fs(\000)p Ft(d)3028 637 y Fr(\024)22
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y(\015)1056 1594 y FA(0)1078 1611 y Ft(;)11 b Fp(\003)1154
1594 y FA(0)1181 1588 y Fv(,)32 b(where)e Fu(\015)1527
1558 y Fs(0)1579 1588 y Fr(2)f Fv(\(0)p Fu(;)e Fv(1],)32
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1783 y FA(0)1897 1799 y Ft(;)11 b Fp(\003)1973 1783 y
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%%Page: 21 21
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4437 y(\014)694 4487 y(\014)722 4482 y Fu(A)784 4448
y Fs(1)855 4482 y Fv(\()p Fu(a)p Fv(\))963 4494 y Ft(b)992
4502 y Fn(1)1025 4494 y Ft(b)1054 4502 y Fn(2)1091 4387
y Fm(\014)1091 4437 y(\014)1091 4487 y(\014)1118 4482
y Fu(;)1183 4387 y Fm(\014)1183 4437 y(\014)1183 4487
y(\014)1210 4482 y Fu(A)1272 4448 y Fp(\001)1331 4482
y Fv(\()p Fu(\034)5 b(;)14 b(a)p Fv(\))1517 4494 y Ft(b)1546
4502 y Fn(1)1579 4494 y Ft(b)1608 4502 y Fn(2)1645 4387
y Fm(\014)1645 4437 y(\014)1645 4487 y(\014)1696 4482
y Fr(\024)23 b(h)p Fu(A)p Fr(i)1910 4494 y Ft(\015)1949
4478 y FA(0)1971 4494 y Ft(;)p Fp(\003)2036 4478 y FA(0)2063
4482 y Fu(e)2102 4448 y Fs(\000)p Ft(\015)2193 4423 y
FA(0)2215 4448 y Fs(j)p Ft(b)2264 4456 y Fn(1)2296 4448
y Fs(\000)p Ft(b)2377 4456 y Fn(2)2409 4448 y Fs(j)2456
4482 y Fr(\024)g Fu(\026e)2633 4448 y Fs(\000)p Ft(\015)2724
4423 y FA(0)2745 4448 y Fs(j)p Ft(b)2794 4456 y Fn(1)2827
4448 y Fs(\000)p Ft(b)2908 4456 y Fn(2)2940 4448 y Fs(j)2964
4482 y Fu(;)180 b Fv(\(3.10\))515 4690 y(if)28 b Fu(s)630
4702 y Fp(1)667 4690 y Fu(;)14 b(s)743 4702 y Fp(2)803
4690 y Fr(2)24 b Fu(O)947 4660 y Ft(a)945 4711 y(\034)987
4690 y Fv(\(\003)1077 4660 y Fs(0)1101 4690 y Fv(\))k(\(cf.)37
b(\(2.3\)\).)639 4790 y(Let)28 b(us)g(set)1483 4889 y
Fl(z)1525 4901 y Ft(s)1556 4909 y Fn(1)1616 4889 y Fv(=)1850
4811 y Fm(X)1704 4993 y Ft(s)p Fs(2)p Fp(\006)1827 4977
y Fn(+)1874 4993 y Fs(n)p Ft(O)1960 4977 y Ff(a)1958
5010 y(\034)1997 4993 y Fp(\(\003)2068 4977 y FA(0)2091
4993 y Fp(\))2130 4889 y Fl(A)2202 4901 y Ft(s)2233 4909
y Fn(1)2266 4901 y Ft(s)2302 4889 y Fl(y)2352 4901 y
Ft(s)2388 4889 y Fu(:)1905 5255 y Fv(21)p eop end
%%Page: 22 22
TeXDict begin 22 21 bop 515 523 a Fv(By)28 b(\(3.9\))g(w)n(e)f(ha)n(v)n
(e)1562 623 y Fr(j)p Fl(z)1627 635 y Ft(s)1658 643 y
Fn(1)1695 623 y Fv(\()p Fu(t)p Fv(\))p Fr(j)d(\024)f
Fv(\010\()p Fu(\034)5 b(;)14 b(b)2130 635 y Fp(1)2167
623 y Fu(;)g(b)2240 635 y Fp(2)2276 623 y Fv(\))p Fu(:)836
b Fv(\(3.11\))639 772 y(Using)28 b(the)g(in)n(tro)r(duced)f(notations)g
(w)n(e)g(write)g(the)h(equation)f(for)g Fl(y)j Fv(as)529
999 y(_)515 1000 y Fl(y)565 1012 y Ft(s)596 1020 y Fn(1)656
1000 y Fv(=)928 921 y Fm(X)744 1103 y Ft(s)p Fp(=)p Ft(\034)7
b(a)p Fp(+)p Ft(c)p Fs(2)p Ft(O)1078 1086 y Ff(a)1076
1120 y(\034)1113 1103 y Fp(\(\003)1184 1086 y FA(0)1207
1103 y Fp(\))1247 908 y Fm(\020)1297 1000 y Fu(A)1359
966 y Fs(1)1429 1000 y Fv(\()p Fu(a)p Fv(\))1537 1012
y Ft(b)1566 1020 y Fn(1)1599 1012 y Ft(c)1646 1000 y
Fv(+)13 b Fu(\034)1769 966 y Fs(\000)p Fp(1)1859 1000
y Fu(A)1921 966 y Fp(\001)1980 1000 y Fv(\()p Fu(\034)5
b(;)14 b(a)p Fv(\))2166 1012 y Ft(b)2195 1020 y Fn(1)2228
1012 y Ft(c)2262 908 y Fm(\021)2311 1000 y Fl(y)2361
1012 y Ft(s)2411 1000 y Fv(+)f Fl(z)2531 1012 y Ft(s)2562
1020 y Fn(1)2599 1000 y Fu(;)97 b Fl(y)q Fv(\(0\))24
b(=)e Fu(l)3014 966 y Ft(s)3045 974 y Fn(2)3095 883 y
Fm(\022)3198 949 y Fv(1)3198 1049 y(0)3281 883 y Fm(\023)3356
1000 y Fu(:)515 1277 y Fv(Let)27 b(us)h(seek)f(the)h(solution)f
Fl(y)j Fv(of)d(this)h(equation)f(as)g(an)g(ansatz)1384
1460 y Fl(y)1434 1472 y Ft(\034)7 b(a)p Fp(+)p Ft(b)1588
1480 y Fn(1)1624 1460 y Fv(\()p Fu(t)p Fv(\))24 b(=)e
Fl(y)1880 1425 y Fp(0)1879 1480 y Ft(b)1908 1488 y Fn(1)1945
1460 y Fv(\()p Fu(t)p Fv(\))d(+)g Fu(\034)2187 1425 y
Fs(\000)p Fp(1)2276 1460 y Fl(y)2327 1425 y Fp(1)2326
1480 y Ft(b)2355 1488 y Fn(1)2392 1460 y Fv(\()p Fu(t)p
Fv(\))p Fu(;)658 b Fv(\(3.12\))515 1642 y(where)27 b
Fl(y)806 1612 y Fp(0)805 1666 y Ft(b)834 1674 y Fn(1)894
1642 y Fv(=)c Fl(y)1033 1612 y Fp(0)1071 1642 y Fv(\()p
Fu(a)p Fv(\))1179 1654 y Ft(b)1208 1662 y Fn(1)1245 1642
y Fu(;)41 b Fl(y)1360 1612 y Fp(1)1359 1666 y Ft(b)1388
1674 y Fn(1)1449 1642 y Fv(=)22 b Fl(y)1587 1612 y Fp(1)1625
1642 y Fv(\()p Fu(a;)14 b(\034)9 b Fv(\))1815 1654 y
Ft(b)1844 1662 y Fn(1)1882 1642 y Fv(,)28 b(and)f Fl(y)2145
1612 y Fp(0)2206 1642 y Fv(=)c Fr(f)p Fl(y)2387 1612
y Fp(0)2386 1666 y Ft(b)2424 1642 y Fv(\()p Fu(t)p Fv(\))p
Fu(;)14 b(b)23 b Fr(2)h Fq(Z)2748 1612 y Ft(d)2787 1642
y Fr(g)j Fv(satis\014es)1289 1885 y(_)1274 1886 y Fl(y)1325
1851 y Fp(0)1386 1886 y Fv(=)c Fu(A)1536 1851 y Fs(1)1607
1886 y Fv(\()p Fu(a)p Fv(\))p Fl(y)1766 1851 y Fp(0)1804
1886 y Fu(;)97 b Fl(y)1975 1851 y Fp(0)2013 1886 y Fv(\(0\))23
b(=)g Fu(l)2257 1851 y Ft(b)2286 1859 y Fn(2)2336 1769
y Fm(\022)2438 1835 y Fv(1)2438 1935 y(0)2521 1769 y
Fm(\023)2596 1886 y Fu(:)548 b Fv(\(3.13\))515 2118 y(Due)28
b(to)f(\(3.10\))o(,)h(\(3.5\))g(and)f(Theorem)g(3.2,)1024
2255 y Fm(\014)1024 2305 y(\014)1024 2355 y(\014)1052
2351 y Fl(y)1103 2316 y Fp(0)1102 2371 y Ft(c)1141 2351
y Fv(\()p Fu(t)p Fv(\))19 b Fr(\000)f Fu(\016)1374 2363
y Ft(c;b)1453 2371 y Fn(2)1503 2234 y Fm(\022)1605 2300
y Fv(1)1605 2400 y(0)1688 2234 y Fm(\023)1763 2255 y(\014)1763
2305 y(\014)1763 2355 y(\014)1814 2351 y Fr(\024)k Fu(C)6
b(\026)p Fr(j)p Fu(t)p Fr(j)p Fu(e)2131 2316 y Fs(\000)p
Ft(\015)t Fs(j)p Ft(c)p Fs(\000)p Ft(b)2353 2324 y Fn(2)2385
2316 y Fs(j)2409 2351 y Fu(;)97 b(c;)14 b(b)2638 2363
y Fp(2)2697 2351 y Fr(2)24 b Fq(Z)2831 2316 y Ft(d)3167
2351 y Fv(\(3.14\))515 2583 y(\(as)j(b)r(efore,)g Fr(j)p
Fu(t)p Fr(j)c(\024)g Fv(1\).)639 2683 y(F)-7 b(or)27
b Fl(y)839 2653 y Fp(1)905 2683 y Fv(w)n(e)g(get)g(the)h(equation)840
2885 y(_)826 2886 y Fl(y)877 2851 y Fp(1)876 2906 y Ft(b)905
2914 y Fn(1)965 2886 y Fv(=)1053 2807 y Fm(X)1098 2981
y Ft(c)1186 2794 y Fm(\020)1236 2886 y Fu(A)1298 2851
y Fs(1)1369 2886 y Fv(\()p Fu(a)p Fv(\))1477 2898 y Ft(b)1506
2906 y Fn(1)1539 2898 y Ft(c)1591 2886 y Fv(+)18 b Fu(\034)1719
2851 y Fs(\000)p Fp(1)1809 2886 y Fu(A)1871 2851 y Fp(\001)1930
2886 y Fv(\()p Fu(\034)5 b(;)14 b(a)p Fv(\))2116 2898
y Ft(b)2145 2906 y Fn(1)2178 2898 y Ft(c)2212 2794 y
Fm(\021)2261 2886 y Fl(1)2309 2901 y Ft(O)2361 2884 y
Ff(a)2359 2917 y(\034)2397 2901 y Fp(\(\003)2468 2884
y FA(0)2491 2901 y Fp(\))2521 2886 y Fv(\()p Fu(\034)9
b(a)20 b Fv(+)e Fu(c)p Fv(\))p Fl(y)2864 2851 y Fp(1)2863
2906 y Ft(c)871 3110 y Fv(+)950 3031 y Fm(X)994 3205
y Ft(c)1083 3110 y Fu(A)1145 3076 y Fp(\001)1204 3110
y Fv(\()p Fu(\034)5 b(;)14 b(a)p Fv(\))1390 3122 y Ft(b)1419
3130 y Fn(1)1452 3122 y Ft(c)1486 3110 y Fl(1)1534 3125
y Ft(O)1586 3108 y Ff(a)1584 3141 y(\034)1622 3125 y
Fp(\(\003)1693 3108 y FA(0)1716 3125 y Fp(\))1746 3110
y Fv(\()p Fu(\034)9 b(a)19 b Fv(+)f Fu(c)p Fv(\))c Fl(y)2102
3076 y Fp(0)2101 3130 y Ft(c)2173 3110 y Fv(+)k Fu(\034)9
b Fl(z)2343 3122 y Ft(s)2374 3130 y Fn(1)2412 3110 y
Fu(;)97 b Fl(y)2583 3076 y Fp(1)2621 3110 y Fv(\(0\))23
b(=)f(0)p Fu(:)265 b Fv(\(3.15\))515 3367 y(Due)28 b(to)f(\(3.10\))o(,)
h(\(3.14\))f(and)h(\(1.9\))o(,)956 3479 y Fm(\014)956
3529 y(\014)956 3579 y(\014)998 3496 y(X)1043 3670 y
Ft(c)1131 3575 y Fu(A)1193 3541 y Fp(\001)1252 3575 y
Fv(\()p Fu(\034)5 b(;)14 b(a)p Fv(\))1438 3587 y Ft(b)1467
3595 y Fn(1)1500 3587 y Ft(c)1534 3575 y Fl(1)1582 3590
y Ft(O)1634 3573 y Ff(a)1632 3606 y(\034)1671 3590 y
Fp(\(\003)1742 3573 y FA(0)1764 3590 y Fp(\))1794 3575
y Fv(\()p Fu(\034)9 b(a)20 b Fv(+)e Fu(c)p Fv(\))p Fl(y)2137
3541 y Fp(0)2136 3595 y Ft(c)2175 3479 y Fm(\014)2175
3529 y(\014)2175 3579 y(\014)2226 3575 y Fr(\024)k Fu(C)6
b(\026e)2467 3541 y Fs(\000)p Ft(\015)t Fs(j)p Ft(b)2607
3549 y Fn(1)2639 3541 y Fs(\000)p Ft(b)2720 3549 y Fn(2)2752
3541 y Fs(j)2776 3575 y Fu(\015)2824 3539 y Fs(\000)p
Ft(d)2819 3599 y Fp(\001)2915 3575 y Fu(;)515 3832 y
Fv(and)27 b(due)h(to)f(\(3.11\),)1341 3932 y Fr(j)p Fu(\034)9
b Fl(z)1451 3944 y Ft(s)1482 3952 y Fn(1)1520 3932 y
Fv(\()p Fu(t)p Fv(\))p Fr(j)24 b(\024)f Fu(C)6 b(\026\015)1912
3896 y Fs(\000)p Ft(d)p Fs(\000)p Fp(1)1907 3956 y(\001)2087
3932 y Fv(\003)2145 3898 y Fp(2)2182 3932 y Fu(e)2221
3898 y Fs(\000)p Ft(\015)t Fs(j)p Ft(b)2361 3906 y Fn(1)2392
3898 y Fs(\000)p Ft(b)2473 3906 y Fn(2)2506 3898 y Fs(j)2530
3932 y Fu(:)515 4081 y Fv(So)32 b(denoting)h(b)n(y)f
Fa(Z)1152 4093 y Ft(b)1181 4101 y Fn(1)1218 4081 y Fv(\()p
Fu(t)p Fv(\))i(the)f(sum)g(of)g(the)g(last)g(t)n(w)n(o)f(terms)g(in)h
(the)h(r.h.s.)52 b(of)33 b(equation)515 4181 y(\(3.15\))o(,)28
b(w)n(e)f(ha)n(v)n(e)f(from)i(the)g(last)f(t)n(w)n(o)g(estimates)g
(that)1361 4363 y Fr(j)p Fa(Z)1434 4375 y Ft(b)1463 4383
y Fn(1)1500 4363 y Fv(\()p Fu(t)p Fv(\))p Fr(j)d(\024)e
Fu(C)6 b(\026)p Fv(\003)1901 4329 y Fp(2)1939 4363 y
Fu(\015)1987 4328 y Fs(\000)p Ft(d)p Fs(\000)p Fp(1)1982
4388 y(\001)2162 4363 y Fu(e)2201 4329 y Fs(\000)p Ft(\015)t
Fs(j)p Ft(b)2341 4337 y Fn(1)2372 4329 y Fs(\000)p Ft(b)2453
4337 y Fn(2)2486 4329 y Fs(j)2509 4363 y Fu(:)635 b Fv(\(3.16\))639
4546 y(W)-7 b(e)28 b(re-write)f(\(3.15\))g(as)594 4753
y(_)579 4754 y Fl(y)630 4719 y Fp(1)629 4774 y Ft(b)691
4754 y Fv(=)779 4675 y Fm(X)824 4849 y Ft(c)913 4662
y Fm(\020)962 4686 y(\000)1000 4754 y Fu(A)1062 4719
y Fs(1)1133 4754 y Fv(\()p Fu(a)p Fv(\))1241 4766 y Ft(bc)1323
4754 y Fv(+)18 b Fu(\034)1451 4719 y Fs(\000)p Fp(1)1541
4754 y Fu(A)1603 4719 y Fp(\001)1662 4754 y Fv(\()p Fu(\034)5
b(;)14 b(a)p Fv(\))1848 4766 y Ft(bc)1911 4686 y Fm(\001)1949
4754 y Fl(1)1997 4769 y Ft(O)2049 4752 y Ff(a)2047 4785
y(\034)2085 4769 y Fp(\(\003)2156 4752 y FA(0)2179 4769
y Fp(\))2209 4754 y Fv(\()p Fu(\034)9 b(a)20 b Fv(+)e
Fu(c)p Fv(\))2501 4662 y Fm(\021)2551 4754 y Fl(y)2602
4719 y Fp(1)2601 4774 y Ft(c)2658 4754 y Fv(+)g Fa(Z)2791
4766 y Ft(b)2824 4754 y Fu(;)97 b Fl(y)2995 4719 y Fp(1)3033
4754 y Fv(\(0\))23 b(=)g(0)p Fu(:)3167 4928 y Fv(\(3.17\))1905
5255 y(22)p eop end
%%Page: 23 23
TeXDict begin 23 22 bop 515 523 a Fv(Let)31 b Fr(S)723
486 y Ft(t)748 494 y Fn(2)717 543 y Ft(t)742 551 y Fn(1)816
523 y Fv(b)r(e)g(the)h(\015o)n(w-maps)d(of)i(the)h(homogeneous)d(v)n
(ersion)g(of)i(this)g(equation)g(\(i.e.,)h(of)515 623
y(equation)27 b(\(3.17\))g(with)h Fa(Z)23 b Fv(=)f(0\).)37
b(Then)28 b(due)g(to)f(\(3.10\))o(,)h(\(3.7\))f(and)h(Theorem)f(3.2,)
1231 733 y Fm(\014)1231 782 y(\014)1231 832 y(\014)1259
736 y(\020)1309 828 y Fr(S)1365 791 y Ft(t)1390 799 y
Fn(2)1359 848 y Ft(t)1384 856 y Fn(1)1445 828 y Fr(\000)18
b Fv(id)1597 736 y Fm(\021)1647 886 y Ft(b)1676 894 y
Fn(1)1709 886 y Ft(b)1738 894 y Fn(2)1774 733 y Fm(\014)1774
782 y(\014)1774 832 y(\014)1825 828 y Fr(\024)23 b Fu(C)6
b(\026)p Fr(j)p Fu(t)p Fr(j)p Fu(e)2143 794 y Fs(\000)p
Fp(\()p Ft(\015)t Fp(+)p Ft(\015)2346 802 y Fn(\001)2395
794 y Fp(\))p Fs(j)p Ft(b)2470 802 y Fn(1)2502 794 y
Fs(\000)p Ft(b)2583 802 y Fn(2)2615 794 y Fs(j)2639 828
y Fu(:)515 1066 y Fv(Since)29 b(b)n(y)h(\(3.17\))58 b
Fl(y)1173 1036 y Fp(1)1211 1066 y Fv(\()p Fu(t)p Fv(\))27
b(=)1422 999 y Fm(R)1478 1019 y Ft(t)1461 1095 y Fp(0)1521
1066 y Fr(S)1577 1036 y Ft(t)1571 1089 y(\022)1609 1066
y Fa(Z)p Fv(\()p Fu(\022)r Fv(\))14 b Fu(d\022)r Fv(,)31
b(then)e(due)h(to)f(\(3.16\))o(,)h(the)g(remark)e(to)h(Theo-)515
1165 y(rem)e(1.5)g(and)g(\(3.7\))h(w)n(e)f(ha)n(v)n(e)626
1346 y Fr(j)p Fl(y)700 1312 y Fp(1)699 1366 y Ft(b)728
1374 y Fn(1)765 1346 y Fv(\()p Fu(t)p Fv(\))p Fr(j)c(\024)g
Fu(C)6 b(\026)p Fr(j)p Fu(t)p Fr(j)p Fv(\(1)19 b(+)f
Fu(\026\015)1458 1310 y Fs(\000)p Ft(d)1453 1370 y Fp(\001)1548
1346 y Fv(\))p Fu(\015)1628 1310 y Fs(\000)p Ft(d)p Fs(\000)p
Fp(1)1623 1370 y(\001)1803 1346 y Fv(\003)1861 1312 y
Fp(2)1898 1346 y Fu(e)1937 1312 y Fs(\000)p Ft(\015)t
Fs(j)p Ft(b)2077 1320 y Fn(1)2109 1312 y Fs(\000)p Ft(b)2190
1320 y Fn(2)2222 1312 y Fs(j)2269 1346 y Fr(\024)23 b
Fu(C)2416 1358 y Fp(1)2453 1346 y Fu(\026)p Fr(j)p Fu(t)p
Fr(j)p Fu(\015)2627 1310 y Fs(\000)p Ft(d)p Fs(\000)p
Fp(1)2622 1370 y(\001)2803 1346 y Fv(\003)2861 1312 y
Fp(2)2897 1346 y Fu(e)2936 1312 y Fs(\000)p Ft(\015)t
Fs(j)p Ft(b)3076 1320 y Fn(1)3108 1312 y Fs(\000)p Ft(b)3189
1320 y Fn(2)3221 1312 y Fs(j)3245 1346 y Fu(:)3167 1445
y Fv(\(3.18\))639 1545 y(By)k(\(3.8\),)52 b(\()p Fu(S)1103
1515 y Ft(t)1147 1545 y Fr(\000)15 b Fv(id\))1328 1557
y Fs(\006)p Ft(s)1411 1565 y Fn(1)1444 1557 y Ft(s)1475
1565 y Fn(2)1535 1545 y Fv(=)22 b Fu(y)1663 1557 y Fs(\006)p
Ft(s)1746 1565 y Fn(1)1783 1545 y Fv(\()p Fu(t)p Fv(\))15
b Fr(\000)g Fu(\016)2009 1557 y Fs(\006)p Ft(s)2092 1565
y Fn(1)2124 1557 y Ft(;s)2175 1565 y Fn(2)2212 1545 y
Fu(:)40 b Fv(Using)27 b(\(3.12\))o(,)f(\(3.14\))f(and)h(\(3.18\))515
1645 y(w)n(e)h(get)g(that)629 1753 y Fm(\022)758 1820
y Fv(\()p Fu(S)846 1790 y Ft(t)893 1820 y Fr(\000)19
b Fv(id\))1078 1832 y Ft(s)1109 1840 y Fn(1)1142 1832
y Ft(s)1173 1840 y Fn(2)732 1919 y Fv(\()p Fu(S)820 1889
y Ft(t)868 1919 y Fr(\000)f Fv(id\))1052 1931 y Fs(\000)p
Ft(s)1135 1939 y Fn(1)1168 1931 y Ft(s)1199 1939 y Fn(2)1277
1753 y Fm(\023)1361 1870 y Fv(=)23 b Fl(y)1500 1836 y
Fp(0)1538 1870 y Fv(\()p Fu(a)p Fv(\))1646 1882 y Ft(b)1675
1890 y Fn(1)1712 1870 y Fv(\()p Fu(t)p Fv(\))c Fr(\000)f
Fu(\016)1945 1882 y Ft(b)1974 1890 y Fn(1)2007 1882 y
Ft(;b)2056 1890 y Fn(2)2106 1753 y Fm(\022)2208 1820
y Fv(1)2208 1919 y(0)2292 1753 y Fm(\023)2371 1870 y
Fv(+)g Fu(\034)2499 1836 y Fs(\000)p Fp(1)2589 1870 y
Fl(y)2640 1836 y Fp(1)2678 1870 y Fv(\()p Fu(a;)c(\034)9
b Fv(\))2868 1882 y Ft(b)2897 1890 y Fn(1)2935 1870 y
Fv(\()p Fu(t)p Fv(\))p Fu(;)115 b Fv(\(3.19\))515 2096
y(where)678 2210 y Fm(\014)678 2260 y(\014)678 2310 y(\014)706
2306 y Fl(y)757 2271 y Fp(0)795 2306 y Fv(\()p Fu(a)p
Fv(\))903 2318 y Ft(b)932 2326 y Fn(1)969 2306 y Fv(\()p
Fu(t)p Fv(\))19 b Fr(\000)f Fu(\016)1202 2318 y Ft(b)1231
2326 y Fn(1)1263 2318 y Ft(;b)1312 2326 y Fn(2)1363 2189
y Fm(\022)1465 2255 y Fv(1)1465 2355 y(0)1548 2189 y
Fm(\023)1623 2210 y(\014)1623 2260 y(\014)1623 2310 y(\014)1665
2306 y Fu(;)1729 2235 y Fm(\014)1729 2285 y(\014)1757
2306 y Fl(y)1808 2271 y Fp(1)1846 2306 y Fv(\()p Fu(a;)c(\034)9
b Fv(\))2036 2318 y Ft(b)2065 2326 y Fn(1)2103 2306 y
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%%Page: 24 24
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%%Page: 25 25
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b Fv(+)d Fr(j)p Fv(\()p Fu(x)h Fr(\000)f Fu(L;)c(y)s
Fv(\))p Fr(j)22 b Fv(=)h Fu(K)6 b(r)21 b Fv(+)2366 1765
y Fm(p)p 2449 1765 520 4 v 81 x Fu(r)2488 1822 y Fp(2)2544
1846 y Fv(+)d Fu(L)2684 1822 y Fp(2)2739 1846 y Fr(\000)g
Fv(2)p Fu(Lx;)106 b(r)26 b Fv(=)d Fr(j)p Fu(z)t Fr(j)p
Fu(:)515 2029 y Fv(No)n(w)k(let)h(us)f(write)h(the)g(set)f
Fu(Q)1481 2041 y Ft(\034)1550 2029 y Fv(as)1217 2224
y Fu(Q)1283 2236 y Ft(\034)1347 2224 y Fv(=)1488 2145
y Fm([)1435 2323 y Fp(0)p Fs(\024)p Ft(t)p Fs(\024)p
Ft(\034)1648 2224 y Fr(O)1714 2236 y Ft(t)1743 2224 y
Fu(;)180 b Fr(O)2012 2236 y Ft(t)2064 2224 y Fv(=)23
b Fr(f)p Fu(z)j Fr(j)d Fv(\010\()p Fu(z)t Fv(\))g(=)f
Fu(t)p Fr(g)p Fu(:)515 2509 y Fv(A)34 b(h)n(yp)r(ersurface)f
Fr(O)1169 2521 y Ft(t)1233 2509 y Fv(is)h(formed)g(b)n(y)g(p)r(oin)n
(ts)g Fu(z)k Fv(=)33 b(\()p Fu(x;)14 b(y)s Fv(\),)37
b(satisfying)2791 2440 y Fr(p)p 2860 2440 V 69 x Fu(r)2899
2485 y Fp(2)2955 2509 y Fv(+)18 b Fu(L)3095 2485 y Fp(2)3151
2509 y Fr(\000)g Fv(2)p Fu(Lx)515 2609 y Fv(=)k Fu(t)d
Fr(\000)f Fu(K)6 b(r)r Fv(.)37 b(F)-7 b(or)27 b(an)n(y)g(suc)n(h)g(a)g
(p)r(oin)n(t)h Fu(z)j Fv(w)n(e)c(ha)n(v)n(e)1800 2791
y Fu(t)c Fr(\025)g Fu(K)6 b(r)16 b(;)1114 b Fv(\(4.2\))515
2974 y(and)1439 3074 y(2)p Fu(Lx)22 b Fv(=)h Fu(r)1734
3039 y Fp(2)1790 3074 y Fv(+)18 b Fu(L)1930 3039 y Fp(2)1986
3074 y Fr(\000)g Fv(\()p Fu(t)g Fr(\000)g Fu(K)6 b(r)r
Fv(\))2380 3039 y Fp(2)2432 3074 y Fu(:)515 3223 y Fv(Since)28
b Fr(\000)p Fu(r)d Fr(\024)e Fu(x)g Fr(\024)g Fu(r)r
Fv(,)28 b(then)1295 3406 y Fr(\000)p Fv(2)p Fu(Lr)d Fr(\024)d
Fu(r)1647 3372 y Fp(2)1704 3406 y Fv(+)c Fu(L)1844 3372
y Fp(2)1899 3406 y Fr(\000)g Fv(\()p Fu(t)h Fr(\000)f
Fu(K)6 b(r)r Fv(\))2294 3372 y Fp(2)2355 3406 y Fr(\024)22
b Fv(2)p Fu(Lr)n(:)515 3588 y Fv(The)j(second)g(inequalit)n(y)f
(implies)i(\()p Fu(r)16 b Fr(\000)e Fu(L)p Fv(\))1864
3558 y Fp(2)1924 3588 y Fr(\024)22 b Fv(\()p Fu(t)14
b Fr(\000)g Fu(K)6 b(r)r Fv(\))2314 3558 y Fp(2)2351
3588 y Fv(,)26 b(and)f(using)h(\(4.2\))f(w)n(e)g(get)g(that)1416
3771 y Fr(\000)p Fv(\()p Fu(t)18 b Fr(\000)g Fu(K)6 b(r)r
Fv(\))24 b Fr(\024)e Fu(r)g Fr(\000)c Fu(L)k Fr(\024)h
Fu(t)18 b Fr(\000)g Fu(K)6 b(r)n(:)515 3954 y Fv(By)27
b(the)h(\014rst)f(of)h(these)g(inequalities,)f Fu(K)6
b(r)21 b Fr(\000)d Fu(r)25 b Fr(\024)e Fu(t)c Fr(\000)f
Fu(L)p Fv(.)36 b(Or)1740 4174 y Fu(r)26 b Fr(\024)1917
4118 y Fu(t)18 b Fr(\000)g Fu(L)p 1901 4155 220 4 v 1901
4231 a(K)24 b Fr(\000)18 b Fv(1)2130 4174 y Fu(;)1055
b Fv(\(4.3\))515 4392 y(since)27 b Fu(K)i(>)22 b Fv(1.)37
b(Therefore)26 b Fr(O)1449 4404 y Ft(t)1502 4392 y Fv(=)c
Fr(;)55 b Fv(if)28 b Fu(t)23 b(<)g(L)p Fv(.)36 b(Accordingly)-7
b(,)1558 4574 y Fu(F)12 b Fv(\()p Fu(\034)d Fv(\))24
b(=)f(0)82 b(if)i Fu(\034)33 b(<)22 b(L:)515 4757 y Fv(If)28
b Fu(\034)33 b Fr(\025)22 b Fu(L)p Fv(,)27 b(then)h(\(4.3\))g(implies)g
(that)g Fu(Q)1777 4769 y Ft(\034)1841 4757 y Fr(\032)22
b(fj)p Fu(z)t Fr(j)g(\024)h Fv(\()p Fu(\034)28 b Fr(\000)18
b Fu(L)p Fv(\))p Fu(=)p Fv(\()p Fu(K)24 b Fr(\000)18
b Fv(1\))p Fr(g)p Fv(.)36 b(So)1206 4939 y Fu(F)12 b
Fv(\()p Fu(\034)d Fv(\))24 b(=)e(mes)14 b Fu(Q)1710 4951
y Ft(\034)1774 4939 y Fr(\024)23 b Fq({)1917 4951 y Ft(d)1970
4939 y Fv(\()p Fu(K)h Fr(\000)18 b Fv(1\))2254 4905 y
Fs(\000)p Ft(d)2344 4939 y Fv(\()p Fu(\034)29 b Fr(\000)18
b Fu(L)p Fv(\))2613 4905 y Ft(d)2665 4939 y Fu(;)1905
5255 y Fv(25)p eop end
%%Page: 26 26
TeXDict begin 26 25 bop 515 523 a Fv(where)33 b Fq({)816
535 y Ft(d)889 523 y Fv(is)h(the)g(v)n(olume)f(of)h(a)f(1-ball)g(in)h
Fu(R)2000 493 y Ft(d)2039 523 y Fv(.)55 b(Due)35 b(to)e(this)h
(estimate)g(and)g(\(4.1\))f(w)n(e)515 623 y(ha)n(v)n(e)1143
831 y Fu(')p Fv(\()p Fu(\015)5 b Fv(\))23 b Fr(\024)g
Fq({)1475 843 y Ft(d)1528 831 y Fu(\015)5 b Fv(\()p Fu(K)24
b Fr(\000)18 b Fv(1\))1860 797 y Fs(\000)p Ft(d)1964
718 y Fm(Z)2047 739 y Fs(1)2010 907 y Ft(L)2131 831 y
Fu(e)2170 797 y Fs(\000)p Ft(\015)t(\034)2302 831 y Fv(\()p
Fu(\034)28 b Fr(\000)18 b Fu(L)p Fv(\))2570 797 y Ft(d)2622
831 y Fu(d\034)1332 1058 y Fv(=)23 b Fq({)1475 1070 y
Ft(d)1528 1058 y Fu(\015)5 b Fv(\()p Fu(K)24 b Fr(\000)18
b Fv(1\))1860 1024 y Fs(\000)p Ft(d)1950 1058 y Fu(e)1989
1024 y Fs(\000)p Ft(\015)t(L)2143 945 y Fm(Z)2226 965
y Fs(1)2189 1133 y Fp(0)2310 1058 y Fu(e)2349 1024 y
Fs(\000)p Ft(\015)t(x)2481 1058 y Fu(x)2528 1024 y Ft(d)2581
1058 y Fu(dx)1332 1284 y Fv(=)23 b Fq({)1475 1296 y Ft(d)1528
1284 y Fu(\015)1576 1250 y Fs(\000)p Ft(d)1666 1284 y
Fv(\()p Fu(K)h Fr(\000)18 b Fv(1\))1950 1250 y Fs(\000)p
Ft(d)2041 1284 y Fu(e)2080 1250 y Fs(\000)p Ft(\015)t(L)2233
1171 y Fm(Z)2316 1192 y Fs(1)2279 1360 y Fp(0)2400 1284
y Fu(e)2439 1250 y Fs(\000)p Ft(y)2531 1284 y Fu(y)2575
1250 y Ft(d)2627 1284 y Fu(dy)f(:)515 1509 y Fv(That)27
b(is)1574 1608 y Fu(')p Fv(\()p Fu(\015)5 b Fv(\))24
b Fr(\024)f Fu(C)1911 1620 y Ft(d)1950 1608 y Fu(\015)1998
1573 y Fs(\000)p Ft(d)1993 1633 y Fp(\001)2088 1608 y
Fu(e)2127 1574 y Fs(\000)p Ft(\015)t Fs(j)p Ft(a)p Fs(j)2296
1608 y Fu(:)515 1758 y Fv(This)30 b(estimate)g(w)n(as)f(pro)n(v)n(ed)f
(for)i Fu(a)c Fr(6)p Fv(=)h(0)j(and)g Fu(\015)h(>)c Fv(0.)43
b(By)30 b(con)n(tin)n(uit)n(y)g(it)g(also)f(holds)h(for)515
1858 y Fu(\015)d Fv(=)c(0.)36 b(So)28 b(\(1.8\))f(is)h(pro)n(v)n(ed)e
(if)i Fu(a)23 b Fr(6)p Fv(=)f(0.)639 1957 y(If)28 b Fu(a)23
b Fv(=)g(0,)k(then)h(the)g(l.h.s.)37 b(in)28 b(\(1.8\))f(is)h(b)r
(ounded)g(b)n(y)886 2187 y Fu(C)965 2074 y Fm(Z)1012
2262 y Fg(R)1054 2246 y Ff(d)1106 2187 y Fu(e)1145 2152
y Fs(\000)p Ft(\015)t Fs(j)p Ft(z)r Fs(j\000)p Fp(\()p
Ft(\015)t Fp(+)p Ft(\015)1513 2160 y Fn(\001)1561 2152
y Fp(\))p Fs(j)p Ft(z)r Fs(j)1678 2187 y Fu(dz)f Fv(=)o
Fu(C)1910 2199 y Fp(1)1962 2074 y Fm(Z)2045 2094 y Fs(1)2008
2262 y Fp(0)2129 2187 y Fu(r)2168 2152 y Ft(d)p Fs(\000)p
Fp(1)2292 2187 y Fu(e)2331 2152 y Fs(\000)p Fp(\(2)p
Ft(\015)t Fp(+)p Ft(\015)2567 2160 y Fn(\001)2616 2152
y Fp(\))p Ft(r)2693 2187 y Fu(dr)1787 2413 y Fv(=)o Fu(C)1910
2425 y Fp(1)1948 2413 y Fv(\(2)p Fu(\015)c Fv(+)18 b
Fu(\015)2214 2425 y Fp(\001)2272 2413 y Fv(\))2304 2379
y Fs(\000)p Ft(d)2409 2300 y Fm(Z)2492 2321 y Fs(1)2455
2489 y Fp(0)2576 2413 y Fu(x)2623 2379 y Ft(d)p Fs(\000)p
Fp(1)2747 2413 y Fu(e)2786 2379 y Fs(\000)p Ft(x)2894
2413 y Fu(dx;)515 2642 y Fv(and)27 b(\(1.8\))g(also)g(holds.)639
2792 y(T)-7 b(o)27 b(pro)n(v)n(e)f(\(1.9\))i(w)n(e)f(\014rst)g
(estimate)h(the)g(l.h.s.)37 b(b)n(y)27 b(the)h(in)n(tegral)811
3017 y Fu(C)870 3029 y Fp(1)921 2904 y Fm(Z)1018 3017
y Fu(e)1057 2983 y Ft(\015)t Fs(j)p Ft(z)r Fs(\000)p
Ft(b)p Fs(j\000)p Fp(\()p Ft(\015)t Fp(+)p Ft(\015)1454
2991 y Fn(\001)1501 2983 y Fp(\))p Fs(j)p Ft(z)r Fs(j)1619
3017 y Fu(dz)e Fv(=)d Fu(C)1874 3029 y Fp(1)1926 2904
y Fm(Z)2022 3017 y Fu(e)2061 2983 y Fs(\000)p Ft(\015)t
Fp(\()p Ft(K)t Fs(j)p Ft(z)r Fs(j\000j)p Ft(z)r Fs(\000)p
Ft(b)p Fs(j)p Fp(\))2561 3017 y Fu(dz)k Fv(=:)22 b Fu(C)2839
3029 y Fp(1)2877 3017 y Fu( )s Fv(\()p Fu(\015)5 b Fv(\))14
b Fu(:)515 3236 y Fv(Denoting)1006 3419 y(\010\()p Fu(z)t
Fv(\))23 b(=)f Fu(K)6 b Fr(j)p Fu(z)t Fr(j)17 b(\000)i(j)p
Fu(z)i Fr(\000)d Fu(b)p Fr(j)p Fu(;)97 b(F)12 b Fv(\()p
Fu(\034)d Fv(\))24 b(=)f(mes)13 b Fr(f)p Fu(z)26 b Fr(j)d
Fv(\010\()p Fu(z)t Fv(\))g Fr(\024)g Fu(\034)9 b Fr(g)14
b Fu(;)515 3602 y Fv(w)n(e)27 b(see)g(that)h(\010)23
b Fr(\025)g(\000j)p Fu(b)p Fr(j)k Fv(and)g(that)1441
3831 y Fu( )s Fv(\()p Fu(\015)5 b Fv(\))24 b(=)e Fu(\015)1783
3718 y Fm(Z)1866 3739 y Fs(1)1829 3907 y(\000j)p Ft(b)p
Fs(j)1967 3831 y Fu(e)2006 3797 y Fs(\000)p Ft(\015)t(\034)2138
3831 y Fu(F)12 b Fv(\()p Fu(\034)d Fv(\))14 b Fu(d\034)24
b(:)515 4075 y Fv(Analysis)j(of)g(this)h(Laplace)f(in)n(tegral)f(is)i
(similar)e(to)i(that)g(of)34 b(\(4.1\))27 b(an)g(w)n(e)h(omit)f(it.)515
4350 y Fw(References)515 4531 y Fv([EK05])39 b(H.)28
b(L.)g(Eliasson)e(and)h(S.)h(B.)g(Kuksin,)f Fo(KAM)j(for)g(NLS)p
Fv(,)e(Preprin)n(t)e(\(2005\).)515 4697 y([HS78])53 b(P)-7
b(.)41 b(R.)g(Halmos)f(and)h(V.)g(S.)g(Sunders,)j Fo(Bounde)l(d)e(inte)
l(gr)l(al)h(op)l(er)l(ators)f(on)g Fu(L)3342 4667 y Fp(2)806
4797 y Fo(sp)l(ac)l(es)p Fv(,)29 b(Springer-V)-7 b(erlag,)25
b(Berlin,)i(1978.)1905 5255 y(26)p eop end
%%Trailer

userdict /end-hook known{end-hook}if
%%EOF
---------------0604270415828--