% Jean-Pierre Demailly % Universit\'e de Grenoble I, Institut Fourier % \'Equations Diff\'erentielles et Analyse Num\'erique % Collection Grenoble-Sciences, Presses Universitaires de Grenoble \newif\ifcolorized \colorizedfalse \newif\iftocfirst \tocfirsttrue \input hyperbasics.tex \ifcolorized \magnification=1200 \hsize=13cm\hoffset=0cm \vsize=18.8cm\voffset=0.3cm \else %% \magnification=1000 \hsize=13cm\hoffset=1.5cm \vsize=18.8cm\voffset=2cm \magnification=1200 \hsize=13cm\hoffset=0cm \vsize=18.8cm\voffset=0.3cm \fi \pretolerance=500 \tolerance=1000 \brokenpenalty=5000 \parindent=0mm \parskip=5pt plus 1pt minus 1pt \baselineskip=12.5pt \catcode`\@=11 \def\makeheadline{\vbox to\z@{\vskip-25.4\p@ \line{\vbox to8.5\p@{}\the\headline}\vss}\nointerlineskip} \def\makefootline{\baselineskip28\p@\lineskiplimit\z@\line{\the\footline}} \catcode`\@=12 % new fonts definitions \font\twentyfourss=cmss10 at 24.8832pt \font\twentyss=cmss10 at 20.736pt \font\seventeenss=cmss10 at 17.28pt \font\fourteenss=cmss10 at 14.4pt \font\twelvess=cmss10 at 12pt \font\tenss=cmss10 \font\sevenss=cmss10 at 7pt \font\twentyfourbss=cmssbx10 at 24.8832pt \font\twentybss=cmssbx10 at 20.736pt \font\seventeenbss=cmssbx10 at 17.28pt \font\fourteenbss=cmssbx10 at 14.4pt \font\twelvebss=cmssbx10 at 12pt \font\tenbss=cmssbx10 \font\sevenbss=cmssbx10 at 7pt \font\twelveline=line10 at 12pt \font\twentyrm=cmr10 at 20.736pt \font\seventeensy=cmsy10 at 17.28pt \font\seventeenrm=cmr10 at 17.28pt \font\seventeenbf=cmbx10 at 17.28pt \font\seventeeni=cmmi10 at 17.28pt \font\fourteensy=cmsy10 at 14.4pt \font\fourteenrm=cmr10 at 14.4pt \font\fourteenbf=cmbx10 at 14.4pt \font\fourteeni=cmmi10 at 14.4pt \font\twelvesy=cmsy10 at 12pt \font\twelvebf=cmbx10 at 12pt \font\twelvei=cmmi10 at 12pt \font\eightbf=cmbx10 at 8pt \font\sixbf=cmbx10 at 6pt \font\twelvei=cmmi10 at 12pt \font\eighti=cmmi10 at 8pt \font\sixi=cmmi10 at 6pt \font\twelverm=cmr10 at 12pt \font\eightrm=cmr10 at 8pt \font\sixrm=cmr10 at 6pt \font\eightsy=cmsy8 \font\sixsy=cmsy6 \font\eightit=cmti8 \font\eighttt=cmtt8 \font\eightsl=cmsl8 \font\seventeenbsy=cmbsy10 at 17.28pt \font\fourteenbsy=cmbsy10 at 14.4pt \font\twelvebsy=cmbsy10 at 12pt \font\tenbsy=cmbsy10 \font\eightbsy=cmbsy8 \font\sevenbsy=cmbsy7 \font\sixbsy=cmbsy6 \font\fivebsy=cmbsy5 \font\tenmsa=msam10 \font\eightmsa=msam8 \font\sevenmsa=msam7 \font\fivemsa=msam5 \newfam\msafam \textfont\msafam=\tenmsa \scriptfont\msafam=\sevenmsa \scriptscriptfont\msafam=\fivemsa \def\msa{\fam\msafam\tenmsa} \font\fourteenmsb=msbm10 at 14.4pt \font\tenmsb=msbm10 \font\eightmsb=msbm8 \font\sevenmsb=msbm7 \font\fivemsb=msbm5 \newfam\msbfam \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb \def\Bbb{\fam\msbfam\tenmsb} \font\fourteenCal=eusm10 at 14.4pt \font\twelveCal=eusm10 at 12pt \font\tenCal=eusm10 \font\sevenCal=eusm7 \font\fiveCal=eusm5 \newfam\Calfam \textfont\Calfam=\tenCal \scriptfont\Calfam=\sevenCal \scriptscriptfont\Calfam=\fiveCal \def\Cal{\fam\Calfam\tenCal} \font\teneuf=eusm10 \font\teneuf=eufm10 \font\seveneuf=eufm7 \font\fiveeuf=eufm5 \newfam\euffam \textfont\euffam=\teneuf \scriptfont\euffam=\seveneuf \scriptscriptfont\euffam=\fiveeuf \def\euf{\fam\euffam\teneuf} \font\seventeenbfit=cmmib10 at 17.28pt \font\fourteenbfit=cmmib10 at 14.4pt \font\twelvebfit=cmmib10 at 12pt \font\tenbfit=cmmib10 \font\eightbfit=cmmib8 \font\sevenbfit=cmmib7 \font\sixbfit=cmmib6 \font\fivebfit=cmmib5 \newfam\bfitfam \textfont\bfitfam=\tenbfit \scriptfont\bfitfam=\sevenbfit \scriptscriptfont\bfitfam=\fivebfit \def\bfit{\fam\bfitfam\tenbfit} \def\gS{\hbox{\teneuf S}} % changing font sizes \catcode`\@=11 \def\eightpoint{% \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \def\rm{\fam\z@\eightrm}% \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \def\mit{\fam\@ne\eighti}% \def\oldstyle{\fam\@ne\eighti}% \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \textfont\msbfam=\eightmsb \def\Bbb{\fam\msbfam\eightmsb}% \abovedisplayskip=9pt plus 2pt minus 6pt \abovedisplayshortskip=0pt plus 2pt \belowdisplayskip=9pt plus 2pt minus 6pt \belowdisplayshortskip=5pt plus 2pt minus 3pt \smallskipamount=2pt plus 1pt minus 1pt \medskipamount=4pt plus 2pt minus 1pt \bigskipamount=9pt plus 3pt minus 3pt \normalbaselineskip=9pt \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \let\bigf@ntpc=\eightrm \let\smallf@ntpc=\sixrm \normalbaselines\rm} \catcode`\@=12 \def\eightpointbf{% \textfont0=\eightbf \scriptfont0=\sixbf \scriptscriptfont0=\fivebf \textfont1=\eightbfit \scriptfont1=\sixbfit \scriptscriptfont1=\fivebfit \textfont2=\eightbsy \scriptfont2=\sixbsy \scriptscriptfont2=\fivebsy \eightbf \baselineskip=10pt} \def\tenpointbf{% \textfont0=\tenbf \scriptfont0=\sevenbf \scriptscriptfont0=\fivebf \textfont1=\tenbfit \scriptfont1=\sevenbfit \scriptscriptfont1=\fivebfit \textfont2=\tenbsy \scriptfont2=\sevenbsy \scriptscriptfont2=\fivebsy \bf} \def\twelvepointbf{% \textfont0=\twelvebf \scriptfont0=\eightbf \scriptscriptfont0=\sixbf \textfont1=\twelvebfit \scriptfont1=\eightbfit \scriptscriptfont1=\sixbfit \textfont2=\twelvebsy \scriptfont2=\eightbsy \scriptscriptfont2=\sixbsy \twelvebss \baselineskip=14.4pt} \def\fourteenpointbf{% \textfont0=\fourteenbf \scriptfont0=\tenbf \scriptscriptfont0=\sevenbf \textfont1=\fourteenbfit \scriptfont1=\tenbfit \scriptscriptfont1=\sevenbfit \textfont2=\fourteenbsy \scriptfont2=\tenbsy \scriptscriptfont2=\sevenbsy \fourteenbss \baselineskip=17.28pt} \def\seventeenpointbf{% \textfont0=\seventeenbf \scriptfont0=\twelvebf \scriptscriptfont0=\eightbf \textfont1=\seventeenbfit\scriptfont1=\twelvebfit\scriptscriptfont1=\eightbfit \textfont2=\seventeenbsy \scriptfont2=\twelvebsy \scriptscriptfont2=\eightbsy \seventeenbss \baselineskip=20.736pt} \def\twelvepoint{% \textfont0=\twelverm \scriptfont0=\eightrm \scriptscriptfont0=\sixrm \textfont1=\twelvei \scriptfont1=\eighti \scriptscriptfont1=\sixi \textfont2=\twelvesy \scriptfont2=\eightsy \scriptscriptfont2=\sixsy \twelvess \baselineskip=14.4pt} \def\fourteenpoint{% \textfont0=\fourteenrm \scriptfont0=\tenrm \scriptscriptfont0=\sevenrm \textfont1=\fourteeni \scriptfont1=\teni \scriptscriptfont1=\seveni \textfont2=\fourteensy \scriptfont2=\tensy \scriptscriptfont2=\sevensy \fourteenss \baselineskip=17.28pt} \def\seventeenpoint{% \textfont0=\seventeenrm \scriptfont0=\twelverm \scriptscriptfont0=\eightrm \textfont1=\seventeeni \scriptfont1=\twelvei \scriptscriptfont1=\eighti \textfont2=\seventeensy \scriptfont2=\twelvesy \scriptscriptfont2=\eightsy \seventeenss \baselineskip=20.736pt} \def\twentypoint{% \textfont0=\twentyrm \scriptfont0=\fourteenrm \scriptscriptfont0=\eightrm \textfont1=\twentyi \scriptfont1=\fourteeni \scriptscriptfont1=\eighti \textfont2=\twentysy \scriptfont2=\fourteensy \scriptscriptfont2=\eightsy \twentyss \baselineskip=24.8832pt} % main item macros \newdimen\srdim \srdim=\hsize \newdimen\irdim \irdim=\hsize \def\NOSECTREF#1{\noindent\hbox to \srdim{\null\dotfill ???(#1)}} \def\SECTREF#1{\noindent\hbox to \srdim{\csname REF\romannumeral#1\endcsname}} \def\INDREF#1{\noindent\hbox to \irdim{\csname IND\romannumeral#1\endcsname}} \newlinechar=`\^^J \def\openauxfile{ \immediate\openin1\jobname.aux \ifeof1 \message{^^JCAUTION\string: you MUST run TeX a second time^^J} \let\sectref=\NOSECTREF \let\indref=\NOSECTREF \else \input \jobname.aux \message{^^JCAUTION\string: if the file has just been modified you may have to run TeX twice^^J} \let\sectref=\SECTREF \let\indref=\INDREF \fi \message{to get correct page numbers displayed in Contents or Index Tables^^J} \immediate\openout1=\jobname.aux \let\END=\end \def\end{\immediate\closeout1\END}} \newcount\notthispage \notthispage=1 \newbox\titlebox \setbox\titlebox\hbox{\hfil} \newbox\sectionbox \setbox\sectionbox\hbox{\hfil} \newbox\chapterbox \setbox\chapterbox\hbox{\hfil} \newbox\bookbox \setbox\bookbox\hbox{\hfil} \newbox\ebookbox \setbox\ebookbox\hbox{\hfil} \def\fakeskip{$\rlap{\phantom{\vrule height18pt depth2pt width0pt}}$} \def\blackline{\rlap{\vrule width \hsize height -4.7pt depth 5pt}} \def\upperblackline{\rlap{\vrule width \hsize height 11pt depth -10.7pt}} \def\foliotoc{\ifnum\pageno=\notthispage \hfil \else \ifodd\pageno \blackline% {\tenss \copy\chapterbox% \hfill\romannumeral\pageno}\else\blackline% {\tenss\romannumeral\pageno\hfill\copy\titlebox}\fi\fi} \def\foliochap{\ifnum\pageno=\notthispage \hfil \else \ifodd\pageno \blackline% {\tenss \copy\sectionbox% \hfill\number\pageno}\else\blackline% {\tenss\number\pageno\hfill\copy\chapterbox}\fi\fi} \def\folioref{\ifnum\pageno=\notthispage \hfil \else \ifodd\pageno \blackline% {\tenss \copy\chapterbox% \hfill\number\pageno}\else\blackline% {\tenss\number\pageno\hfill\copy\titlebox}\fi\fi} \def\foliopapebook{\blackline{\copy\bookbox\hfill\copy\ebookbox}} \footline={\hfil} \let\forceheader\eject \def\blankline{\phantom{}\hfil\vskip0pt} \def\chapterspacing{\phantom{$\ $}\vskip2.5cm} \def\titlerunning#1{\setbox\titlebox\hbox{\tenss #1}} \def\chapterrunning#1{\notthispage=\pageno \setbox\chapterbox\hbox{\tenss #1}} \def\title#1{\noindent\hfil$\smash{\hbox{\seventeenpointbf #1}}$\hfil \titlerunning{#1}\medskip} \def\titleleft#1{\noindent$\smash{\hbox{\seventeenpointbf #1}}$\hfil \titlerunning{#1}\medskip} \def\supersection#1{% \par\vskip1.5cm\penalty -100 \vbox{\baselineskip=17.28pt\noindent{{\fourteenpointbf #1}}} \vskip3pt \penalty 500 \titlerunning{#1}} \newcount\numbersection \numbersection=-1 \def\sectionrunning#1{\setbox\sectionbox\hbox{\tenss #1} \write1{\string\def\string\REF \romannumeral\numbersection\string{% \noexpand#1 \string\dotfill\string\space\number\pageno\string}}} \def\RGBColor#1#2{\special{color push rgb #1}#2\special{color pop}} \ifcolorized \def\maincolor#1{\RGBColor{0 0 0.5}{#1}} \else \def\maincolor#1{#1} \fi \def\colorstate#1{\noindent\maincolor{{\bf #1.}}} \def\section#1{% \removelastskip \vskip1cm\penalty -100 \vbox{\baselineskip=17.28pt\noindent{\maincolor{\seventeenpoint #1\vskip0pt}}} \vskip1pt \penalty 500 \advance\numbersection by 1 \sectionrunning{#1}} \def\subsection#1{% \removelastskip \vskip0.5cm\penalty -100 \vbox{\noindent{\maincolor{\fourteenpoint #1\vskip0pt}}} \penalty 500} \newcount\numberindex \numberindex=0 \def\index#1#2{% \advance\numberindex by 1 \write1{\string\def \string\IND #1% \romannumeral\numberindex \string{% \noexpand#2 \string\dotfill \space \number\numbersection, p.\string\ \space\number\pageno \string}}} \newcount\numberchap \iftocfirst \numberchap=-2 \else \numberchap=-1 \fi \def\CHpage{ \immediate\write1{\string\def\string\CH \romannumeral\numberchap\string{% \number\pageno\string}}} \def\chapterjump{ \vfill\eject \ifodd\pageno \else {\headline={\hfil}\null\vskip0pt\vfill\eject} \fi \advance\numberchap by 1 \CHpage } \def\numpage#1{~\dotfill~#1} \newdimen\dpp \newbox\claimbox \setbox\claimbox\hbox{\hfil} \long\def\claim#1#2\endclaim{\par\vskip 5pt\noindent {{\tenpointbf\maincolor{#1.}}\ {\it #2}\vskip-18pt} \strut\kern\hsize \ifcolorized \special{" gsave 0.8 0.8 0.1 setrgbcolor -1 -1 scale 0 6 moveto 7 0 rlineto 0 -3 rlineto -4 0 rlineto 0 -4 rlineto -3 0 rlineto closepath fill grestore}% \else \special{" gsave 0.667 setgray -1 -1 scale 0 6 moveto 7 0 rlineto 0 -3 rlineto -4 0 rlineto 0 -4 rlineto -3 0 rlineto closepath fill grestore}% \fi \par\vskip11pt} \long\def\exo#1#2\endexo{\par\vskip 5pt\noindent {\tenpointbf\maincolor{#1.}}\ {\rm #2}\par} \def\ceqno(#1){\eqno{{\maincolor{(#1)}}}} \def\cleqno(#1){\leqno{{\maincolor{(#1)}}}} \def\eqleqno#1\hfill#2{\leqno \hbox to 0.0001pt{\rlap{\rlap{\maincolor{#1}}% \kern\hsize\llap{\maincolor{#2}}}}} \def\joinrel{\mathrel{\mkern-3.5mu}} \def\llra{\relbar\joinrel\relbar\joinrel\longrightarrow} \def\llmapsto{\mapstochar\relbar\joinrel\relbar\joinrel\longrightarrow} \def\vlra#1|{\mathrel{\hbox to#1mm{\rightarrowfill}}} % Usual sets of numbers \def\bC{{\Bbb C}} \def\bK{{\Bbb K}} \def\bN{{\Bbb N}} \def\bP{{\Bbb P}} \def\bQ{{\Bbb Q}} \def\bR{{\Bbb R}} \def\bZ{{\Bbb Z}} % Calligraphic capital letters \def\cA{{\Cal A}} \def\cC{{\Cal C}} \def\cD{{\Cal D}} \def\cE{{\Cal E}} \def\cF{{\Cal F}} \def\cG{{\Cal G}} \def\cI{{\Cal I}} \def\cL{{\Cal L}} \def\cN{{\Cal N}} \def\cP{{\Cal P}} \def\cR{{\Cal R}} \def\cS{{\Cal S}} \def\cT{{\Cal T}} \def\cV{{\Cal V}} \def\tC{\smash{\tilde C}} \def\tvi{{\vrule height 10pt depth 5pt width 0pt}} \def\tv{\tvi\vrule} \def\hexnbr#1{\ifnum#1<10 \number#1\else \ifnum#1=10 A\else\ifnum#1=11 B\else\ifnum#1=12 C\else \ifnum#1=13 D\else\ifnum#1=14 E\else\ifnum#1=15 F\fi\fi\fi\fi\fi\fi\fi} \def\msatype{\hexnbr\msafam} \def\msbtype{\hexnbr\msbfam} \mathchardef\smallsetminus="2\msbtype72 \let\ssm\smallsetminus \mathchardef\restr="3\msatype16 \mathchardef\supsetneqq="3\msbtype25 \mathchardef\subsetneq="3\msbtype28 \mathchardef\supsetneq="3\msbtype29 \mathchardef\leqslant="3\msatype36 \mathchardef\geqslant="3\msatype3E \mathchardef\complement="0\msatype7B \let\ge=\geqslant \let\le=\leqslant \let\dsp=\displaystyle \def\VVert{{|}\kern-1.2pt{|}\kern-1.2pt{|}} \def\square{\maincolor{\hfill \hbox{ \vrule height 1.453ex width 0.093ex depth 0ex \vrule height 1.5ex width 1.3ex depth -1.407ex\kern-0.1ex \vrule height 1.453ex width 0.093ex depth 0ex\kern-1.35ex \vrule height 0.093ex width 1.3ex depth 0ex}}} \def\qed{\phantom{$\quad$}$\square$} \def\frac#1#2{{#1\over #2}} \def\dfrac#1#2{{#1\over #2}} \def\fracs#1#2{{\scriptstyle{#1\over#2}}} \def\fracss#1#2{{\scriptscriptstyle{#1\over#2}}} \catcode`@=11 \def\ccmalign#1#2{\null\,\vcenter{\normalbaselines\m@th \ialign{$\displaystyle ##$\hfil&&\kern#1$\displaystyle ##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} #2\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,} \def\cmalign#1{\null\,\vcenter{\normalbaselines\m@th \ialign{\hfil$##$&&$##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,} \catcode`@=12 \def\coloritem#1{\item{\maincolor{#1}}} \def\itemv{% \item{\llap{$\square\kern7pt$}\llap{$\raise0.6pt\hbox{$\times$}\kern6pt$}}} \def\itemf{% \item{\llap{$\square\kern7pt$}}} \def\bul{$\scriptstyle\bullet$} \def\bigzero{\hbox{~\seventeenrm 0~~~}} \def\branch{\raise2pt\hbox{ \vbox{\hbox{\vrule height 0.4pt depth 0pt width 12.2pt} \hbox{\raise-11.6pt\hbox{\twelveline\char"00}}}}} \def\?{\hbox{$\,$}} \def\lguil{\hbox{% \raise1pt\hbox{$\scriptscriptstyle\langle\!\langle\kern1pt$}}} \def\rguil{\hbox{% \raise1pt\hbox{$\kern1pt\scriptscriptstyle\rangle\!\rangle$}}} \def\sqind#1{\kern1.5pt\rlap{\raise5pt\hbox{$\scriptstyle#1$}}\kern-1.5pt} \def\KH{{\rm KH}} \def\Riemann{{\rm Riemann}} \def\eq{\mathop{\rm =}} \def\Bij{\mathop{\rm Bij}} \def\tr{\mathop{\rm tr}} \def\Log{\mathop{\rm Log}} \def\Mat{\mathop{\rm Mat}} \def\Ker{\mathop{\rm Ker}} \def\Vect{\mathop{\rm Vect}} \def\rang{\mathop{\rm rang}\nolimits} \def\Inv{\mathop{\rm Inv}} \def\Sp{\mathop{\rm Sp}\nolimits} \def\Arccos{\mathop{\rm Arccos}} \def\Arcsin{\mathop{\rm Arcsin}} \def\Arctan{\mathop{\rm Arctan}} \def\Argcosh{\mathop{\rm Argcosh}} \def\Argsinh{\mathop{\rm Argsinh}} \def\cotan{\mathop{\rm cotan}} \def\cotanh{\mathop{\rm cotanh}} \def\supess{\mathop{\rm sup\,ess}} \def\infess{\mathop{\rm inf\,ess}} \def\card{\mathop{\rm card}} \def\Id{\mathop{\rm Id}\nolimits} \def\Re{\mathop{\rm Re}} \def\Im{\mathop{\rm Im}} \def\Jac{\mathop{\rm Jac}} \def\Supp{\mathop{\rm Supp}} \def\oscil{\mathop{\rm oscil}} \def\vol{\mathop{\rm vol}} \def\aspect{\mathop{\rm aspect}} \def\aire{\mathop{\rm aire}} \def\longueur{\mathop{\rm longueur}} \def\diam{\mathop{\rm diam}} \def\sign{\mathop{\rm sign}} \def\grad{\mathop{\rm grad}} \def\rot{\mathop{\rm rot}} \def\div{\mathop{\rm div}} \def\Var{\mathop{\rm Var}} \def\pf{p\kern-1pt f} \def\dplus{\mathrel{\dot +}} \def\note#1#2#3{\footnote{}% {\baselineskip=8pt\leftskip=5.4mm\rlap{\strut\kern-5.4mm${}^{\rm(#2)}$}% \hyperdef#1{}{}{}{\eightpoint #3}\vskip-15pt}% ${}^{\rm(\hyperref#1{#2})}$} \def\\{\hfil\break} \def\demi{\textstyle{1\over 2}} \def\ovl{\overline} \def\ovr{\overrightarrow} \def\ul#1{$\underline{\smash{\hbox{#1}}}$} \def\build#1^#2_#3{\mathop{#1}\limits^{#2}_{#3}} \def\demo{\noindent{\it D\'emonstration.}\ } \newbox\formulabox \setbox\formulabox\hbox{\hfil} \newdimen\wdd \newdimen\htt \def\boxed#1#2{\setbox\formulabox\hbox{$\displaystyle #2$} \wdd=8pt \advance \wdd by \wd\formulabox \htt=4pt \advance \htt by \ht\formulabox \dpp=4pt \advance \dpp by \dp\formulabox \ifcolorized \smash{\rlap{\hbox{\RGBColor{1 0.92 0.94}{\kern-4pt\vrule width \wdd height \htt depth \dpp}}}}\else% \smash{\rlap{\hbox{\RGBColor{0.85 0.85 0.85}{\kern-4pt\vrule width \wdd height \htt depth \dpp}}}}\fi% \copy\formulabox} \def\boxit#1#2{\hbox{\vrule \vbox{\hrule\kern#1 \vtop{\hbox{\kern#1 #2\kern#1}% \kern#1\hrule}}% \vrule}} \def\boxmat#1#2{\boxit{#1}{$#2$}} % inclusion of PostScript files \special{header=mdrlib.ps} \long\def\InsertFig#1 #2 #3 #4\EndFig{\par \hbox{\hskip #1mm$\vbox to#2mm{\vfil\special{" #3}}#4$}} \long\def\InsertPSFile#1 #2 #3 #4 #5 #6\EndFig{\par \hbox{\hskip #1mm$\vbox to#2mm{% \vfil\special{psfile=#5 hscale=#3 vscale=#4}}#6$}} \long\def\InsertImage#1 #2 #3 #4 #5 #6 #7 #8\EndFig{\par \hbox{\hskip #1mm$\vbox to#2mm{% \vfil\special{psfile="`img2eps file #7 height #4 mm width #3 mm gamma #5 angle #6}}#8$}} \long\def\LabelTeX#1 #2 #3\ELTX{\rlap{\kern#1mm\raise#2mm\hbox{#3}}} \def\psrotate#1#2{\special{ps: /a{pop pop}def gsave #1 neg rotate}#2 \special{ps: grestore /a{moveto}bind def}} % bibibliography \def\bibitem#1&% {\hangindent=1.2cm\hangafter=1 \noindent\rlap{\hbox{\eightpointbf #1}}\kern1.2cm{\rm #2}{\it #3}{\rm #4.}} %%% Texte principal \setbox\bookbox\hbox{\tenss \maincolor{% Analyse num\'erique et \'equations diff\'erentielles}} \setbox\ebookbox\hbox{\tenss \maincolor{% \href{https://grenoble-sciences.ujf-grenoble.fr/pap-ebook/demailly/3-integration-numerique}{\RGBColor{0 0 1}{$\strut\,\Rightarrow$ site web compagnon$\,$}}}} \footline={\upperblackline \maincolor{% \tenss Solution des probl\`emes du chap.~3\hfill \copyright\ Grenoble Sciences, J.-P.~Demailly $\scriptstyle \bullet$ \folio}} \openauxfile \headline={\foliopapebook} \notthispage=\pageno \strut\vskip4mm \maincolor{% \title{Solution des probl\`emes du chapitre 3}} \titlerunning{\maincolor{% Solutions des probl\`emes du chapitre 3}} \vskip10mm \maincolor{{\bf 6.11.}} On consid\`ere une m\'ethode de quadrature \'el\'ementaire obtenue par interpolation d'une fonction $f\in\cC^{l+2}([\alpha,\beta])$ en des points $\tau_j\in[\alpha,\beta]$, $0\le j\le l$, et on utilise la m\'ethode des diff\'erences divis\'ees pour \'evaluer l'erreur d'interpolation $x\mapsto f(x)-p_l(x)$.\medskip {\bf(a)} Montrons par r\'ecurrence sur $k$ la formule g\'en\'erale $$ f[x_0,x_1\ldots,x_k]=\sum_{j=0}^k{f(x_j)\over\prod_{i\ne j}(x_j-x_i)}. $$ La formule est vraie pour $k=0$ (le produit est alors vide et vaut $1$ par convention). Pour $k=1$, on a bien aussi $$ f[x_0,x_1]={f(x_1)-f(x_0)\over x_1-x_0}= {f(x_0)\over x_0-x_1}-{f(x_1)\over x_1-x_0}. $$ Supposons la formule vraie \`a l'ordre $k-1$, $k\ge 1$. Alors par d\'efinition (voir II.1.3) $$ \eqalign{ f[x_0,x_1\ldots,x_k]&= {f[x_1,x_2\ldots,x_k]-f[x_0,x_1\ldots,x_{k-1}]\over x_k-x_0}\cr &= {1\over x_k-x_0}\Bigg( \sum_{j=1}^k{f(x_j)\over\prod_{i>0,i\ne j}(x_j-x_i)} -\sum_{j=0}^{k-1}{f(x_j)\over\prod_{i0,i\ne j}(x_j-x_i)}- {1\over\prod_{i0$ pour $x\in{}]-1,1[$. Une int\'egration par parties dans l'expression de $E(f)$ obtenue au (d) donne $$ E(f)=\Big[w(x)f[\tau_0,\tau_1,\ldots,\tau_l,x]\Big]_{-1}^1- \int_{-1}^1w(x)\,{d\over dx}f[\tau_0,\tau_1,\ldots,\tau_l,x]\,dx, $$ et, compte tenu de ce que $w(-1)=w(1)=0$, l'identit\'e du (c) implique $$ E(f)=-\frac{1}{(l+2)!}\int_{-1}^1w(x)\,f^{(l+2)}(\xi_x)\,dx.\leqno(**) $$ Les fonctions mises en jeu sont continues (et m\^eme $\cC^{l+1}$). La formule de la moyenne pour le poids $w(x)>0$ montre qu'il existe $\eta\in[-1,1]$ tel que $$ E(f)=-{1\over(l+2)!}\,f^{(l+2)}(\eta)\,A\quad\hbox{o\`u}\quad A=\int_{-1}^1w(x)\,dx>0. $$ Une nouvelle int\'egration par parties donne $$ A=[t\,w(t)]_{-1}^1-\int_{-1}^1t\,w'(t)\,dt= -\int_{-1}^1t\prod_{j=0}^l(t-\tau_j)\,dt. $$ Le changement de variable $t={u\over n}$, $u\in[-n,n]$, $\tau_{n+j}={j\over n}$, $-n\le j\le n$, donne maintenant $$ A=-{1\over n^{2n+3}}\int_{-n}^nu\prod_{j=-n}^{+n}(u-j)= -{1\over n^{2n+3}}\int_{-n}^nu^2(u^2-1)\ldots (u^2-n^2)\,du. $$ En prenant $f(x)=x^k$, $k\le l+2$, on voit gr\^ace \`a $(**)$ que l'erreur $E(f)$ est nulle pour $k\le l+1$ et strictement n\'egative (\'egale \`a $-A$) pour $k=l+1$. La m\'ethode de Newton-Cotes NC${}_l$ est donc d'ordre $N=l+1$ exactement. On sait dans ces conditions que l'erreur s'exprime \`a l'aide du noyau de Peano $K_N=K_{l+1}$ par la formule $$ E(f)=\frac{1}{(l+1)!}\int^1_{-1}K_{l+1}(t)f^{(l+2)}(t)dt. $$ En prenant $f(x)=x^{l+2}$ il vient $E(f)=-A$ et $\int_ {-1}^1K_{l+1}(t)\,dt=-A/(l+2)$. {\bf(f)} Supposons que le noyau de Peano ne soit pas de signe constant. C'est une fonction polynomiale par morceaux, donc les points o\`u le signe change sont en nombre fini~$m$. Il existe une fonction continue $g_\varepsilon\in\cC([-1,1])$ comprise entre $-1$ et $1$ telle que $$g_\varepsilon(t)=\pm 1~\hbox{ou}~0=\hbox{signe de}~K_{l+1}(t)$$ en dehors d'un intervalle de longueur $\varepsilon/m$ autour de chaque point o\`u le signe de $K_{l+1}$ change. On en d\'eduit $$\int_{-1}^1K_{l+1}(t)g_\varepsilon(t)\,dt\ge \int_{-1}^1|K_{l+1}(t)|\,dt - C\varepsilon\quad\hbox{o\`u}\quad C=\sup_{[-1,1]}|K_N|.$$ Choisissons une fonction $f_\varepsilon$ telle que $f_\varepsilon^{(l+2)}= g_\varepsilon$. Alors l'erreur d'interpolation v\'erifierait d'une part $E(f_\varepsilon)=\frac{1}{(l+2)!}g_\varepsilon(\eta) A$, $\eta\in[-1,1]$, donc $|E(f_\varepsilon)|\le \frac{1}{(l+2)!}A$, et d'autre part $$ E(f_\varepsilon)=\frac{1}{(l+1)!}\int^1_{-1}K_{l+1}(t)g_\varepsilon(t)dt\ge \frac{1}{(l+1)!}\int_{-1}^1|K_{l+1}(t)|\,dt - \frac{1}{(l+1)!}C\varepsilon. $$ Cependant, si $K_{l+1}$ change de signe, on a $$ \int_{-1}^1|K_{l+1}(t)|\,dt>\left|\int_{-1}^1K_{l+1}(t)\,dt\right|=\frac{1}{l+2}A, $$ ce qui est une contradiction pour $\varepsilon$ assez petit. Par cons\'equent $K_{l+1}(t)$ est de signe constant et comme son int\'egrale est n\'egative on en conclut que $K_{l+1}(t)\le 0$ sur $[-1,1]$. {\bf(g)} Dans le cas o\`u $l=2n+1$ est impair avec un pas $h={2\over 2n+1}$, on peut reprendre les calculs pr\'ec\'edents et observer que l'on a apr\`es simplification de $(x-\tau_l)$~: $$ f(x)-p_l(x)=\prod_{j=0}^{l-1}(x-\tau_j)\,\big( f[\tau_0,\tau_1,\ldots,\tau_{l-1},x]-f[\tau_0,\tau_1,\ldots,\tau_{l-1},\tau_l] \big). $$ On prend cette fois $w(x)=\int_{-1}^x\prod_{j=0}^{l-1}(t-\tau_j)\,dt$. L'erreur d'interpolation est alors donn\'ee par $$ \eqalign { E(f)&=\int_{-1}^1\big(f(x)-p_l(x)\big)\,dx\cr &=\int_{-1}^1w'(x)\big(f[\tau_0,\tau_1,\ldots,\tau_{l-1},x]- f[\tau_0,\tau_1,\ldots,\tau_{l-1},\tau_l]\big)\,dx.\cr} $$ Une int\'egration par parties donne $$ E(f)= -\int_{-1}^1w(x){d\over dx}f[\tau_0,\tau_1,\ldots,\tau_{l-1},x]\,dx, $$ et on en d\'eduit comme dans (f) (avec le point d'interpolation $\tau_l$ en moins) qu'il existe $\eta\in[\alpha,\beta]$ tel que $$ E(f)=-{1\over(l+1)!}\,f^{(l+1)}(\eta)\,A,\qquad A=\int_{-1}^1w(x)\,dx. $$ Il r\'esulte de la question (f) que $w(x)>0$ sur $]-1,1-h[$~: en effet $\tau_0,\tau_1,\ldots,\tau_{l-1}$ forment une subdivision de pas constant de $[-1,1-h]$ en $2n$ intervalles, et on peut simplement faire un changement de variable affine pour ramener l'intervalle $[-1,1-h]$ \`a $[-1,1]$. On voit ainsi que $w(1-h)=0$, avec une sym\'etrie non \'evidente par rapport au point $-h/2$. Comme l'int\'egrande de $w(x)$ est positif aussi sur l'intervalle restant $]1-h,1]$, on a encore $w(x)>0$ sur $]1-h,1]$. Les arguments de (f) et (g) restent applicables, au remplacement pr\`es de $l$ par $(l-1)$, et on trouve ainsi que $$ E(f)=-\frac{1}{(l+1)!}\int_{-1}^1w(x)\,f^{(l+1)}(\xi_x)\,dx $$ pour une certaine fonction $\xi_x\in[-1,1]$. La m\'ethode est donc d'ordre $l$ exactement. Le noyau de Peano est de signe constant, n\'egatif ou nul, et on a $$ \int_{-1}^1K_l(t)\,dt=-\frac{1}{l+1}A,\qquad A=\int_{-1}^1w(x)dx>0. $$ Enfin, une int\'egration par parties fournit $$ A=[(x-1)w(x)]_{-1}^1-\int_{-1}^1(x-1)w'(x)\,dx=-\int_{-1}^1\prod_{j=0}^l(x-\tau_j)\,dx. $$ Le changement de variable et d'indice $t=(2u-1)/(2n+1)$, $u\in[-n,n+1]$, $\tau_{j+n}=(2j-1)/(2n+1)$, $-n\le j\le n+1$, fournit l'expression explicite $$ \eqalign{ A&=-{2^{2n+3}\over (2n+1)^{2n+3}}\int_{-n}^{n+1}\prod_{j=-n}^{n+1}(u-j)\,du\cr &={2^{2n+3}\over (2n+1)^{2n+3}}\int_{-n}^{n+1}u(n+1-u)(u^2-1)\ldots(u^2-n^2)\,du \cr} $$ (sous cette forme, il n'est pas \'evident que $A>0$~!) \end % Local Variables: % TeX-command-default: "uTeX" % End: