%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % This file is divided into 2 parts by a ---cut here--- line. % % The 1st part is a preprint in plain TeX (279K). % The 2nd part is a uuencoded file (2119K) containing the source % code of a F77 program and 2 input data files. % % To extract the program and data files: % - Save everything that immediately follows the ---cut here--- line % into a separate file, say supplement.uu % - Type: uudecode supplement.uu % This produces the compressed file Section.8.tar.gz % - On a Unix system, type: gunzip -c Section.8.tar.gz | tar xvf - % This produces a directory Section.8/ containing the program % and data files. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \magnification \magstep1 \newdimen\papwidth \newdimen\papheight \papwidth=16truecm \papheight=22truecm \voffset=0.8truecm \hoffset=0.1truecm \hsize=\papwidth \vsize=\papheight \advance\vsize by 0.2truecm \advance\topskip by 0.2truecm \newdimen\texpscorrection \texpscorrection=0truecm %must be 0.15truecm in ps_fonts \def\em{\sl} %will be italic in reality %============================================================= % Fonts %================================================================= \font\unige=cmbx10 scaled\magstep3 \font\huge=cmbx10 scaled\magstep2 \font\xbold=cmbx10 scaled\magstep1 \font\mbold=cmmi10 scaled\magstep1 \font\sub=cmbx10 \font\comptr=cmtt10 \font\smallcomptr=cmtt8 \font\plgt=cmtt10 at 10truept \font\tencm=cmcsc10 scaled 900 \font\twelverm=cmr12 \font\twelvei=cmmi12 \font\twelvesy=cmsy10 scaled\magstep1 \font\twelveex=cmex10 scaled\magstep1 \font\twelveit=cmti12 \font\twelvett=cmtt12 \font\twelvebf=cmbx12 \font\twelvesl=cmsl12 \font\tenmsb=msbm10 \font\sevenmsb=msbm7 \font\fivemsb=msbm5 \newfam\msbfam \font\ninerm=cmr9 \font\ninesy=cmsy9 \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightsl=cmsl8 \font\eightit=cmti8 \font\sixrm=cmr6 \font\sixbf=cmbx6 \font\sixi=cmmi6 \font\sixsy=cmsy6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% the following fonts force true %%%% %%% computer modern behaviour %%%% %%% they are used to override %%%% %%% the ps-fonts for math symbols %%%% %%% like \dot \ne .... %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newfam\truecmr \newfam\truecmsy \font\twelvetruecmr=cmr10 scaled\magstep1 \font\twelvetruecmsy=cmsy10 scaled\magstep1 \font\tentruecmr=cmr10 \font\tentruecmsy=cmsy10 \font\eighttruecmr=cmr8 \font\eighttruecmsy=cmsy8 \font\seventruecmr=cmr7 \font\seventruecmsy=cmsy7 \font\sixtruecmr=cmr6 \font\sixtruecmsy=cmsy6 \font\fivetruecmr=cmr5 \font\fivetruecmsy=cmsy5 %%%% add the definitions for 10pt %%%%%%%% \textfont\truecmr=\tentruecmr \scriptfont\truecmr=\seventruecmr \scriptscriptfont\truecmr=\fivetruecmr \textfont\truecmsy=\tentruecmsy \scriptfont\truecmsy=\seventruecmsy \scriptscriptfont\truecmr=\fivetruecmr \scriptscriptfont\truecmsy=\fivetruecmsy %%%%% size changes%%%%%% \newskip\ttglue \def \eightpoint{\def\rm{\fam0\eightrm}% switch to 8-point type \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \tt \ttglue=.5em plus.25em minus.15em \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \normalbaselineskip=9pt \let\sc=\sixrm \let\big=\eightbig \normalbaselines\rm \textfont\truecmr=\eighttruecmr \scriptfont\truecmr=\sixtruecmr \scriptscriptfont\truecmr=\fivetruecmr \textfont\truecmsy=\eighttruecmsy \scriptfont\truecmsy=\sixtruecmsy } \catcode`@=12 %%%%%constant subscript positions%%%%% \fontdimen16\tensy=2.7pt \fontdimen13\tensy=4.3pt \fontdimen17\tensy=2.7pt \fontdimen14\tensy=4.3pt \fontdimen18\tensy=4.3pt \fontdimen16\eightsy=2.7pt \fontdimen13\eightsy=4.3pt \fontdimen17\eightsy=2.7pt \fontdimen14\eightsy=4.3pt %%%%%%%%%%%%%%%%%%%%%%%% % footnote %%%%%%%%%%%%%%%%%%%%%%%% \catcode`@=11 \def\footnote#1{\let\@sf\empty % parameter #2 (the text) is read later \ifhmode\edef\@sf{\spacefactor\the\spacefactor}\/\fi #1\@sf\vfootnote{#1}} \def\vfootnote#1{\insert\footins\bgroup\eightpoint \interlinepenalty\interfootnotelinepenalty \splittopskip\ht\strutbox % top baseline for broken footnotes \splitmaxdepth\dp\strutbox \floatingpenalty\@MM \leftskip\z@skip \rightskip\z@skip \spaceskip\z@skip \xspaceskip\z@skip \textindent{#1}\footstrut\futurelet\next\fo@t} \def\fo@t{\ifcat\bgroup\noexpand\next \let\next\f@@t \else\let\next\f@t\fi \next} \def\f@@t{\bgroup\aftergroup\@foot\let\next} \def\f@t#1{#1\@foot} \def\@foot{\strut\egroup} \def\footstrut{\vbox to\splittopskip{}} \skip\footins=\bigskipamount % space added when footnote is present \count\footins=1000 % footnote magnification factor (1 to 1) \dimen\footins=8in % maximum footnotes per page \catcode`@=12 % at signs are no longer letters %============================================================== % Titles %================================================================= \count5=0 %section \count6=1 %equation \count7=1 %claim \count8=1 %subsection \def\title#1\par{\centerline{\huge #1}} \def\abstract#1\par{\noindent{\sub Abstract.} #1 \par} \def\proof{\medskip\noindent{\bf Proof.\ }} \def\preuve{\medskip\noindent{\bf Preuve.\ }} \def\qed{\hfill\vskip 1pt \line{\hfill\vrule height 1.8ex width 2ex depth +.2ex \ \ \ \ \ \ }\bigskip} \def\remark{\medskip\noindent{\bf Remark.\ }} \def\notation{\medskip\noindent{\bf Notations.\ }} \def\equs(#1){(\csname e#1\endcsname)} \def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax} \def\equation(#1){\ifx\draft\undefined \eqno(\tag(#1)) \else \eqno{{\rm #1}(\tag(#1))}\fi} \def\equ(#1){\ifundefined{e#1}$\spadesuit$#1\else (\csname e#1\endcsname)\fi} \def\equat(#1){\ifx\draft\undefined \hfil(\tag(#1)) \else \hfil{{\rm #1}(\tag(#1))}\fi} \def\aequation(#1){\ifx\draft\undefined \eqno(\atag(#1)) \else \eqno{{\rm #1}(\atag(#1))}\fi} \def\aequat(#1){\ifx\draft\undefined \hfil(\atag(#1)) \else \hfil{{\rm #1}(\atag(#1))}\fi} \def\equationp(#1){\ifx\draft\undefined \eqno(\csname e#1\endcsname') \else \eqno{{\rm #1}(\csname e#1\endcsname')}\fi} \def\equp(#1){\ifundefined{e#1}$\spadesuit$#1\else (\csname e#1\endcsname')\fi} \def\newdef #1,#2,#3 {\ifundefined{#1#2} \expandafter\xdef\csname #1#2\endcsname{#3}\else \write16{!!!!!doubly defined #1,#2}\fi} \def\tag(#1){\number\count5.\number\count6 \newdef e,#1,\number\count5.\number\count6 \global\advance\count6 by 1} \def\atag(#1){{\rm A.}\number\count6 \newdef e,#1,A.\number\count6 \global\advance\count6 by 1} \def\claim #1(#2) #3\par{ \vskip.1in\medbreak\noindent {\ifx\draft\undefined\else\llap{(#2)}\fi \bf #1\ \number\count5.\number\count7.\ }{\sl #3}\par \newdef c,#2,#1~{\number\count5.\number\count7} \global\advance\count7 by 1 \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi} \def\claimrm #1(#2) #3\par{ \vskip.1in\medbreak\noindent {\ifx\draft\undefined\else\llap{(#2)}\fi \bf #1\ \number\count5.\number\count7.\ }{\rm #3}\par \newdef c,#2,#1~{\number\count5.\number\count7} \global\advance\count7 by 1 \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi} \def\clm(#1){\ifundefined{c#1}$\spadesuit$#1\else\csname c#1\endcsname\fi} \def\section#1\par{\vskip0pt plus.3\vsize\penalty-75 \vskip0pt plus -.3\vsize\bigskip\bigskip \global\advance\count5 by 1 \message{#1}\leftline {\xbold \number\count5.\ #1} \count6=1 \count7=1 \count8=1 \nobreak\smallskip\noindent} \def\sectionnonr#1\par{\vskip0pt plus.3\vsize\penalty-75 \vskip0pt plus -.3\vsize\bigskip\bigskip \global\advance\count5 by 1 \message{#1}\leftline {\xbold #1} \count6=1 \count7=1 \count8=1 \nobreak\smallskip\noindent} \def\sec(#1){\ifundefined{s#1}$\spadesuit$#1\else Section \csname s#1\endcsname\fi} \def\subsection#1\par{\vskip0pt plus.2\vsize\penalty-75 \vskip0pt plus -.2\vsize\bigskip\bigskip \message{#1}\leftline{\sub \number\count5.\number\count8.\ #1} \global\advance\count8 by 1 \nobreak\smallskip\noindent} %====================================================== %figure psfile height (in cm) width (in cm) caption (will be centered) %================================================================= \newcount\FIGUREcount \FIGUREcount=0 \newdimen\figcenter \def\figure #1 #2 #3 #4\cr{\null\ifundefined{fig#1}\global \advance\FIGUREcount by 1\newdef fig,#1,{Fig.~\number\count5.\number\FIGUREcount}\fi \write16{ FIG \number\count5.\number\FIGUREcount: #1} {\goodbreak\figcenter=\hsize\relax \advance\figcenter by -#3truecm \divide\figcenter by 2 \midinsert\vskip #2truecm\noindent\hskip\figcenter \special{psfile=#1}\vskip 0.8truecm\noindent \vbox{\eightpoint\noindent {\bf\fig(#1)}: #4}\endinsert}} \def\figurewithtex #1 #2 #3 #4 #5\cr{\null\ifundefined{fig#1}\global \advance\FIGUREcount by 1\newdef fig,#1,{Fig.~\number\count5.\number\FIGUREcount}\fi \write16{ FIG \number\count5.\number\FIGUREcount: #1} {\goodbreak\figcenter=\hsize\relax \advance\figcenter by -#4truecm \divide\figcenter by 2 \midinsert\vskip #3truecm\noindent\hskip\figcenter \special{psfile=#1}{\hskip\texpscorrection\input #2 } \vskip 0.8truecm\noindent \vbox{\eightpoint\noindent {\bf\fig(#1)}: #5}\endinsert}} \def\fig(#1){\ifundefined{fig#1}\global \advance\FIGUREcount by 1\newdef fig,#1,{Fig.~\number\count5.\number\FIGUREcount} \fi \csname fig#1\endcsname\relax} %================================================================= % Macros %================================================================= \def\\{\backslash} \def\ssmallskip{\vskip 0pt} \def\supess{\mathop{\rm sup\,ess}} \def\...{{$\cdot\cdot\cdot$}} \footline{\ifnum\pageno=0\hss\else\hss\tenrm\folio\hss\fi} \let\cl=\centerline \def\mean{{\rm I\kern-.18em E}} \def\natural{{\rm I\kern-.18em N}} \def\integer{{\rm Z\kern-.32em Z}} \def\real{{\rm I\kern-.2em R}} \def\dn{|\kern-.08em\|} \def\complex{\kern-.1em{\raise.47ex\hbox{ $\scriptscriptstyle |$}}\kern-.40em{\rm C}} \def\I{|\hskip-0.08em |\hskip-0.08em |} \def\Id{{\rm I}} \def\Int{\int\limits_{-\infty}^\infty\!} \def\HT{_{\scriptscriptstyle HT}} \def\UV{_{\scriptscriptstyle UV}} \def\IR{_{\scriptscriptstyle IR}} \def\AA{{\cal A}} \def\BB{{\cal B}} \def\CC{{\cal C}} \def\DD{{\cal D}} \def\EE{{\cal E}} \def\FF{{\cal F}} \def\GG{{\cal G}} \def\HH{{\cal H}} \def\II{{\cal I}} \def\JJ{{\cal J}} \def\KK{{\cal K}} \def\LL{{\cal L}} \def\MM{{\cal M}} \def\NN{{\cal N}} \def\OO{{\cal O}} \def\PP{{\cal P}} \def\QQ{{\cal Q}} \def\RR{{\cal R}} \def\SS{{\cal S}} \def\TT{{\cal T}} \def\UU{{\cal U}} \def\VV{{\cal V}} \def\WW{{\cal W}} \def\XX{{\cal X}} \def\YY{{\cal Y}} \def\ZZ{{\cal Z}} \def\restr#1{|_{\textstyle#1}} \def\t#1{{\hbox{\comptr#1}}} \def\lampka{{\lambda^+,\kappa}} \def\lampde{{\lambda^+,\delta}} \def\lamp{{\lambda^+}} \def\lamm{{\lambda^-}} \def\zeet{{\zeta\eta}} \def\sigtau{{\sigma\tau}} \def\mn{{\mu\nu}} \def\nm{{\nu\mu}} \def\gabe{{\gamma\beta}} \def\lamka{{\lambda,\kappa}} \def\lamde{{\lambda,\delta}} \def\alal{{\alpha\alpha}} \def\albe{{\alpha\beta}} \def\beal{{\beta\alpha}} \def\stdba#1#2{{\rm std}({\cal B}_{#1#2})} \def\std#1{{\rm std}(#1)} \def\banach#1#2{{\cal B}_{#1#2}} \def\veps{\varepsilon} \def\supp{\mathop{\rm supp}\nolimits} \def\Ker{\mathop{\rm Ker}\nolimits} \def\tflog{\mathop{\rm Log}\nolimits} \def\log{\mathop{\rm log}\nolimits} %================================================================= % BIBLIOGRAPHY %================================================================= \def\period{\unskip.\spacefactor3000 { }} \newbox\nobox \newbox\bybox \newbox\paperbox \newbox\yrbox \newbox\jourbox \newbox\pagesbox \newbox\volbox \newbox\preprintbox \newbox\toappearbox \newbox\bookbox \newbox\bybookbox \newbox\publisherbox \newbox\inprintbox \def\refclear{ \setbox\nobox=\null \gdef\isno{F} \setbox\bybox=\null \gdef\isby{F} \setbox\paperbox=\null \gdef\ispaper{F} \setbox\yrbox=\null \gdef\isyr{F} \setbox\jourbox=\null \gdef\isjour{F} \setbox\pagesbox=\null \gdef\ispages{F} \setbox\volbox=\null \gdef\isvol{F} \setbox\preprintbox=\null \gdef\ispreprint{F} \setbox\toappearbox=\null \gdef\istoappear{F} \setbox\inprintbox=\null \gdef\isinprint{F} \setbox\bookbox=\null \gdef\isbook{F} \gdef\isinbook{F} \setbox\bybookbox=\null \gdef\isbybook{F} \setbox\publisherbox=\null \gdef\ispublisher{F} } \def\ref{\refclear\bgroup} \def\no {\egroup\gdef\isno{T}\setbox\nobox=\hbox\bgroup} \def\by {\egroup\gdef\isby{T}\setbox\bybox=\hbox\bgroup} \def\paper{\egroup\gdef\ispaper{T}\setbox\paperbox=\hbox\bgroup} \def\yr{\egroup\gdef\isyr{T}\setbox\yrbox=\hbox\bgroup} \def\jour{\egroup\gdef\isjour{T}\setbox\jourbox=\hbox\bgroup} \def\pages{\egroup\gdef\ispages{T}\setbox\pagesbox=\hbox\bgroup} \def\vol{\egroup\gdef\isvol{T}\setbox\volbox=\hbox\bgroup\bf} \def\preprint{\egroup\gdef \ispreprint{T}\setbox\preprintbox=\hbox\bgroup} \def\toappear{\egroup\gdef \istoappear{T}\setbox\toappearbox=\hbox\bgroup} \def\inprint{\egroup\gdef \isinprint{T}\setbox\inprintbox=\hbox\bgroup} \def\book{\egroup\gdef\isbook{T}\setbox\bookbox=\hbox\bgroup\em} \def\publisher{\egroup\gdef \ispublisher{T}\setbox\publisherbox=\hbox\bgroup} \def\inbook{\egroup\gdef\isinbook{T}\setbox\bookbox=\hbox\bgroup\em} \def\bybook{\egroup\gdef\isbybook{T}\setbox\bybookbox=\hbox\bgroup} \def\endref{\egroup \sfcode`.=1000 \if T\isno \item{[\unhbox\nobox\unskip]} \else \noindent \fi \if T\isby \unhbox\bybox\unskip: \fi \if T\ispaper \unhbox\paperbox\unskip\period \fi \if T\isbook {\it\unhbox\bookbox\unskip}\if T\ispublisher, \else. \fi\fi \if T\isinbook In {\it\unhbox\bookbox\unskip}\if T\isbybook, \else\period \fi\fi \if T\isbybook (\unhbox\bybookbox\unskip)\period \fi \if T\ispublisher \unhbox\publisherbox\unskip \if T\isjour, \else\if T\isyr \ \else\period \fi\fi\fi \if T\ispreprint Pre\-print\period \fi \if T\isjour \unhbox\jourbox\unskip\ \fi \if T\istoappear (to appear)\period \fi \if T\isvol \unhbox\volbox\unskip\if T\ispages, \else\ \fi\fi \if T\ispages \unhbox\pagesbox\unskip\ \fi \if T\isyr (\unhbox\yrbox\unskip)\period \fi \if T\isinprint (in print)\period \fi \filbreak } %================================================================= % PSDUMP.TEX %================================================================= % TeX macros for dumping included Postscript to files. % Adapted from Knuth's \answer macro in the TeXbook. % Requires Plain TeX. Maybe other flavors will work too? % Jamie Stephens, jamies@math.utexas.edu, 28 Nov 94 \def\endofps{EndOfTheIncludedPostscriptMagicCookie} \chardef\other=12 \newwrite\psdumphandle \outer\def\psdump#1{\par\medbreak \immediate\openout\psdumphandle=#1 \copytoblankline} \def\copytoblankline{\begingroup\setupcopy\copypsline} \def\setupcopy{\def\do##1{\catcode`##1=\other}\dospecials \catcode`\\=\other \obeylines} {\obeylines \gdef\copypsline#1 {\def\next{#1}% \ifx\next\endofps\let\next=\endgroup % \else\immediate\write\psdumphandle{\next} \let\next=\copypsline\fi\next}} \outer\def\closepsdump{ \immediate\closeout\psdumphandle} % EXAMPLE (remove the leading % signs to make it work): % %\psdump{example.ps}These three lines %are going be dumped "as is" %to the file example.ps %EndOfTheIncludedPostscriptMagicCookie %\closepsdump %================================================================ \psdump{fth1.ps}%! /jpedict 250 dict def jpedict begin /Cou /Times-Roman findfont 50.000000 scalefont 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m 202 208 l s 202 208 m 203 208 l s 203 208 m 204 208 l s 204 208 m 204 209 l s 204 209 m 205 209 l s 205 209 m 206 209 l s 206 209 m 207 209 l s 207 209 m 208 210 l s 208 210 m 209 210 l s 209 210 m 210 210 l s 210 210 m 211 211 l s 211 211 m 211 211 l s 211 211 m 212 211 l s 212 211 m 213 211 l s 213 211 m 214 212 l s 214 212 m 215 212 l s 215 212 m 216 212 l s 216 212 m 217 212 l s 217 212 m 218 213 l s 218 213 m 219 213 l s 219 213 m 219 213 l s 219 213 m 220 214 l s 220 214 m 221 214 l s 221 214 m 222 214 l s 222 214 m 223 214 l s 223 214 m 224 215 l s 224 215 m 225 215 l s 225 215 m 226 215 l s 226 215 m 227 215 l s 227 215 m 227 216 l s 227 216 m 228 216 l s 228 216 m 229 216 l s 229 216 m 230 217 l s 230 217 m 231 217 l s 231 217 m 232 217 l s 232 217 m 233 217 l s 233 217 m 234 218 l s 234 218 m 235 218 l s 235 218 m 236 218 l s 236 218 m 236 218 l s 236 218 m 237 219 l s 237 219 m 238 219 l s 238 219 m 239 219 l s 239 219 m 240 220 l s 240 220 m 241 220 l s 241 220 m 242 220 l s 242 220 m 243 220 l s 243 220 m 244 221 l s 244 221 m 245 221 l s 245 221 m 246 221 l s 246 221 m 247 221 l s 247 221 m 248 222 l s 248 222 m 248 222 l s 248 222 m 249 222 l s 249 222 m 250 223 l s 250 223 m 251 223 l s 251 223 m 252 223 l s 252 223 m 253 223 l s 253 223 m 254 224 l s 254 224 m 255 224 l s 255 224 m 256 224 l s 256 224 m 257 224 l s 257 224 m 258 225 l s 258 225 m 259 225 l s 259 225 m 260 225 l s 260 225 m 261 226 l s 261 226 m 262 226 l s 262 226 m 263 226 l s 263 226 m 264 226 l s 264 226 m 264 227 l s 264 227 m 265 227 l s 265 227 m 266 227 l s 266 227 m 267 227 l s 267 227 m 268 228 l s 268 228 m 269 228 l s 269 228 m 270 228 l s 270 228 m 271 229 l s 271 229 m 272 229 l s 272 229 m 273 229 l s 273 229 m 274 229 l s 274 229 m 275 230 l s 275 230 m 276 230 l s 276 230 m 277 230 l s 277 230 m 278 230 l s 278 230 m 279 231 l s 279 231 m 280 231 l s 280 231 m 281 231 l s 281 231 m 282 232 l s 282 232 m 283 232 l s 283 232 m 284 232 l s 284 232 m 284 232 l s 284 232 m 285 233 l s 285 233 m 286 233 l s 286 233 m 287 233 l s 287 233 m 288 233 l s 288 233 m 289 234 l s 289 234 m 290 234 l s 290 234 m 291 234 l s 291 234 m 292 235 l s 292 235 m 293 235 l s 293 235 m 294 235 l s 294 235 m 295 235 l s 295 235 m 296 236 l s 296 236 m 297 236 l s 297 236 m 298 236 l s 298 236 m 299 236 l s 299 236 m 300 237 l s 300 237 m 300 237 l s 300 237 m 301 237 l s 301 237 m 302 238 l s 302 238 m 303 238 l s 303 238 m 304 238 l s 304 238 m 305 238 l s 305 238 m 306 239 l s 306 239 m 307 239 l s 307 239 m 308 239 l s 308 239 m 309 240 l s 309 240 m 310 240 l s 310 240 m 310 240 l s 310 240 m 311 240 l s 311 240 m 312 241 l s 312 241 m 313 241 l s 313 241 m 314 241 l s 314 241 m 315 241 l s 315 241 m 316 242 l s 316 242 m 317 242 l s 317 242 m 317 242 l s 317 242 m 318 243 l s 318 243 m 319 243 l s 319 243 m 320 243 l s 320 243 m 321 243 l s 321 243 m 322 244 l s 322 244 m 323 244 l s 323 244 m 323 244 l s 323 244 m 324 244 l s 324 244 m 325 245 l s 325 245 m 326 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356 257 m 357 258 l s 357 258 m 357 258 l s 357 258 m 358 258 l s 358 258 m 358 258 l s 358 258 m 359 259 l s 359 259 m 359 259 l s 359 259 m 360 259 l s 360 259 m 360 259 l s 360 259 m 361 260 l s 361 260 m 361 260 l s 361 260 m 362 260 l s 362 260 m 362 261 l s 362 261 m 362 261 l s 362 261 m 363 261 l s 363 261 m 363 261 l s 363 261 m 364 262 l s 364 262 m 364 262 l s 364 262 m 364 262 l s 364 262 m 365 262 l s 365 262 m 365 263 l s 365 263 m 365 263 l s 365 263 m 366 263 l s 366 263 m 366 264 l s 366 264 m 366 264 l s 366 264 m 367 264 l s 367 264 m 367 264 l s 367 264 m 367 265 l s 367 265 m 368 265 l s 368 265 m 368 265 l s 368 265 m 368 265 l s 368 265 m 368 266 l s 368 266 m 369 266 l s 369 266 m 369 266 l s 369 266 m 369 267 l s 369 267 m 369 267 l s 369 267 m 370 267 l s 370 267 m 370 267 l s 370 267 m 370 268 l s 370 268 m 370 268 l s 370 268 m 370 268 l s 370 268 m 370 268 l s 370 268 m 371 269 l s 371 269 m 371 269 l s 371 269 m 371 269 l s 371 269 m 371 270 l s 371 270 m 371 270 l s 371 270 m 371 270 l s 371 270 m 371 270 l s 371 270 m 371 271 l s 371 271 m 372 271 l s 372 271 m 372 271 l s 372 271 m 372 271 l s 372 271 m 372 272 l s 372 272 m 372 272 l s 372 272 m 372 272 l s 372 272 m 372 273 l s 372 273 m 372 273 l s 372 273 m 372 273 l s 372 273 m 372 273 l s 372 273 m 372 274 l s 372 274 m 372 274 l s 372 274 m 372 274 l s 372 274 m 372 274 l s 372 274 m 372 275 l s 372 275 m 372 275 l s 372 275 m 372 275 l s 372 275 m 372 276 l s 372 276 m 372 276 l s 372 276 m 372 276 l s 372 276 m 372 276 l s 372 276 m 372 277 l s 372 277 m 371 277 l s 371 277 m 371 277 l s 371 277 m 371 277 l s 371 277 m 371 278 l s 371 278 m 371 278 l s 371 278 m 371 278 l s 371 278 m 371 279 l s 371 279 m 371 279 l s 371 279 m 370 279 l s 370 279 m 370 279 l s 370 279 m 370 280 l s 370 280 m 370 280 l s 370 280 m 370 280 l s 370 280 m 370 281 l s 370 281 m 369 281 l s 369 281 m 369 281 l s 369 281 m 369 281 l s 369 281 m 369 282 l s 369 282 m 369 282 l s 369 282 m 368 282 l 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m 350 295 l s 350 295 m 350 295 l s 350 295 m 349 295 l s 349 295 m 349 296 l s 349 296 m 348 296 l s 348 296 m 348 296 l s 348 296 m 347 296 l s 347 296 m 347 297 l s 347 297 m 346 297 l s 346 297 m 345 297 l s 345 297 m 345 297 l s 345 297 m 344 298 l s 344 298 m 344 298 l s 344 298 m 343 298 l s 343 298 m 343 299 l s 343 299 m 342 299 l s 342 299 m 341 299 l s 341 299 m 341 299 l s 341 299 m 340 300 l s 340 300 m 340 300 l s 340 300 m 339 300 l s 339 300 m 338 300 l s 338 300 m 338 301 l s 338 301 m 337 301 l s 337 301 m 336 301 l s 336 301 m 336 302 l s 336 302 m 335 302 l s 335 302 m 335 302 l s 335 302 m 334 302 l s 334 302 m 333 303 l s 333 303 m 333 303 l s 333 303 m 332 303 l s 332 303 m 331 303 l s 331 303 m 331 304 l s 331 304 m 330 304 l s 330 304 m 329 304 l s 329 304 m 329 305 l s 329 305 m 328 305 l s 328 305 m 327 305 l s 327 305 m 327 305 l s 327 305 m 326 306 l s 326 306 m 325 306 l s 325 306 m 325 306 l s 325 306 m 324 306 l s 324 306 m 323 307 l s 323 307 m 322 307 l s 322 307 m 322 307 l s 322 307 m 321 308 l s 321 308 m 320 308 l s 320 308 m 320 308 l s 320 308 m 319 308 l s 319 308 m 318 309 l s 318 309 m 318 309 l s 318 309 m 317 309 l s 317 309 m 316 309 l s 316 309 m 315 310 l s 315 310 m 315 310 l s 315 310 m 314 310 l s 314 310 m 313 311 l s 313 311 m 312 311 l s 312 311 m 312 311 l s 312 311 m 311 311 l s 311 311 m 310 312 l s 310 312 m 310 312 l s 310 312 m 309 312 l s 309 312 m 308 312 l s 308 312 m 307 313 l s 307 313 m 307 313 l s 307 313 m 306 313 l s 306 313 m 305 314 l s 305 314 m 304 314 l s 304 314 m 304 314 l s 304 314 m 303 314 l s 303 314 m 302 315 l s 302 315 m 301 315 l s 301 315 m 301 315 l s 301 315 m 300 315 l s 300 315 m 299 316 l s 299 316 m 298 316 l s 298 316 m 298 316 l s 298 316 m 297 317 l s 297 317 m 296 317 l s 296 317 m 296 317 l s 296 317 m 295 317 l s 295 317 m 294 318 l s 294 318 m 293 318 l s 293 318 m 293 318 l s 293 318 m 292 319 l s 292 319 m 291 319 l s 291 319 m 290 319 l s 290 319 m 290 319 l s 290 319 m 289 320 l s 289 320 m 288 320 l s 288 320 m 287 320 l s 287 320 m 287 320 l s 287 320 m 286 321 l s 286 321 m 285 321 l s 285 321 m 284 321 l s 284 321 m 284 322 l s 284 322 m 283 322 l s 283 322 m 282 322 l s 282 322 m 281 322 l s 281 322 m 281 323 l s 281 323 m 280 323 l s 280 323 m 279 323 l s 279 323 m 278 323 l s 278 323 m 278 324 l s 278 324 m 277 324 l s 277 324 m 276 324 l s 276 324 m 275 325 l s 275 325 m 275 325 l s 275 325 m 274 325 l s 274 325 m 273 325 l s 273 325 m 273 326 l s 273 326 m 272 326 l s 272 326 m 271 326 l s 271 326 m 270 326 l s 270 326 m 270 327 l s 270 327 m 269 327 l s 269 327 m 268 327 l s 268 327 m 267 328 l s 267 328 m 267 328 l s 267 328 m 266 328 l s 266 328 m 265 328 l s 265 328 m 265 329 l s 265 329 m 264 329 l s 264 329 m 263 329 l s 263 329 m 263 329 l s 263 329 m 262 330 l s 262 330 m 261 330 l s 261 330 m 260 330 l s 260 330 m 260 331 l s 260 331 m 259 331 l s 259 331 m 258 331 l s 258 331 m 258 331 l s 258 331 m 257 332 l s 257 332 m 256 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406 m 243 407 l s 243 407 m 244 407 l s 244 407 m 244 407 l s 244 407 m 245 407 l s 245 407 m 245 408 l s 245 408 m 245 408 l s 245 408 m 246 408 l s 246 408 m 246 408 l s 246 408 m 247 409 l s 247 409 m 247 409 l s 247 409 m 248 409 l s 248 409 m 248 410 l s 248 410 m 249 410 l s 249 410 m 249 410 l s 249 410 m 250 410 l s 250 410 m 250 411 l s 250 411 m 250 411 l s 250 411 m 251 411 l s 251 411 m 251 411 l s 251 411 m 252 412 l s 252 412 m 252 412 l s 252 412 m 253 412 l s 253 412 m 253 413 l s 253 413 m 254 413 l s 254 413 m 254 413 l s 254 413 m 255 413 l s 255 413 m 255 414 l s 255 414 m 255 414 l s 255 414 m 256 414 l s 256 414 m 256 414 l s 256 414 m 257 415 l s 257 415 m 257 415 l s 257 415 m 258 415 l s 258 415 m 258 416 l s 258 416 m 259 416 l s 259 416 m 259 416 l s 259 416 m 260 416 l s 260 416 m 260 417 l s 260 417 m 261 417 l s 261 417 m 261 417 l s 261 417 m 261 417 l s 261 417 m 262 418 l s 262 418 m 262 418 l s 262 418 m 263 418 l s 263 418 m 263 419 l s 263 419 m 264 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284 431 m 284 431 l s 284 431 m 284 432 l s 284 432 m 285 432 l s 285 432 m 285 432 l s 285 432 m 286 432 l s 286 432 m 286 433 l s 286 433 m 287 433 l s 287 433 m 287 433 l s 287 433 m 287 434 l s 287 434 m 288 434 l s 288 434 m 288 434 l s 288 434 m 289 434 l s 289 434 m 289 435 l s 289 435 m 289 435 l s 289 435 m 290 435 l s 290 435 m 290 435 l s 290 435 m 291 436 l s 291 436 m 291 436 l s 291 436 m 291 436 l s 291 436 m 292 437 l s 292 437 m 292 437 l s 292 437 m 293 437 l s 293 437 m 293 437 l s 293 437 m 293 438 l s 293 438 m 294 438 l s 294 438 m 294 438 l s 294 438 m 295 439 l s 295 439 m 295 439 l s 295 439 m 295 439 l s 295 439 m 296 439 l s 296 439 m 296 440 l s 296 440 m 297 440 l s 297 440 m 297 440 l s 297 440 m 297 440 l s 297 440 m 298 441 l s 298 441 m 298 441 l s 298 441 m 298 441 l s 298 441 m 299 442 l s 299 442 m 299 442 l s 299 442 m 300 442 l s 300 442 m 300 442 l s 300 442 m 300 443 l s 300 443 m 301 443 l s 301 443 m 301 443 l s 301 443 m 301 443 l s 301 443 m 302 444 l s 302 444 m 302 444 l s 302 444 m 302 444 l s 302 444 m 303 445 l s 303 445 m 303 445 l s 303 445 m 304 445 l s 304 445 m 304 445 l s 304 445 m 304 446 l s 304 446 m 305 446 l s 305 446 m 305 446 l s 305 446 m 305 446 l s 305 446 m 306 447 l s 306 447 m 306 447 l s 306 447 m 306 447 l s 306 447 m 307 448 l s 307 448 m 307 448 l s 307 448 m 307 448 l s 307 448 m 308 448 l s 308 448 m 308 449 l s 308 449 m 308 449 l s 308 449 m 309 449 l s 309 449 m 309 449 l s 309 449 m 309 450 l s 309 450 m 310 450 l s 310 450 m 310 450 l s 310 450 m 310 451 l s 310 451 m 311 451 l s 311 451 m 311 451 l s 311 451 m 311 451 l s 311 451 m 312 452 l s 312 452 m 312 452 l s 312 452 m 312 452 l s 312 452 m 313 452 l s 313 452 m 313 453 l s 313 453 m 313 453 l s 313 453 m 314 453 l s 314 453 m 314 454 l s 314 454 m 314 454 l s 314 454 m 315 454 l s 315 454 m 315 454 l s 315 454 m 315 455 l s 315 455 m 316 455 l s 316 455 m 316 455 l s 316 455 m 316 455 l s 316 455 m 317 456 l s 317 456 m 317 456 l 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end} def %%EndProlog $F2psBegin 0 setlinecap 0 setlinejoin -22 36 translate 0.0 101.0 translate 0.360 -0.360 scale 1.000 setlinewidth % Polyline n 859 239 m 759 279 l 719 379 l 819 339 l 859 239 l gs col-1 s gr % Polyline n 579 239 m 619 339 l 719 379 l 679 279 l 579 239 l gs col-1 s gr % Polyline n 719 99 m 679 199 l 579 239 l 619 139 l 719 99 l gs col-1 s gr % Polyline n 719 99 m 819 139 l 859 239 l 759 199 l 719 99 l gs col-1 s gr % Polyline n 319 99 m 179 239 l 319 379 l 459 239 l 319 99 l gs col-1 s gr 0.500 setlinewidth % Polyline n 479 239 m 559 239 l gs col-1 s gr n 551.000 237.000 m 559.000 239.000 l 551.000 241.000 l gs 2 setlinejoin col-1 s gr 1.000 setlinewidth % Polyline n 59 99 m 59 379 l gs col-1 s gr 0.500 setlinewidth % Polyline n 79 239 m 159 239 l gs col-1 s gr n 151.000 237.000 m 159.000 239.000 l 151.000 241.000 l gs 2 setlinejoin col-1 s gr $F2psEnd EndOfTheIncludedPostscriptMagicCookie \closepsdump %====================================================== % Paper %====================================================== \voffset=1.2cm %\hoffset=0.15cm \newdimen\normalskip \newdimen\normalindent \newdimen\greatbaselineskip \normalskip=5pt \normalindent=20pt \greatbaselineskip=5.25mm \normalbaselineskip=\baselineskip %====================================================== % Page zero %====================================================== \pageno=0 \hsize=16.0truecm\vsize=22.5truecm \null\vskip 1.2cm \title Computer--Assisted Proofs for Fixed Point Problems in \title Sobolev Spaces \vskip 1cm \centerline {\bf Alain Schenkel$^{1,}$\footnote{$^\dagger$}{The research of these authors was supported in part by the Fonds National Suisse.}, Jan Wehr$^{2,}$\footnote{$^\ddagger$}{The research of this author was supported in part by the NSF Probability and Mathematical Statistics grant 9706915.}, and Peter Wittwer$^{3,}$$^\dagger$} \vskip 0.5cm {\noindent\eightrm\baselineskip=10pt \raise 1mm\hbox{$\scriptstyle 1$}$\,$Department of Mathematics, Helsinki University, P.O.~Box~4, 00014 Helsinki, Finland\hfill\break \raise 1mm\hbox{$\scriptstyle 2$}$\,$Interdisciplinary Center for Mathematical and Computer Modeling, Warsaw University,\hfill\break \hbox{\ \ }Pawinskiego~5a, Warszawa~02~106, Poland\footnote{$\,^*$}{Permanent address: Department of Mathematics, University of Arizona, Tucson AZ~85721, USA.}\hfill\break \raise 1mm\hbox{$\scriptstyle 3$}$\,$D\'epartement de Physique Th\'eorique, Universit\'e de Gen\`eve, CH-1211 Gen\`eve 4, Switzerland\hfill\break \par} \vskip 1.2cm \abstract In this paper we extend the technique of computer--assisted proofs to fixed point problems in Sobolev spaces. Up to now the method was limited to the case of spaces of analytic functions. The possibility to work with Sobolev spaces is an important progress and opens up many new domains of applications. Our discussion is centered around a concrete problem that arises in the theory of critical phenomena and describes the phase transition in a hierarchical system of random resistors. For this problem we have implemented in particular the convolution product based on the Fast Fourier Transform (FFT) algorithm with rigorous error estimates. \vskip 0.5cm \noindent {\bf Key words:} computer--assisted proofs, constructive analysis in Sobolev spaces, phase transitions in random media, discrete convolutions are convolutions of splines. \vfill\eject %====================================================== % Contents %====================================================== \pageno=+1 \def\points{\quad \dotfill \quad} \parindent=18pt \parskip=0pt \noindent{\xbold Contents} \vskip\normalskip {\sub\item{1} Introduction} \points 2 \itemitem{1.1} The Model \points 3 \itemitem{1.2} Main Result \points 6 \itemitem{1.3} Computer--Assisted Proofs \points 10 \vskip\normalskip {\sub \item{2} Organization of the Proof} \points 11 \vskip\normalskip {\sub \item{3} Constructive Analysis in $\BB_\albe$} \points 17 \itemitem{3.1} Operations Involving Real and Complex Numbers \points 18 \itemitem{3.2} Standard Sets of $\BB_\albe$ \points 21 \vskip\normalskip {\sub \item{4} Operations Involving Functions} \points 24 \itemitem{4.1} Elementary Operations \points 24 \itemitem{4.2} The Scaling Operator \points 26 \itemitem{4.3} The Operator $T$ \points 27 \itemitem{4.4} The Convolution \points 28 \itemitem{4.5} The Identity \points 31 \vskip\normalskip {\sub \item{5} The Maps $\NN_\lambda$}\points 32 \itemitem{5.1} A Bound on $\NN_\lambda$ \points 32 \itemitem{5.2} Existence of the Family of Fixed Points: First Estimate \points 35 \itemitem{5.3} Existence of the Fixed Point $f_{\lambda^*}$ \points 36 \vskip\normalskip {\sub \item{6} Contractivity Properties of $D\NN_\lambda$} \points 38 \itemitem{6.1} Oscillatory Functions \points 40 \itemitem{6.2} Functions with Support Near the Origin \points 43 \itemitem{6.3} Functions with Support Near Infinity \points 45 \itemitem{6.4} Piecewise Constant Functions \points 47 \vskip\normalskip {\sub \item{7} The Tangent Maps $D\MM_\lamka$}\points 50 \itemitem{7.1} Decomposition of the Operator Norm \points 50 \itemitem{7.2} The Operator $M$ \points 52 \itemitem{7.3} Existence of the Family of Fixed Points: Second Estimate \points 55 \vskip\normalskip {\sub \item{8} The Program} \vskip\normalskip {\sub Acknowledgments} \points 58 \vskip\normalskip {\sub Appendix} \points 59 %{\sub Appendix} \points 59 \vskip\normalskip {\sub References} \points 64 \parskip=\normalskip \parindent=\normalindent %====================================================== % Section 1 %====================================================== \section Introduction In this paper we study certain aspects of a model that describes the conductivity in a disordered material. A disordered material is often modeled in statistical mechanics by what is known as a network of random resistors. One would like to be able to describe, for example, what happens in a $d$--dimensional cubic lattice where each link represents a resistor whose resistance is a random variable. One of the quantities of interest is the effective conductivity of such a network. This conductivity could be defined, for instance, by taking the limit as $L$ goes to infinity of the conductivity $\sigma(L)$ measured on a cube of side length $L$. More generally, a network of random resistors is defined by giving a graph and a sequence $\Sigma_{ij}$ of random variables with $0\leq$ $\Sigma_{ij}\leq \infty$. $\Sigma_{ij}$ describes the conductivity of the link in the graph that connects the vertices $i$ and $j$. For simplicity one considers in general families of independent identically distributed random variables. Such models have been widely studied over the last decades, see for example [Z, K1, Be1, K2, Be2, BW, SW, BSW, K3, Be3, BO, SS] and for a purely probabilistic approach to the problem [Bl, EB, W1-2]. The model which we study in this paper belongs to a class of models where one permits the links to be perfect insulators, that is $q\equiv P(\Sigma _{ij}=0)>0,$ but for which on the other hand $\Sigma_{ij}<\infty.$ This situation is interesting, as it presents the phenomenon of percolation: For $q$ close to one, the links that are conducting form disconnected finite sets and the effective conductivity of the network is trivially zero. This is not any longer the case for $q$ close to zero, where the conductivity depends on the resistivity of the connecting links. As a consequence, a phase transition occurs in the effective conductivity for a certain critical value of the parameter $q$. The introduction of the parameter $q$ permits therefore, through the mechanism of percolation, to obtain a model that models a critical phenomenon. The study of phase transitions has progressed enormously with the arrival of renormalization group methods. Different types of renormalization schemes have been proposed to describe the phase transition on regular lattices of random resistors, in particular transformations of the Migdal--Kadanoff type [K3, BO, Bl], and so--called exact renormalization schemes on hierarchical lattices [SW, Be3, SS]. Some of the difficulties that have to be overcome in a renormalization group study disappear when one considers hierarchical models. Moreover, these models typically seem to provide good approximations to more complicated systems [BO]. In Section~1.1 we describe the hierarchical lattice of random resistors for which we have studied the phase transition in this paper. This lattice has been proposed by [SW] as an approximation to the square lattice in two dimensions. In Section~1.2, we state our main result. \subsection The Model The hierarchical network that we study in this paper is constructed recursively as indicated in Fig.~1.2. \figure fth2.ps 6.0 12.0 The hierarchical lattice at order 2.\cr Consider the mapping on graphs that consists of replacing every link by two pairs of links. If we start with a graph consisting of one link connecting two sites, then after applying this operation we end up with four sites and after $n$ applications we end up with a graph of $4^{n}$ links. For $n\geq1$, the network of random resistors that we consider is obtained after $n$ iterations of the procedure outlined above, and consists of resistors with conductivities described by $4^{n}$ independent copies $\Sigma_{0}^{(k)}$ of a random variable $\Sigma_{0}$. The random variables $\Sigma_{0}^{(k)}$ are therefore $i.i.d.\,$. The choice of a hierarchical geometry permits to give a simple formulation of the effective conductivity of the network, when measured between the two vertices of the initial link, in terms of a map. Indeed, using the composition laws for conductors connected in series and in parallel, the conductivity of a circuit that consists of four resistors with conductivities $\sigma_{1},$ $\sigma_{2},$ $\sigma_{3},$ $\sigma_{4}$ arranged in a loop is $$ \sigma=D_c(\sigma_1,\sigma_2,\sigma_3,\sigma_4)\equiv {1\over{1\over\sigma_1}+{1\over\sigma_2}}+ {1\over{1\over\sigma_3}+{1\over\sigma_4}}. \equation(defmoydiam) $$ If the conductivities $\sigma_{1},$ $\sigma_{2},$ $\sigma_{3},$ $\sigma_{4}$ are random, this is also the case for the effective conductivity $\sigma.$ Therefore, by applying this nonlinear average to each of the loops of the hierarchical network of order $n$, we obtain the random variables that describe the conductivities of the links of the network at the level $n-1.$ Since the random variables that were given on the network of order $n$ were $i.i.d.,$ the $4^{n-1}$ random variables that are obtained for the network of order $n-1$ are also $i.i.d.\,$. They are independent copies of the random variable given by $$ \Sigma_1= D_c(\Sigma^{(1)}_0,\Sigma^{(2)}_0,\Sigma^{(3)}_0,\Sigma^{(4)}_0). $$ By applying successively the nonlinear average $D_{c},$ one can therefore go up the hierarchy to compute the effective conductivity $\Sigma_{n}$ of the network of order $n.$ We note that by applying one more time the average $D_{c}$ to four independent copies of $\Sigma_{n},$ we obtain the conductivity $\Sigma_{n+1}$ of the hierarchical network of order $n+1$ made up from resistors with conductivities given by $4^{n+1}$ independent copies of the random variable $\Sigma_{0}.$ We are interested in the limit as $n\rightarrow\infty$ of the sequence $\Sigma_{n}$ that we have just defined. This limit corresponds to the effective conductivity of our hierarchical network in the infinite volume limit. It is not difficult to see that our model is an approximation to the renormalization on a simple square lattice in $d=2$ dimensions. In [SW] a detailed discussion of this approximation can be found. See also [BO]. The questions that arise naturally are the following. First of all, a phenomenon of self--averaging should lead to a deterministic effective conductivity for the infinite network. Next, it is interesting to know in what way this conductivity depends on the conductivity of each link of the network, that is on the distribution of the initial random variable $\Sigma_{0}.$ Finally, once the convergence of the effective conductivity is established, one can study the fluctuations of $\Sigma_{n}$ around this limit. \medskip A certain amount of information can easily be obtained by studying the parameter $$ p_n=P(\Sigma_n>0). \equation(defpn) $$ Recall that we permit the links to be perfect insulators with a nonzero probability. Here, $p_{0}\equiv p$ is the probability that a resistor of the original (infinite) network is not broken. From \equ(defmoydiam) it is not difficult to see that the conductivity of a diamond circuit is nonzero with probability $$ \tilde p=g(p)\equiv1-(1-p^2)^2, \equation(defg) $$ where $p$ is the probability that each of the for resistors has a nonzero conductivity. The function $g$ is characteristic for percolation problems [G]. The function $g$ is increasing on the interval $[0,1]$ and has, in addition to the fixed points at zero and one, a unique unstable fixed point $p_{c}$ in the interval $(0,1).$ The value of $p_{c}$ is, $$ p_c={\sqrt5-1\over2}. \equation(defpc) $$ There are therefore three cases. If $pp_{c},$ one has $\lim_{n\rightarrow\infty}p_{n}=1.$ This means that with probability one there is a path made from resistors with nonzero conductivity that connects the two sites of the lattice for $n=0.$ In other words, the percolation threshold of the network is given by $p_{c}.$ We note that this does not imply that the effective conductivity is nonzero. However, it has been proved in [W1] that for $p>p_{c}$ the sequence $\Sigma_{n}$ converges with probability one to a constant $\sigma^{\ast}(p) $, and in [Sh] that this constant is strictly positive. Therefore, the percolation threshold corresponds exactly to the phase transition of the effective conductivity. At the critical point $p=p_{c},$ one has $p_{n}=p_{c}$ for all $n.$ This means that the probability $P(\Sigma_{n}=0)$ is invariant. In the following we are going to be interested in the part of the distribution of $\Sigma_{n} $ supported on $(0,\infty).$ An argument based on the study of the expectation of $\Sigma_{n}$ shows, however, that the effective conductivity of the network in the infinite volume limit is still zero in this case. Indeed, if we denote by $E(X)$ the expectation of a random variable $X,$ one has $$ E(\Sigma_{n+1})=E\bigl(D_c(\Sigma^{(1)}_n,\dots,\Sigma^{(4)}_n)\bigr) =E\Bigl({2\over{1\over\Sigma^{(1)}_n}+{1\over\Sigma^{(2)}_n}}\Bigr), $$ where we have used independence of the random variables. Since, in addition we have the following inequality between the arithmetic mean and the geometric mean, $$ {2\over{1\over x}+{1\over y}}\leq{x+y\over2}, $$ and since the left hand side of this expression is equal to zero if $x$ or $y $ are zero, we can bound $E(\Sigma_{n+1})$ by the expectation of the average of $\Sigma_{n}^{(1)}$ and $\Sigma_{n}^{(2)}$ over the set where the random variables are strictly positive. Therefore, one obtains $$ E(\Sigma_{n+1})\leq p_nE(\Sigma_n). $$ Since $p_{n}=p_c$ for all $n,$ we have $E(\Sigma_{n})\rightarrow0$ as $n\rightarrow\infty,$ and the sequence $\Sigma_{n}$ converges to zero in probability. \medskip The goal of this paper is to describe how the $\Sigma_{n}$ converge to zero at the critical point, that is to describe the fluctuations of the $\Sigma_{n}$ around their limiting value. A numerical study [SW] of the probability densities of the random variables $\Sigma_{n}$ indicates that if one normalizes $\Sigma_{n}$ with an appropriate factor $\mu_{n}$ that fixes the expectation of the random variable $\mu_{n}\Sigma_{n}\ $at the expectation of $\Sigma_{0},$ the sequence $\mu_{n} \Sigma_{n}\ $converges in distribution to a multiple of a (universal) random variable $\Sigma_*$, and $$ \lim_{n\rightarrow\infty}{\mu_{n+1}\over\mu_n}=\lambda^*\approx1.756, $$ independently of the choice of $\Sigma_{0}.$ This means that the fluctuations of the effective conductivity of the network present a certain universality in the limit $n\rightarrow\infty$: the limiting probability densities for different initial random variables $\Sigma_{0}$ distinguish themselves only by a change of scale. Furthermore, the probability density for the positive values of $\Sigma_{\ast}$ decays faster than exponentially at zero and infinity. Therefore, at the critical point, the behavior of the fluctuations distinguishes itself from the supercritical case for which a perturbative computation [SS] indicates that the sequence of properly normalized random variables $\Sigma_{n}$ converges to a normal distribution (a proof of this fact will appear in [WW]). Also, since $\lambda^*<2$, conductance fluctuations can be thought of as anomalously large compared to the supercritical case. In this work we address the question of the existence of a positive real number $\lambda^{\ast}$ and a random variable $\Sigma_{\ast}$ such that $$ \Sigma_*=\lambda^*D_c(\Sigma^{(1)}_*,\dots,\Sigma^{(4)}_*). \equation(jjunk) $$ We note that the map $D_{c}$ is homogeneous of degree one and the random variables $\lambda\Sigma_{\ast}$ are therefore solutions of \equ(jjunk) for all $\lambda>0.$ The number $\lambda^{\ast}$ gives the dynamics of the renormalization group $D_{c}$ at the critical point on the whole of the set $\lambda\Sigma_{\ast}$ and is related to the critical exponent $t$ that describes the phase transition of the effective conductivity, $$ \sigma^*(p)\sim(p-p_c)^t,\quad p>p_c, $$ through the formula, cf. [SW], $$ t={\log\lambda^*\over\log g'(p_c)}\,. $$ \subsection Main Result In order to study the fluctuations of the effective conductivity of our hierarchical network at the critical point, we work in the framework of functional analysis. The part of the distribution of the random variables $\Sigma_{n}$ that interests us is the one that is supported on $(0,\infty).$ One assumes that this part of the distribution of $\Sigma_{0}$ is absolutely continuous with respect to Lebesgue measure and one derives the functional equation for the probability densities that correspond to the nonlinear average \equ(defmoydiam). It turns out to be simpler to work with the resistivities instead of the conductivities. One considers therefore the random variables $\Upsilon$ for the resistivity given by $$ \Upsilon={1\over\Sigma}. $$ If $\sigma$ is the density of the random variable $\Sigma,$ then the density $\rho$ of $\Upsilon$ is given by $$ \rho(x)=T(\sigma)(x)\equiv{1\over x^2}\sigma\bigl({1\over x}\bigr). \equation(deft) $$ The average $D_{c}$ for the conductivities can be rewritten for the resistivities as $$ D_r(r_1,\dots,r_4)={1\over{1\over r_1+r_2}+{1\over r_3+r_4}}. \equation(defmoydiamr) $$ The functional equation for the density ${\cal D}_{r}(\rho)$ of the average \equ(defmoydiamr) of four independent copies of a random variable $\Upsilon$ with density $\rho$ is therefore given in terms of the map $T$ and the convolution operator. Indeed, the probability density $\rho$ of a sum of two random variables with densities $\rho_{1}$ and $\rho_{2}$ is given by the convolution of $\rho_{1}$ and $\rho_{2},$ that is by $$ \rho(x)=(\rho_1*\rho_2)(x)=\int_0^x\rho_1(y)\rho_2(x-y)\,dy. \equation(defconv) $$ Therefore, it follows from \equ(defmoydiamr) that $$ \DD_r(\rho)=T\bigr(T(\rho*\rho)*T(\rho*\rho)\bigr). \equation(defef1) $$ One observes that, formally, the Dirac--densities $\delta(x-a)$ are fixed points of this transformation for all $a\geq0.$ They correspond to the limiting densities in the non--critical cases. In order to obtain an equation for the probability densities with support on $(0,\infty)$ we determine the contributions of the four resistors to the finite value of $D_{r}(r_{1},\dots r_{4}).$ By inspecting \equ(defmoydiamr), one observes that they are of two types: either $r_{1}+r_{2}=\infty$ and $r_{3}+r_{4}<\infty$ (with the corresponding symmetric case $r_{1}+r_{2}<\infty$ and $r_{3} +r_{4}=\infty),$ or all of the four resistivities are finite. At the critical point $p_{c},$ one determines easily that the probability of the first case is given by $$ c_1\equiv 2p_c(1-p_c^2)=0.763..., \equation(defc1) $$ whereas the probability of the second case is $$ c_2\equiv p_c^3=0.236...\,. \equation(defc2) $$ Obviously $c_{1}+c_{2}=1.$ Therefore, the operator that acts on the probability densities and corresponds to the finite part of the map $D_{r}$ on the random variables is given at the critical point by $$ \DD(\rho)=c_1(\rho*\rho)+c_2T\bigl(T(\rho*\rho)*T(\rho*\rho)\bigr). \equation(mainmap) $$ In order to rewrite the fixed point problem \equ(jjunk) in terms of the probability densities, one uses that for $\lambda>0$ the probability density of a random variable $\Upsilon/\lambda$ is given by $S_{\lambda}\rho,$ where $\rho$ is the probability density of $\Upsilon$ and where $S_{\lambda}$ is the operator that changes the scale, $$ S_\lambda f(x)\equiv\lambda f(\lambda x)\,,\quad\lambda>0. \equation(defscal) $$ Therefore the fixed point problem \equ(jjunk) can be rewritten as $$ \rho^*=S_{\lambda^*}\DD(\rho^*). \equation(pbmfpd) $$ The proof of the existence of a real $\lambda^{\ast} $and a function $\rho^{\ast}$ satisfying \equ(pbmfpd) which we present here is constructive. In particular, we will be capable of providing explicit bounds on $\lambda^{\ast}$ as well as an approximation to the function $\rho^{\ast}. $ The graph of the approximation that we have obtained this way is represented in Fig.~1.2. \figurewithtex fth1.ps fth1.tex 6.0 12.0 The fixed point $\rho^*$.\cr The operator \equ(mainmap) was studied by mathematically rigorous, computer--assisted, constructive analysis. Before we state with precision the result that is proved in this paper we define the function spaces with which we work. For that we first define the notation that will be used later. \noindent{\bf Notation.} We denote by $\real_{+}$ the set of nonnegative real numbers. The set of positive real numbers will be denoted by $\real_{+}^{\ast}.$ For an interval $I\subseteq\real$, we denote by ${\cal C}^{n}(I)$ the set of functions that are $n$ times continuously differentiable on $I.$ The derivative of a function $f$ of one variable will be denoted by $f^{\prime}. $ For a positive function $\mu$, we denote by $L^{1}(\real_{+},\mu(x)~dx)$ the space of functions defined on $\real_{+}$ and integrable with respect to the measure $\mu(x)~dx.$ Finally $W_{1}^{1}(\real_{+},\mu(x)~dx)$ is the Sobolev space of functions of $L^{1}(\real_{+},\mu(x)~dx)$ with one distributional derivative in $L^{1}(\real_{+},\mu(x)~dx)$. For $r\geq0$ and $x$ in a metric space, $B_r(x)$ will denote the open ball of radius $r$ centered at $x$. \claim Definition(spaces) For $\alpha,$ $\beta\geq0$ and functions $w_{\alpha\beta}$ given by $$ w_\albe(x)=\exp\Bigl({\alpha\over x}+\beta x\Bigr), \equation(defpoids) $$ we define ${\cal B}_{\alpha\beta}$ to be the Banach space $L^{1}(\real _{+},w_{\alpha\beta}(x)~dx).$ We denote the norm of $f\in{\cal B} _{\alpha\beta}$ by $\left\| f\right\| _{a\beta}$, that is, $$ \|f\|_\albe=\int_0^\infty w_\albe(x)\,|f(x)|\,dx. \equation(defnorm) $$ We furthermore define $$ \BB=\bigcap_{\scriptstyle\alpha>0\atop\scriptstyle\beta>0}\BB_\albe. \equation(definclusion) $$ \claimrm Remark(rembanach) It is clear that we have the inclusion ${\cal B}_{\sigma\tau} \subseteq{\cal B}_{\alpha\beta}$ for $\sigma\geq\alpha$ and $\tau\geq\beta. $ The inclusion is strict unless $\sigma=\alpha$ and $\tau=\beta.$ We will also need the following definitions of the mass and expectation. \claim Definition(masseesp) We define the mass $M(f)$ of a function $f\in L^{1}(\real_{+})$ by $$ M(f)=\int_0^\infty f(x)\,dx. \equation(defmasse) $$ If $f\in L^{1}(\real_{+},(1+\left| x\right| )~dx),$ we define the expectation $E(f)$ by $$ E(f)=\int_0^\infty xf(x)\,dx. \equation(defesp) $$ We can now state the principal result of this paper. \claim Theorem(mainresult) There exists a real number $\lambda^{\ast}$ and a function $f^{\ast}\in{\cal B}$ that satisfy the equation $$ S_{\lambda^*}\DD(f^*)=f^*. $$ In addition $f^{\ast}$ has the following two properties\vskip0pt\noindent \hglue 5cm$(1)$ $M(f^*)=1$,\hfill\break \hglue 5cm$(2)$ $f^*\in\CC^\infty(\real^+)$.\vskip0pt\noindent Note that this theorem does not imply that the fixed point $f^{\ast}$ is a probability density, since $f^{\ast}$ is not necessarily a positive function. While we see strong numerical evidences for positivity of the fixed point $f^{\ast}$, we have no proof of this fact. Before terminating this section, we summarize in the following lemma some of the properties that are satisfied by the maps that are contained in \equ(pbmfpd). \claim Lemma(proprapp) The maps $S_{\lambda}$ form a multiplicative group, i.e., $S_{1}=I,$ and $S_{\lambda_{1}}S_{\lambda_{2}}=S_{\lambda_{1}\lambda_{2}}.$ Moreover, for $f$ and $g$ integrable functions one has $$\eqalignno{ &S_\lambda(f*g)=S_\lambda f*S_\lambda g,&\equat(prjj1)\cr &S_\lambda T=TS_{1/\lambda}.&\equat(prjj2)\cr }$$ If $f,$ $g\in L^{1}(\real_{+},(1+|x|)~dx),$ then the mass and expectation satisfy the following identities $$\eqalign{ &M(f)=M(S_\lambda f)=M(Tf)\,,\cr &M(f*g)=M(f)M(g)\,,\cr &E(S_\lambda f)={1\over\lambda}E(f)\,,\cr &E(f*g)=M(f)E(g)+E(f)M(g)\,.\cr }\equation(proprme) $$ Even though the proof of \clm(mainresult) needs in part a computer for its proof, the properties $(1)$ and $(2)$ follow directly from the existence of the fixed point. The regularity of the fixed point will be established later, cf.~Proposition~2.3, whereas property $(1)$ is proved in the following lemma. \claim Lemma(mass1) Let $f\in L^{1}(\real_{+})$ and let $\lambda>0$ arbitrary. If $f$ satisfies $S_{\lambda}{\cal D}(f)=f,$ then $M(f)=1.$ \proof Using the relations \equ(proprme) one finds using $M(f)=M(S_{\lambda} {\cal D}(f))$ $$ M(f)=c_1M(f)^2+c_2M(f)^4. $$ Using the monotonicity of the function $x\mapsto c_{2}x^{3}+c_{1}x-1,$ one verifies that if $c_{1}+c_{2}=1$, then the only zero is given by $x_{0}=1.$ Therefore, $M(f)=1.$ \qed \subsection Computer--Assisted Proofs The rest of the proof of \clm(mainresult) is the main part of this paper. It is based on a very large number of inequalities proved rigorously with the help of a computer. The use of a computer for proving theorems in analysis has become standard by now. This method, which allows to do constructive functional analysis on a computer, has been developed by O.E. Lanford in his seminal paper [L1], and has then been generalized by [EKW2]. This technique of proof has since then been applied to problems of various origin. See for example [BS, C, CC, dlL, EKW1-2, EW1-2, FL, FS, KP, KSW, KW1-7, L1-3, LR1-3, M, R, Se1-2, St]. The proofs constructed up to now have in common that they all deal with spaces of analytic functions. One important novelty of the work presented here is that a proof is constructed for function spaces of $L^{1}$--type. The basic ideas underlying the proof remain the same, but the generalization to $L^{1}$ spaces uses approximation methods which are typical in numerical analysis, and we will explain how to control discretization errors in this context. A computer--assisted proof is complete once the program has come to an end without a ``domain error''. %====================================================== % Section 2 %====================================================== \section Organization of the Proof \noindent In order to prove existence of a fixed point $f^*$ for the map $S_{\lambda^*}\DD$, we will rely on the contraction mapping principle. The following argument shows, however, that $S_{\lambda^*}\DD$ cannot even be hyperbolic due to the presence of a symmetry. Recall that the nonlinear average $D_r$ is homogeneous of order $1$. This causes the scaling operator $S_\lambda$ to commute with $\DD$ for every $\lambda>0$. Hence, existence of a fixed point $f^*$ implies existence of a one parameter family of fixed points $\{S_\lambda f^*\}_{\lambda>0}$. In the case $E(f^*)\not=0$, this family can be parameterized by the expectation $E(S_\lambda f^*)=E(f^*)/\lambda$. We first remove this symmetry and make the fixed point problem hyperbolic by introducing the family of maps $\{\NN_\lambda\}_{\lambda>0}$ defined by $$ \NN_\lambda(f)=S_\lambda\Bigl(c_\lambda(f)f*f+c_2T\bigl(T(f*f)*T(f*f)\bigr) \Bigr)\,,\equation(mapnl) $$ where $c_\lambda(f)$ is such that $$ E\bigl(\NN_\lambda(f)\bigr)=1\,. \equation(condonc1) $$ The expression inside the outer brackets on the RHS of \equ(mapnl) differs from the map $\DD$ only by the coefficient $c_\lambda(f)$. The following remark relates the maps $S_\lambda\DD$ and $\NN_\lambda$: If $c_\lambda(f_\lambda)=c_1$ for some fixed $\lambda>0$ and some $f_\lambda$, then $S_\lambda\DD(f_\lambda)=\NN_\lambda(f_\lambda)$ by definition of $\DD$ and $\NN_\lambda$. This leads to the following criterion for the existence of a fixed point of $S_{\lambda^*}\DD$. \claim Lemma(liennln) If there exist a real $\lambda^*>0$ and a fixed point $f_{\lambda^*}$ of $\NN_{\lambda^*}$ with $c_{\lambda^*}(f_{\lambda^*})=c_1$, then $f_{\lambda^*}$ is solution of the functional equation $f=S_{\lambda^*}\DD(f)$. To prove existence of $\lambda^*$ and $f_{\lambda^*}$, we will study the family $\{\NN_\lambda\}$ in a neighborhood $(\lambda^-,\lambda^+)$ of our best numerical value for $\lambda^*$. For each value of $\lambda$ in this neighborhood, the contraction mapping principle will be used to prove existence of a fixed point $f_\lambda$ of $\NN_\lambda$. The maps $\NN_\lambda$ are hyperbolic but not contracting in the neighborhood of their fixed point, due to an unstable direction that, roughly speaking, crosses transversally the manifold of functions with total mass equal to one. To cope with this problem, we will adopt later a standard strategy that consists of applying a variant of Newton's method. Once the existence of a fixed point $f_\lambda=\NN_\lambda(f_\lambda)$ is established for all $\lambda\in[\lamm,\lamp]$, we will show that $c_{\lambda^-}(f_\lamm)0$. However, $c_\lambda(f)$ is not defined for functions $f$ such that $M(f)E(f)=0$ and we will exclude functions of this type, i.e., functions in $$ \HH=\{f\in L^1(\real^+,(1+|x|)\,dx)\ \bigl|\ M(f)E(f)=0\}\,. \equation(defhh) $$ We now state a result about the domains of definition and target spaces of the maps $\NN_\lambda$. \claim Proposition(existenceofnl) For $\alpha\geq0$, $\beta>0$ and $\lambda>0$, $\NN_\lambda$ is well defined as a map from $\BB_\albe\\\HH$ to $\BB_{\sigma\tau}$ for all $\sigma\leq4\alpha/\lambda$ and $\tau\leq\lambda\beta$. \proof The operators $S_\lambda,T$ and the convolution $f\mapsto f*f$ are well defined on $L^1(\real_+)$. Next, we show that the convolution product maps $\BB_\zeet\times\BB_\zeet$ into $\BB_{(4\zeta)\eta}$. Using Fubini's theorem, we get the following inequalities $$\eqalignno{ \|f*g\|_{(4\zeta)\eta}&\leq\int_0^\infty dx\ w_{(4\zeta)\eta}(x) \int_0^x dy\ |f(y)||g(x-y)|&\cr &=\int_0^\infty dy|f(y)|\int_0^\infty dx\ w_{(4\zeta)\eta}(x+y)|g(x)|&\cr &\leq\sup_{x>0\atop y>0}\Bigl({w_{(4\zeta)\eta}(x+y)\over w_{\zeta\eta}(x)w_\zeet(y)}\Bigr)\|f\|_\zeet\|g\|_\zeet&\cr &=\|f\|_\zeet\|g\|_\zeet\,.& \equat(boundonconv)\cr }$$ The last equality follows from $$ \sup_{x>0\atop y>0}{w_{(4\zeta)\eta}(x+y)\over w_\zeet(x)w_\zeet(y)}= \sup_{x>0\atop y>0}\exp\bigl(-\zeta h(x,y)\bigr)\,, $$ and $$ h(x,y)={x+y\over xy}-{4\over x+y}\geq0 $$ for all $x,y>0$. Next, $S_\lambda$ is bounded as a map from $\BB_\zeet$ to $\BB_{(\zeta/\lambda)(\lambda\eta)}\,$: $$\eqalignno{ \|S_\lambda f\|_{(\zeta/\lambda)(\lambda\eta)}&=\int_0^\infty w_{(\zeta/\lambda)(\lambda\eta)}(x/\lambda)|f(x)|dx&\cr &\leq\sup_{x>0}\Bigl({w_{(\zeta/\lambda)(\lambda\eta)} (x/\lambda)\over w_\zeet(x)}\Bigr)\|f\|_\zeet&\cr &=\|f\|_\zeet\,.&\equat(boundongscal)\cr }$$ Therefore $\NN^1_\lambda$ maps $\BB_\albe$ into $\BB_{(4\alpha/\lambda)(\lambda\beta)}$, and hence into $\BB_{\sigma\tau}$ for $\sigma\leq4\alpha/\lambda$ and $\tau\leq\lambda\beta$. In order to check that $\NN^2_\lambda$ maps $\BB_\albe$ into $\BB_{(4\alpha/\lambda)(\lambda\beta)}$, we first note that $T$ is obviously bounded as a map from $\BB_{\zeta\eta}$ to $\BB_{\eta\zeta}\,$, with $$ \|Tf\|_{\eta\zeta}=\|f\|_\zeet\,, \equation(boundonT) $$ since $w_\zeet(1/x)=w_{\eta\zeta}(x)$ for all $x>0$. Using \equ(boundonconv), \equ(boundonT) and the bound on $\NN^1_\lambda$, one concludes that $\NN^2_\lambda(f)\in\BB_{(4\alpha/\lambda)(4\lambda\beta)} \subset\BB_{(4\alpha/\lambda)(\lambda\beta)}$ for $f\in\BB_\albe$. Finally, the bounds \equ(boundonconv), \equ(boundonT), and the fact that the expectation of a function in $\BB_{\sigma\tau}$ is finite for $\tau>0$ imply that $c_\lambda(f)$ is finite for $f\in\BB_\albe\\\HH$ and $\beta>0$. \qed \clm(existenceofnl) together with \clm(rembanach) immediately imply that every fixed point $f\in\BB_\albe$ of $\NN_\lambda$ with $\alpha,\beta>0$ and $\lambda\in(1,4)$ satisfies $f\in\BB\,$. Furthermore, using the regularization properties of the convolution, one can show that every such fixed point is a smooth function. More precisely, we have the following proposition, whose proof can be found in the appendix. \claim Proposition(existenceofnl2) Let $\alpha,\beta>0$, $\lambda\in(1,4)$, and let $f\in\BB_\albe$ be a fixed point of $\NN_\lambda$. Then $f\in\BB$, $f$ is of class $\CC^\infty(\real_+)$, and $f'\in\BB$. The following theorem implies \clm(mainresult). \claim Theorem(thmfornl) Let $\lambda^-\equiv1.7562035$ and $\lambda^+\equiv1.7562048$. Then, {\parindent=2cm \item{\rm(a)} For some $\alpha,\beta>0$, there is a continuous family $\{f_\lambda\}_{\lambda\in[\lamm,\lamp]}$ of functions in $\BB_\albe\\\HH$ such that $\NN_\lambda(f_\lambda)=f_\lambda$ for all $\lambda\in[\lamm,\lamp]\,$, \item{\rm(b)} $c_{\lambda^-}(f_{\lambda^-})0$, and that $$\eqalignno{ |M(f)|&\leq\sup_{x>0}\Bigl({1\over w_{\sigma\tau}(x)}\Bigr)\|f\|_{\sigma\tau}~, &\equat(boundonmete1)\cr |E(f)|&\leq\sup_{x>0}\Bigl({x\over w_{\sigma\tau}(x)}\Bigr)\|f\|_{\sigma\tau}~. &\equat(boundonmete2)\cr }$$ Hence, $f\mapsto E_2(f)$ is continuous as a map from $\BB_\albe$ to $\real$ for every $\alpha,\beta>0$, using in addition the bounds derived in the proof of \clm(existenceofnl). Therefore $f\mapsto c_\lambda(f)$ is continuous as a map from $\BB_\albe\\\HH$ to $\real$ for every $\alpha,\beta>0$ and $\lambda\in\real$. Next, for each $f\in\BB_\albe\\\HH$ with $\alpha,\beta>0$, the map $\lambda\mapsto c_\lambda(f)$ is continuous. The continuity of $\lambda\mapsto c_\lambda(f_\lambda)$ as a map from $[\lamm,\lamp]$ to $\real$ follows. \qed The proof of \clm(thmfornl) is in part computer--assisted. Once (a) is established, the verification of part (b) involves mainly an explicit calculation, that will be given in Section 5.3. The remainder of this section is devoted to the proof of part~(a). First, in order to simplify further our estimates, we introduce yet another family of operators, closely related to $\{\NN_\lambda\}_{\lambda>0}\,$. This family is defined by $$ \NN_{\lambda,\kappa}=S_\kappa\NN_\lambda~, \equation(defnlk) $$ where $\lambda,\kappa>0$. \def\hfl#1#2{\smash{\mathop{\hbox to 12mm{\rightarrowfill}} \limits^{\displaystyle#1}_{\displaystyle#2}}} \def\vfl#1#2{\llap{$\displaystyle #1$}\left\downarrow \vbox to 6mm{}\right.\rlap{$\displaystyle #2$}} \def\diagram#1{\def\normalbaselines{\baselineskip=0pt \lineskip=10pt\lineskiplimit=1pt} \matrix{#1}} \claim Lemma(liennldnl) Let $\alpha\geq0$ and $\beta,\lambda,\kappa>0$. Then $\NN_{\lambda,\kappa}$ is well defined as a map from $\BB_\albe\\\HH$ to $\BB_{\sigma\tau}$ for every $\sigma\leq4\alpha/\kappa\lambda$ and $\tau\leq\kappa\lambda\beta$, and one has $$ \diagram{ \BB_{\alpha\beta} &\hfl{\NN_{\lambda,\kappa}}{} &\BB_{({4\alpha\over\kappa\lambda})(\kappa\lambda\beta)}\cr \vfl{S_{1/\kappa}}{}&&\vfl{}{S_{1/\kappa}}\cr \BB_{(\kappa\alpha)({\beta\over\kappa})} &\hfl{\NN_{\kappa\lambda}}{} &\BB_{({4\alpha\over\lambda})(\lambda\beta)}\cr } \equation(conjug) $$ \proof First, our previous result on the domains of definition and target spaces of $\NN_\lambda$ implies that the operator $\NN_{\lambda,\kappa}$ is well defined as a map from $\BB_\albe\\\HH$ to $\BB_{(4\alpha/\kappa\lambda)(\kappa\lambda\beta)}$ whenever $\alpha\geq0$ and $\beta,\lambda,\kappa>0$. We now show that $$ c_\lambda(f)=c_{\kappa\lambda}(S_{1/\kappa}f)\,. \equation(idforcl) $$ Using \equ(formulec1), we see that $$ c_{\kappa\lambda}(S_{1/\kappa}f)={\kappa\lambda-c_2 E_2(S_{1/\kappa}f)\over2M(S_{1/\kappa}f) E(S_{1/\kappa}f)}\,, $$ and the relations $M(S_{1/\kappa}f)=M(f)$, $E(S_{1/\kappa}f)=\kappa E(f)$ and $E_2(S_{1/\kappa}f)=\kappa E_2(f)$ lead to \equ(idforcl). Using \equ(idforcl), we compute $$\eqalign{ \NN_{\kappa\lambda}(S_{1/\kappa}f)&=c_{\kappa\lambda}(S_{1/\kappa}f) \NN^1_{\kappa\lambda}(S_{1/\kappa}f)+c_2\NN^2_{\kappa\lambda}(S_{1/\kappa}f)\cr &=c_\lambda(f)\NN^1_\lambda(f)+c_2\NN^2_\lambda(f)\cr &=\NN_\lambda(f)\,,\cr }$$ and conclude by observing that $\NN_\lambda=S_{1/\kappa}\NN_\lamka\,$. \qed From \clm(liennldnl) it follows that the fixed points of $\NN_\lambda$ for $\lambda\in[\lamm,\lamp]$ are related to the fixed points of $\NN_{\lamp,\kappa}$ for $\kappa\in[\lamm/\lamp,1]$. Furthermore, the operators $\NN_{\lambda,\kappa}$ are well defined as maps from $\BB_\albe\\\HH$ to $\BB_\albe$ for $\lambda$ and $\kappa$ satisfying $$ {1\over\lambda}\leq\kappa\leq{4\over\lambda}\,. \equation(condonkappa) $$ This condition is easily seen to hold for $\kappa\in[\lamm/\lamp,1]$ and $\lambda\in[\lamm,\lamp]$. As mentioned earlier, the fixed points $f_\lambda$ of the maps $\NN_\lambda$ are not attractive. Numerically, the two largest eigenvalues of $D\NN_\lambda(f_\lambda)$ are roughly $1.37$ and $0.54$ for $\lambda\approx\lambda^*$. In the context of computer--assisted proofs, the standard way of solving a hyperbolic fixed point problem is to turn it into a fixed point problem for a contraction by proceeding in the following way. We choose an invertible linear map $M$ close to the inverse of $1-D\NN_{\lambda^*}(f_{\lambda^*})\,$ and define $$ \MM_{\lamka}=1+M(\NN_\lamka-1)\,. \equation(defmlk) $$ In Section~7.2, we give a detailed description of $M$ and establish its invertibility. Furthermore, we will see that $M$ is bounded in $\BB_\albe$ for all $\alpha,\beta\geq0$. Hence, $\MM_{\lambda,\kappa}$ is well defined as a map from $\BB_\albe\\\HH$ to $\BB_\albe$ for all $\alpha\geq0$, $\beta>0$, and $\kappa,\lambda$ satisfying \equ(condonkappa). The existence of the continuous family of fixed points $\{f_\lambda\}$ will follow from estimates on the contractions $\MM_{\lamp,\kappa}\,$. These estimates are collected in the following proposition. \claim Proposition(hypapcformlk) Let $\lamp$ and $\lamm$ be defined as in \clm(thmfornl). Then, for $\mu=0.5$, $\nu=0.9$ and $r=9\cdot10^{-4}$, there is a function $f^0_\lamp\in\BB_\mn$ and two positive real numbers $q<1$ and $\veps0$. Finally, we prove the continuity of the family $\{f_\lambda\}$. From \clm(existenceofnl2), it follows that $f_\lambda\in\BB\cap\CC^\infty(\real_+)$ with $f_\lambda'\in\BB$. Since the functions $f_{\lamp,\kappa}=S_\kappa f_{\kappa\lamp}$ have the same properties, $\kappa\mapsto S_{1/\kappa}f_{\lamp,\kappa}$ is continuous as a map from $[\lamm/\lamp,1]$ to $\BB_\albe$ provided that the family $\{f_{\lamp,\kappa}\}_\kappa$ is continuous in $\BB_\mn$. In order to show that this is indeed the case, we check that $\{f_\lampka\}_\kappa$ is continuous at $\kappa=\kappa_0$ for each $\kappa_0\in[\lamm/\lamp,1]$. Let us fix such a $\kappa_0$ and denote $f_0\equiv f_{\lamp,\kappa_0}$. First, since the contraction mapping principle and \clm(hypapcformlk) imply that $f_0$ belongs to the ball $B_{\tilde r}(f^0_\lamp)$, where $\tilde r=\veps/(1-q)0$ satisfying $\tilde\veps0$. This implies that there is a $\delta=\delta(\tilde\varepsilon)$ such that $\MM_{\lamp,\kappa}$ maps the ball $\overline{B_{\tilde\veps}(f_0)}$ into itself for each $\kappa$ with $|\kappa-\kappa_0|<\delta\,$: for $f\in\overline{B_{\tilde\veps}(f_0)}$, it follows from the continuity of the map $\kappa\mapsto\MM_{\lamp,\kappa}(f_0)$ and $q<1$ that $$\eqalign{ \|\MM_{\lamp,\kappa}(f)-f_0\|_\mn &\leq\|\MM_{\lamp,\kappa}(f)-\MM_{\lamp,\kappa}(f_0)\|_\mn+ \|\MM_{\lamp,\kappa}(f_0)-f_0\|_\mn\cr &\leq q\tilde\varepsilon+ \|\MM_{\lamp,\kappa}(f_0)-\MM_{\lamp,\kappa_0}(f_0)\|_\mn\cr &<\tilde\varepsilon\,. }$$ Therefore, the contraction mapping principle implies the existence of a fixed point of $\MM_{\lamp,\kappa}$ in the ball $B_{\tilde\veps}(f_0)$ whenever $|\kappa-\kappa_0|<\delta$. By uniqueness of the fixed points $f_{\lamp,\kappa}$, one concludes that $\|f_\lampka-f_{\lamp,\kappa_0}\|_\mn<\tilde\veps$ for all $\kappa$ satisfying $|\kappa-\kappa_0|<\delta$. \qed \clm(hypapcformlk) reduces the proof of \clm(mainresult) to the verification of the estimates \equ(inequalityc1), \equ(precf0lp) and \equ(normdmlpk). This verification is computer--assisted, and yields $q\approx0.85$ and $\varepsilon\approx1.15\cdot10^{-4}$. The function $f^0_\lamp$ has been numerically determined to be a very good approximation of the fixed point $f_\lamp$. It is given by the linear interpolation of $2^{17}$ positive numbers at well chosen points. Regarding the computation of the norm of the tangent map $D\MM_{\lamp,\kappa}(f)$, we will take advantage of the fact that $D\NN_\lambda(f)$ has very good contraction properties on certain subspaces of finite codimension provided $f$ has some regularity. In particular, the nontrivial action of the operator $M$ can be restricted to a finite dimensional subspace, and the computation of the norm of $D\MM_{\lamp,\kappa}(f)$ essentially requires to explicitly evaluate $D\MM_{\lamp,\kappa}(f)$ on finitely many basis vectors. The remainder of this paper is devoted to the proof of \clm(hypapcformlk) and the verification of inequality \equ(inequalityc1). In Section~3, we review the basic approach of computer--assisted proofs, and extend it to function spaces of $L^{1}$--type. In Sections~4 and 5, we give a detailed account of the rigorous implementation on a computer of the maps $\NN_\lambda$ and of the computation of bound \equ(precf0lp) and inequality \equ(inequalityc1). Section~6 is devoted to the tangent maps $D\NN_\lambda$ and their contraction properties, whereas Section~7 deals with $D\MM_{\lamp,\kappa}$ and the computation of the bound \equ(normdmlpk). {\comptr Section.8} is available as a supplement to this paper, and contains the source code of the program (\t{proof.f}) and two input data files (\t{fpoint.lp} and \t{fpoint.lm}). The program has been written in Fortran~77\footnote{$\,^\dagger$}{Mostly standard Fortran~77. Some extensions that can be found on most popular compilers (SUN(TM), Microsoft(TM)PowerFortran,\dots) are used for convenience. In particular, we make use of the double precision complex data type {\smallcomptr complex*16}.} and consists of a (short) main body and several subroutines ordered in a ``bottom--up'' hierarchy, accordingly to the organization of the paper. Except for Section~3, references to the program are collected in remarks at the end of each section. For a description of the input data files, see Sections~5.2 and 5.3. %====================================================== % Section 3 %====================================================== \section Constructive Analysis in $\BB_\albe$ Computer--assisted proofs rely on the ability, first, to discretize the problem under study in terms of objects that are representable on a computer, and, second, to have a rigorous control on the errors arising from the discretization. We note that in this respect, arithmetic operations are special, since controlling them rigorously requires an explicit knowledge of how rounding is performed by the computer. Nevertheless, we emphasize that the main difficulties related to discretization are usually concerned with the specific transformations involved in the functional equation under study, the control of numerical rounding being typically of no particular relevance. To address discretization issues, one introduces the notions of bounds and standard sets. Denoting, for any set $\Sigma$, by $\PP(\Sigma)$ the set of all subsets of $\Sigma$, we start by defining what we call a bound in the context of computer--assisted proofs. \claim Definition(bornes) Let $\varphi$ be a map from $D_\varphi\subseteq\Sigma$ to $\Sigma'$. Denote by $f$ the set map obtained by lifting $\varphi$ in the canonical way, that is, $D_f=\PP(D_\varphi)$ and $ f(S)=\{s^\prime\in\Sigma^\prime\;|\ s^\prime=\varphi(s){\rm~for~some\ } s\in S\} $ for every $S\in D_f$. We say that a set map $g:\PP(\Sigma)\supseteq D_g\rightarrow\PP(\Sigma')$ is a bound on $\varphi$ if $D_f\supseteq D_g$ and if $f(S)\subseteq g(S)$ for all $S\in D_g\,$. We remark that $g$ being a bound on $\varphi$ means $\varphi(s)\in g(S)$ whenever $s\in S\in D_g\,$. Bounds of this type have the following two properties. First, they make it possible to estimate complicated maps in terms of simpler ones, since the composition of two bounds, if well defined, provides a bound on the composition. Second, they can be implemented on a computer. Indeed, given a set $\Sigma$, we start by specifying a finite collection of sets $\std{\Sigma}\subset\PP(\Sigma)$ that can be represented on the computer with a given data type. The elements of $\std{\Sigma}$ are referred to as the {\it standard sets} of $\Sigma$. Next, given a map $\varphi\colon\;\Sigma\supseteq D_\varphi\to\Sigma^\prime$, we construct a bound on $\varphi$ within the class of maps $g\colon\;\std{\Sigma}\supseteq D_g\to\std{\Sigma^\prime}$. Finally, it is in general possible to characterize the images of $g$ in $\std{\Sigma'}$ constructively and implement this map on the computer. We note that one can usually choose $\std{\Sigma}$ and $\std{\Sigma'}$ specifically adapted to the map $\varphi$ in order to improve the bounds $g$ that can be constructed. Unless specified otherwise, the standard sets for a Cartesian product $\Sigma\times\Sigma'$ will be defined by setting $\std{\Sigma\times\Sigma'}=\std{\Sigma}\times\std{\Sigma'}$. \subsection Operations Involving Real and Complex Numbers In our application of the above mentioned procedure to the case of real numbers, we have followed the approach of [KSW] which is based on the 64 bit IEEE standard for floating point arithmetics. This standard specifies two things: a format for floating point numbers (IEEE numbers) and rules concerning rounding after the operations $+,-,*,/$ and $\sqrt{\ }$. We will not discuss the detail of the implementation, but refer the interested reader to the corresponding section of [KSW]. We first choose a subset $\SS$ of IEEE numbers, the ``safe range'', for which no underflows nor overflows can occur. The standard sets of $\real$ and $\real^*_+$ are defined as follows. \goodbreak \claim Definition(defstdreal) We define $\std{\real}$ as the collection of all (closed) intervals $[a,b]$ with $a\leq b$ elements of $\SS$. We define $\std{\real^*_+}$ as the subset of $\std\real$ made of intervals $[a,b]$ with $a>0$. To represent an interval in $\std{\real}$ on the computer, we use for convenience the data type for complex numbers available in Fortran. Given $a\leq b\in\SS$, the procedure \t{sbound} returns the interval $[a,b]$, whereas, given the interval $[a,b]\in\std{\real}$, \t{rl} and \t{ru}, respectively, returns $a$ and $b$. We add two more functions, \t{siconst} and \t{srconst}, which, given $r\in\SS$ an integral constant, and, respectively, $r\in\SS$ an IEEE number, return the (unique) singleton in $\std{\real}$ containing $r$. By using the IEEE specifications related to the rounding occurring after the operations $+,-,*,/$ and $\sqrt{\ }$, one first writes two functions, \t{rup} and \t{rdown}, which, given $r_c$ the rounded result of any of these operations, compute an upper bound and a lower bound, respectively, on the exact result $r$. If these bounds do not belong to $\SS$, a flag is raised and the program stops. In a straightforward manner, one next constructs bounds in $\std{\real}$ (in the sense of \clm(bornes)) on the maps $x\mapsto -x$ (\t{sneg}), $|x|$ (\t{sabs}), $1/x$ (\t{sinv}), $x^2$ (\t{spower2}), $\sqrt{x}$ (\t{ssqrt}), and $(x,y)\mapsto x+y$ (\t{ssum}), $x-y$ (\t{sdiff}), $x*y$ (\t{sprod}) and $x/y$ (\t{squot}). We will also need a bound on the function $x\mapsto\exp(x)$. The precision with which this function is evaluated is not specified by the IEEE standard. Hence, we make use of the bounds constructed so far and compose them in the following way. First, we use $\exp(nx)=\exp(x)^n$ to restrict the Taylor expansion of $\exp(x)$ to cases where $|x|<0.03$. Next, we compute the first three terms in the expansion and bound the tail by a geometrical series. This bound is implemented in the function \t{sexp}. We note that it is only involved in the computation of the weight $w_\albe$ and is not required to be of great accuracy. Finally, we will need to evaluate for $x$ close to zero and $n=0,\dots,3,$ $$ \tflog_{n}(x)\equiv-(-x)^{-n}\sum_{k=n+1}^\infty{(-x)^k\over k}. \equation(logn) $$ Note that the second factor is just the tail of the Taylor expansion of $\log(1+x)$. In particular, $\tflog_{0}(x)=\log(1+x)$. One easily checks that the inequalities $$ -\sum_{k=1}^m{(-x)^k\over k+n}-\Bigl|{x^{m+1}\over m+n+1}\Bigr|\leq \tflog_n(x) \leq-\sum_{k=1}^m{(-x)^k\over k+n}+\Bigl|{x^{m+1}\over m+n+1}\Bigr| $$ are valid for all $m\geq1$. With $m=4$, a sufficiently accurate bound is constructed in \t{selogne} from the previous inequalities. We end this section with the discussion of a bound on the discrete convolution. For $r=(r_0,\dots,r_{n-1})$ and $s=(s_0,\dots,s_{n-1})\in\real^n$, the discrete convolution $r*s$ is the element of $\real^{2n-1}$ given by $$ (r*s)_k=\sum_{i+j=k}r_is_j, \quad k=0,\dots,2n-2. \equation(discreteconv) $$ Computing $r*s$ according to \equ(discreteconv) involves $\OO(n^2)$ operations, and becomes impractical for large $n$. The standard strategy is to go into Fourier space where the convolution becomes the pointwise product of vectors. The gain in computational time is due to the fact that the discrete Fourier transform can be implemented with an $\OO(n\log_2n)$ algorithm, known as the Fast Fourier Transform (FFT). More precisely, the discrete Fourier transform is a map from $\complex^n$ to $\complex^n$ defined by $$ \bigr(\FF(z)\bigl)_k=\sum_{j=0}^{n-1}\exp\Bigl(i{2\pi kj\over n}\Bigr)z_j, \quad k=0,\dots,n-1, \equation(ftrans) $$ for $z=(z_0,\dots,z_{n-1})\in\complex^n$. The inverse Fourier transform $\FF^{-1}$ is given by $$ \bigr(\FF^{-1}(z)\bigl)_k={1\over n}\sum_{j=0}^{n-1}\exp \Bigl(-i{2\pi kj\over n}\Bigr)z_j, \quad k=0,\dots,n-1. \equation(iftrans) $$ For $r,s\in\real^n$ as above, one has the well known relation $$ (r*s)_k=(\FF^{-1}(\FF(\tilde r)\cdot\FF(\tilde s)))_k, \quad k=0,\dots,2n-2, \equation(dconvolution) $$ where $\tilde r=(r_0,\dots,r_{n-1},0,\dots,0)\in\real^{2n}$ and $\tilde s=(s_0,\dots,s_{n-1},0,\dots,0)\in\real^{2n}$. An efficient implementation of \equ(ftrans) and \equ(iftrans) follows from the observation that if $n$ is odd, then $\FF(z)$ can be decomposed into the sum of the Fourier transform of two vectors in $\complex^{n/2}$. Hence, for $n$ a power of $2$, one can repeat this decomposition $\log_2n$ times until only the Fourier transform of a single complex number remains to be computed. $\FF(z)$ is finally obtained by summing the intermediate Fourier transforms, which needs $\OO(n)$ operations. In order to implement a bound on the discrete Fourier transform, we first need to choose the standard sets of $\complex$. For our purpose, it is sufficient to define them in terms of $\std{\real}$ as follows. \claim Definition(defstdcomplex) We define $\std{\complex}$ to be the collection of all sets $R+i*I$ of the form $$ R+i*I=\bigl\{x+i\cdot y\in\complex\;|\ x\in R,\, y\in I\bigr\} \equation(stdcomplex) $$ with $R$ and $I$ elements of $\std{\real}$. The only operations in $\complex$ involved in the FFT algorithm are the addition and pro\-duct, bounds on which are readily implemented from our bounds acting on $\std{\real}$. One also needs bounds on the trigonometric factors appearing in \equ(ftrans) and \equ(iftrans). From the periodicity properties of the functions $\sin$ and $\cos$, one first notes that it is sufficient to construct a bound on the maps $(l,m)\mapsto\sin(l\pi/m),\cos(l\pi/m)$ where $m\geq4$ is a power of $2$ and $l$ ranges in $\{0,\dots,m/4\}$. The case $l=0$ is trivial. For $l=1$ and $m=4,8,\dots$, one evaluates $\sin$ and $\cos$ recursively: For $m=4$ one has $\cos(\pi/4)=\sin(\pi/4)=1/\sqrt2$, and for $m>4$ a power of 2 one uses the half angle formulas $$ \cos(x/2)=\sqrt{{1+\cos(x)\over2}}\,,\quad \sin(x/2)={1\over2}{\sin(x)\over\cos(x/2)}\,. $$ Finally, for $l>1$, one applies the double angle formulas. \remark Bounds on the Fourier transform and inverse Fourier transform are implemented in the procedure \t{fft} according to the FFT algorithm. We do not enter into the details of this algorithm, and refer the reader to [PFTV], from which the code has been adapted to interval analysis using the bounds described above. Adapting again to interval analysis a code from [PFTV], a bound on the discrete convolution $r\mapsto r*r$ is implemented in the procedure \t{fastconvolution1}, while the general case $(r,s)\mapsto r*s$ is implemented in \t{fastconvolution2}. Those bounds are restricted to vectors whose dimension is a power of $2$. In the sequel, we actually compute the discrete convolution of vectors of the form $r=(0,r_1,\dots,r_n,0)$, $n$ a power of $2$. The first two elements and last two elements of such convolution are trivially zero and are updated directly in the procedures \t{fastconvolution1} and \t{fastconvolution2}. For convenience later on, see Section~4.4, we also add at the beginning and the end of the result one element zero. \subsection Standard Sets of $\BB_\albe$ We now describe the standard sets of the Banach spaces $\BB_\albe$. As mentioned above, the choice of these sets should be adapted to the problem in order to optimize the bounds that one needs to construct. Although functions in $\BB_\albe$ are in general irregular, the fixed points of the maps $\NN_\lambda$ are smooth. Furthermore, these maps are continuous and preserve the regularity. Therefore, we will take for our standard sets of $\BB_\albe$ balls centered at regular functions. To represent a regular function on the computer, we will rely on the approximation scheme of spline interpolation. A spline function of order $n$ is a function in $\CC^{n-1}$ which is piecewise polynomial, each of the polynomials being of degree $n$. For our purpose, it is sufficient to consider splines of order one as the centers of our standard sets, i.e., continuous piecewise affine functions. This choice is a compromise between the quality of the approximation and the simplicity of the bounds that we will have to construct. Note that increasing the order of the interpolation does not lead in general to better approximations. Indeed, for a function $f\in\CC^\infty([a,b\,])$ and a typical partition of $[a,b\,]$ with mesh size $\veps>0$, the associated interpolation of $f$ by a spline $g$ of order $n-1$ satisfies for a norm of $L^1$--type $$ \|f-g\|\approx\veps^n\|f^{(n)}\|. $$ Hence, depending upon the behavior of $f^{(n)}$, it can become better to consider finer partitions of $[a,b\,]$ rather than to increase the order of the interpolation. We now introduce a few objects that will be used to define the standard sets of $\BB_\albe$. \claim Definition(defpartition) For $n\geq2$, we denote by ${\cal P}_n$ the set of all partitions $p$ of $\real^*_+$ of the form $$ p=\{00$. The uniform partitions have been introduced in order to simplify the implementation of a bound on the convolution operator. For $p=\{x_i\}_{i=0}^n\in\PP_n$ and $\lambda>0$, we adopt the convention to denote by $\lambda p$ the partition $\{\lambda x_i\}_{i=0}^n$. Next, we describe more precisely the piecewise affine functions we will work with. \claim Definition(defaffpm) We define ${\cal A}$ to be the set of all functions $\rho\in\CC^0(\real_+)$ for which there is an $n\geq2$ and a partition $p=\{x_i\}_{i=0}^n\in\PP_n$ such that $\rho$ is affine on $[x_{i-1},x_i]$ for $i=1,\ldots,n$ and $\rho(x)=0$ for $x\not\in(x_0,x_n)$. Furthermore, $\AA^u$ denotes the subset of $\AA$ consisting of those functions for which $p$ can be chosen uniform. We note that $\AA,\AA^u\subset\banach\alpha\beta$ for all $\alpha,\beta\geq0$. Given a partition $p=\{x_i\}_{i=0}^n\in\PP_n$ and a set of values $v=\{v_i\}_{i=0}^n\in\real^{n+1}$ satisfying $v_0=v_n=0$, we denote by $\TT_1(p,v)$ the linear interpolation of $(p,v)$ in $\AA$, i.e., $$ \TT_1(p,v)(x)=\left\{\matrix{ v_i+{v_{i+1}-v_i\over x_{i+1}-x_i}(x-x_i)\ {\rm for}\ x\in[x_i,x_{i+1}]\ {\rm and}\ i\in\{0,\dots,n-1\},\cr \cr 0\qquad{\rm otherwise}.\hfill\cr }\right. \equation(interpolation) $$ Conversely, associated with every function $\rho\in\AA$, there is a pair $(p,v)$ in ${\cal P}_n\times\real^{n+1}$, for some $n\geq2$, satisfying $\TT_1(p,v)=\rho$. If $\rho\not\equiv0$ and if one imposes a minimality condition on $n$, then the associated pair $(p,v)$ is unique. \claim Definition(definvss) For $\rho\not\equiv0$ a function in $\AA$, let $$ n(\rho)\equiv\min\{n\geq2\;|\ \exists\ (p,v)\in\PP_n\times\real^{n+1}\ {\rm such\ that}\ \TT_1(p,v)=\rho\}, $$ and define $\pi(\rho)$ to be the (unique) element of $\PP_{n(\rho)}\times\real^{n(\rho)+1}$ satisfying $\TT_1\bigl(\pi(\rho)\bigr)=\rho$. Note that by definition of $\AA$, one has always $\pi(\rho)=(\cdot,\{\rho_i\}_{i=0}^{n(\rho)})$ with $\rho_0=\rho_{n(\rho)}=0$. In order to define the standard sets of $\AA$ and $\AA^u$, we need to choose the standard sets of $\PP_n$ and $\PP_n^u$. \claim Definition(stdpartition) For $n\geq2$, we define $\std{{\cal P}_n}$ to be the collection of all sets $(X_0,\dots,X_n)$ of the form $$ (X_0,\dots,X_n)=\bigl\{\{x_i\}_{i=0}^n\in{\cal P}_n\;|\ x_0\in X_0, \dots,x_n\in X_n\}, \equation(defstdpart1) $$ with $X_0,\dots,X_n$ any increasing sequence of $n+1$ pairwise disjoint elements of $\std{\real^*_+}$. Similarly, we define $\std{{\cal P}^u_n}$ as the collection of all sets $(A,E)$ of the form $$ (A,E)=\bigl\{p\in\PP^u_n\;|\ p=\{a+i\veps\}_{i=0}^n, a\in A\ {\rm and}\ \veps\in E\bigr\}, \equation(defstdpart2) $$ with $A,E\in\std{\real^*_+}$. Note that $\std{\PP^u_n}$ is not a subset of $\std{\PP_n}$. Indeed, the sets $(A,E)$ contain only uniform partitions, whereas there are always non--uniform partitions in each set $(X_0,\dots,X_n)$ which is not a singleton. The standard sets of $\AA$ and $\AA^u$ are defined in terms of $\std{\PP_n}$ and $\std{\PP^u_n}$ as follows. \claim Definition(defstdpwaf) Let $N=2^{20}$. We define $\std\AA$, respectively $\std{\AA^u}$, to be the collec\-tion of all sets $(P,V)$ of the form $$ (P,V)=\bigl\{\rho\in\AA\;|\ \rho=\TT_1(p,v), p\in P\ {\rm and}\ v\in V\bigr\}, \equation(stdpwaf) $$ with $P\in\std{\PP_n}$, respectively $P\in\std{\PP_n^u}$, $V\in\std{\real^{n+1}}$ and $2\leq n\leq N$. Finally, we introduce the standard sets of $\BB_\albe$. They will be of two types, denoted by $\std{\BB_\albe}$ and $\std{\BB_\albe}^u$. \claim Definition(defstdbanach) Let $\alpha\geq0$ and $\beta\geq0$. We define $\std{\BB_\albe}$, respectively $\std{\BB_\albe}^u$, to be the collection of all sets $(P,V,G)$ of the form $$ (P,V,G)=\bigl\{f\in\BB_\albe\;|\ f=\rho+g,\ \rho\in(P,V),\ g\in\BB_\albe\ {\rm and\ }\|g\|_\albe\leq G\bigr\}, \equation(stdbanach) $$ with $(P,V)\in\std\AA$, respectively $(P,V)\in\std{\AA^u}$, and $G\in\SS$, $G\geq0$. Hence, a set $(P,V,G)$ is the union of all balls of radius $G$ that are centered at piecewise affine functions belonging to $(P,V)$. \remark In our program, the data type with which a set $(P,V,G)$ is represented, with $P\in\std{\PP_n}$ and $V\in\std{\real^{n+1}}$, is a $2\times(n+2)$ matrix, say \t{f}, with entries of complex data type. (Recall that we use the Fortran data type for complex numbers to represent the elements of $\std{\real}$ and $\std{\real^*_+}$.) The entries \t{f($1$,$0$)} up to \t{f($1$,$n$)} contain $V$, and \t{f($1$,$n+1$)} contains the interval $[0,G]$. The entries \t{f($0$,$0$)} up to \t{f($0$,$n$)} contain the partition $P$. If $P=(A,E)\in\std{\PP_n^u}$, then \t{f($0$,$0$)}$=A$ and \t{f($0$,$n+1$)}$=E$. If $P\not\in\std{\PP_n^u}$, then \t{f($0$,$n+1$)}$=[0,0]$. Given an integer $n\geq2$ and $a,\varepsilon\in\SS$, $a,\varepsilon>0$, the procedure \t{fzero} returns a standard set $(P,V,G)\in\std{\BB_{\alpha\beta}}^u$ where $G=0$, $V=([0,0],\dots,[0,0])$ and $P$ contains the partition $\{a+i\varepsilon\}_{i=0}^n$. Finally, given $(P,V,G)\in\std{\BB_{\alpha\beta}}$ and an integer $i$, the procedure \t{get\_f\_on\_i} returns two elements of $\std{\real^*_+}$ and two elements of $\std{\real}$ containing respectively $x_{i-1},x_i,\rho_{i-1}$ and $\rho_i$ for all $\rho\in(P,V)$, $\pi(\rho)=(\{x_j\},\{\rho_j\})_{j=0}^n$. We end this section with a few comments about the strategy that we will adopt when constructing bounds on the various maps that enter the definition of $\NN_\lambda$. Some of these maps are linear and preserve $\AA$. Let $\LL$ be such a map. Then, for $f=\rho+g$ with $\rho\in\AA$, the piecewise affine part $\rho$ and the general term $g$ can be treated separately, and since the piecewise affine parts carry the relevant information, it is natural to describe the affine part of $\LL(f)$ by $\LL(\rho)$ and its general term by $\LL(g)$. Moreover, the choice of the standard set image in $\std{\AA}$ containing $\LL(\rho)$ is straightforward. For instance, the product of a function $f\in\BB_\albe$ by a scalar $\lambda\in\real$ is bounded using $\pi(\lambda\rho)=(p,\lambda v)$, where $(p,v)=\pi(\rho)$, and $\|\lambda g\|_\albe=|\lambda|\,\|g\|_\albe$. This bound is implemented in the procedure \t{fmult}. In general, however, the maps that will be considered do not preserve $\AA$. Let $\UU:\BB_\albe\rightarrow\BB_\zeet$ be such a transformation. For $f=\rho+g$ with $\rho\in\AA$ and $g\in\banach\alpha\beta$, we write $$ \UU(\rho+g)=\UU(\rho)+\WW(\rho,g). $$ Since $\UU(\rho)\not\in\AA$, we will consider the linear interpolation $\tilde\rho\in\AA$ of $\UU(\rho)$ at well chosen points. This choice will usually be a compromise between the quality of the approximation and the simplicity of the implementation. One then has $$ \UU(\rho+g)=\tilde\rho+\WW(\rho,g)+\bigl(\UU(\rho)-\tilde\rho\bigr), $$ and the last two terms on the RHS correspond to the general term $\tilde g$ of $\UU(\rho+g)$. They will be bounded using first $$ \|\tilde g\|_\zeet\leq\|\WW(\rho,g)\|_\zeet+\|\UU(\rho)-\tilde\rho\|_\zeet. $$ Next, explicit formulas involving $\rho$, $g$ and the values of $\UU(\rho)$ at the chosen interpolation points, together with the use of interval analysis, will lead to a rigorous upper bound on the previous expression, and hence to the representable $G\in\SS$ entering \clm(defstdbanach). We note that the elements of $\std{\real}$ defining the piecewise affine function $\tilde\rho$ consist in general of intervals of non zero length. Nevertheless, since the bound $G$ has been computed for all reals in those intervals, one can ``close'' each of them by picking arbitrarily one of the representable numbers it contains. This will prevent the standard sets containing the piecewise affine part from ``opening up'' substantially when bounds are composed, in particular when evaluating convolution products. %====================================================== % Section 4 %====================================================== \section Operations Involving Functions In this section, we construct bounds (in the sense of Section~3) on the various maps that enter the definition of the transformations $\NN_\lambda$. Most of the bounds given here follow from direct calculations and are easy to prove. We have grouped some of these calculations in the appendix. In the following, we will usually consider $f\in\banach\alpha\beta$ of the form $f=\rho+g$, where $\rho\in\AA$ will always stand for the piecewise affine part of $f$ and $g\in\banach\alpha\beta$ for the general term of $f$. Furthermore, when not explicitly mentioned otherwise, $n(\rho)$ and $\pi(\rho)=(p,v)$ will be denoted by $n$ and $(\{x_i\}_{i=0}^n,\{\rho_i\}_{i=0}^n)$, respectively. Finally, we denote the interval $[x_{i-1},x_i]$ by $I_i$, $i=1,\dots,n$. \subsection Elementary Operations We start with the map $f\mapsto\|f\|_\albe$, a bound on which is constructed from $\std{\BB_\albe}$ to $\std{\real_+}$ using the triangle inequality and, for the piecewise affine part, the estimate $$\eqalignno{ \|\rho\|_\albe&=\sum_{i=1}^n\int_{I_i}w_\albe(x)\,|\rho(x)|\,dx&\cr &\leq\sum_{i=1}^n\sup_{x\in I_i}\bigl(w_\albe(x)\bigr)(x_i-x_{i-1}) {|\rho_{i-1}|+|\rho_i|\over2}. &\equat(normpl)\cr }$$ The convexity of $w_\albe$ leads to $$ \sup_{x\in I_i}w_\albe(x)=\max\{w_\albe(x_{i-1}),w_\albe(x_i)\}. \equation(supw) $$ We next consider the mass $M(f)$ and the expectation $E(f)$ of a function $f\in\banach\alpha\beta$, for which it will be sufficient to construct bounds from $\std{\BB_\albe}^u$ to $\std\real$. By linearity, we can first treat separately the affine part $\rho$, and by using $\rho_0=\rho_n=0$, a direct calculation yields $$\eqalignno{ &M(\rho)=\varepsilon\sum_{i=1}^{n-1}\rho_{i}, &\equat(masspl)\cr &E(\rho)=\veps\sum_{i=1}^{n-1}\rho_i x_i, &\equat(expectpl)\cr }$$ where $\varepsilon=x_1-x_0$. Using $M(\rho)-|M(g)|\leq M(f)\leq M(\rho)+|M(g)|$ and the corresponding inequality for $E(f)$, we get the desired bounds by estimating the mass and expectation of the general term $g$ with \equ(boundonmete1) and \equ(boundonmete2). The supremum of $1/w_\albe$ appearing in \equ(boundonmete1) is taken at $x_c=\sqrt{\alpha/\beta}$, which leads to $$ \sup_{x>0}\Bigl({1\over w_\albe(x)}\Bigr)=\exp(-2\sqrt{\alpha\beta}). \equation(supofinvw) $$ Similarly, one computes $$ \sup_{x>0}\Bigl({x\over w_\albe(x)}\Bigr)= {1+\sqrt{1+4\alpha\beta}\over2\beta} \exp\bigl(-\sqrt{1+4\alpha\beta}\bigr). \equation(xcforsupofxonvw) $$ We end this section with the discussion of a bound on the addition of two functions $f_1$, $f_2\in\banach\alpha\beta$. Due to the linearity of this map and the fact that the addition of two functions in $\AA$ is again in $\AA$, it is natural to choose for the general term of $f_1+f_2$ the addition of the general terms of those two functions, whose norm is bounded by using the triangle inequality. It then remains to construct a bound on the map $+:\AA\times\AA\rightarrow\AA$. Let $\rho_1,\rho_2\in\AA$ such that $\pi(\rho_1)=(p_1,v_1)\in\PP_n\times\real^{n+1}$ and $\pi(\rho_2)=(p_2,v_2)\in\PP_m\times\real^{m+1}$. If $\rho_1$ and $\rho_2$ have no common nodes, then $(p,w)\equiv\pi(\rho_1+\rho_2)\in\PP_{n+m+1}\times\real^{n+m+2}$, with $p$ being the refined partition made of the ordered union of $p_1$ and $p_2$. The last is valid only if $\rho_1$ and $\rho_2$ have no common nodes and we shall construct a bound whose domain is restricted to such cases. Hence, denoting $p=\{y_i\}_{i=0}^{n+m+1}$ and $w=\{w_i\}_{i=0}^{n+m+1}$, there is, for each $i=0,\dots,n+m+1$, a unique $j$ such that $$ w_i\equiv(\rho_1+\rho_2)(y_i)=\cases{ (v_1)_j+\rho_2((p_1)_j)& if $y_i\in p\cap p_1$,\cr (v_2)_j+\rho_1((p_2)_j)& if $y_i\in p\cap p_2$.\cr } $$ To implement this bound with interval analysis, we must check first that the nodes in $\std{\real^*_+}$ of the standard sets $P_1\in\std{\PP_{n}}$ and $P_2\in\std{\PP_{m}}$ containing the partition $p_1$ and $p_2$ are pairwise disjoint intervals. This implies that every function in $(P_1, V_1)$ is linear on each node of $P_2$, and vice versa. This in turn implies that a bound on the evaluation $\rho_2((p_1)_j)$ and $\rho_1((p_2)_j)$ is readily obtained from \equ(interpolation) using interval analysis. We note finally that when the standard set containing $\rho_1$ ($\rho_2$) is in $\std{\AA^u}$, i.e., with $P_1$ ($P_2$) of the form $(A,E)$, we first proceed to the evaluation of the nodes in terms of $A$ and $E$. Finally, one constructs a bound on the difference of two functions by composing the previous bound with a bound on the unary minus $\banach\alpha\beta\ni f\mapsto-f$ obtained from $||-g||_\albe=||g||_\albe$ and $\pi(-\rho)=(p,-v)$, where $(p,v)=\pi(\rho)$. \remark A bound on the weight $w_{\alpha\beta}$ is computed in the procedure \t{sw}. The inequality \equ(normpl) is implemented in \t{snorm\_pl}, whereas \equ(supw) and \equ(xcforsupofxonvw) are implemented in \t{ssup\_of\_w} and \t{ssup\_of\_x\_over\_w}, respectively. For several intervals $I\subset\real^*_+$, we will need to evaluate later the quantities $\sup_{x\in I}1/w_\albe(x)$, $\int_I w$, and $|I|^{-1}\int_I w$, where $|I|$ denotes the length of $I$. By using the convexity of $w$, a bound on the first quantity is computed in \t{ssup\_of\_winverse}, whereas the other two quantities are bounded in \t{sint\_of\_w}. The other bounds described in this section are implemented in the procedures \t{snorm}, \t{smass}, \t{sexpectation}, \t{fadd} and \t{fdiff}. \subsection The Scaling Operator It will be sufficient for our purpose to construct a bound on $$\eqalign{ S_\lambda:\BB_\albe&\rightarrow\BB_{{\alpha\over4}\gamma} \cr f(x)& \mapsto h(x)=\lambda f(\lambda x),\cr }\equation(lambda) $$ acting from $\std{\BB_\albe}^u$ to $\std{\BB_{(\alpha/4)\gamma}}^u$. We recall that the scaling operators are bounded under constraints which translate in this particular case into $\lambda\leq4$ (if $\alpha>0$) and $\gamma\leq\lambda\beta$. It will be checked by the program that these inequalities are satisfied for the values of $\lambda,\alpha,\beta$ and $\gamma$ we will use. Since $S_\lambda$ is linear and preserves $\AA^u$, we can treat separately the piecewise affine part and the general term. A bound on $S_\lambda:\AA^u\rightarrow\AA^u$ is obtained from the relation $\pi(S_\lambda\rho)=(p/\lambda,\lambda v)$, where $(p,v)=\pi(\rho)$. For the general term we estimate using \equ(supofinvw), $$\eqalignno{ \|S_\lambda g\|_{{\alpha\over4}\gamma}& \leq\sup_{x>0}{w_{{\alpha\over4}\gamma} (x/\lambda)\over w_\albe(x)}\,\|g\|_\albe&\cr &=\exp\Bigl(-2\sqrt{\alpha(1-\lambda/4)(\beta-\gamma/\lambda)}\,\Bigr)\, \|g\|_\albe,& \equat(boundgscal)\cr }$$ the last equality being valid under the conditions on $\lambda,\alpha,\beta$ and $\gamma$ mentioned above. Note that the scaling operator \equ(lambda) is a strict contraction for $\gamma/\beta<\lambda<4$. This will be used to improve the bound on $\NN^1_\lambda(f)=S_\lambda f*S_\lambda f$ that we shall construct later. In \equ(lambda), taking a larger target space, i.e., $\BB_{(\alpha/\sigma)\gamma}$ with $\sigma>4$, would lead to a better contraction. Nevertheless, $\sigma=4$ is the largest value for which $\NN^1_\lambda$ maps $\BB_{\alpha\beta}$ into $\BB_{\alpha\gamma}$, cf. \equ(boundonconv). \remark A bound on $S_\lambda:\AA^u\rightarrow\AA^u$ is implemented in the procedure \t{fscale\_pl}, and the bound \equ(boundgscal) in \t{fscale\_gen}. Those two procedures are called in \t{fscale} to build the desired bound on the operator \equ(lambda). \subsection The Operator $T$ We now construct a bound on the operator $$\eqalign{ T:\BB_\albe&\rightarrow\BB_\beal\cr f(x)&\mapsto h(x)={1\over{x^2}}f({1\over x}),\cr }\equation(inverse) $$ acting from $\std{\BB_\albe}^u$ to $\std{\BB_\beal}$. For $f=\rho+g$ with $\rho\in\AA^u$ and $g\in\BB_\albe$, one has $Tf=T\rho+Tg$. Since $T\rho$ is not piecewise linear, we must first choose a function $\tilde\rho\in\AA$ which approximates $T\rho$. Denoting again $\pi(\rho)=(\{x_i\}_{i=0}^n,\{\rho_i\}_{i=0}^n)$, we consider for $\tilde\rho$ the linear interpolation of $T\rho$ at the nodes $$ \tilde x_i=1/x_{n-i},\quad i=0,\dots,n. \equation(interpopt) $$ Therefore, we define $\tilde\rho$ to be $$ \tilde\rho=\TT_1(\tilde p,\tilde v), \equation(defaffopt) $$ where $\tilde p=\{\tilde x_i\}_{i=0}^n$, and $\tilde v=\{\tilde\rho_i\}_{i=0}^n$ with $$ \tilde\rho_i\equiv(T\rho)(\tilde x_i)=x^2_{n-i}\rho_{n-i}\,, \quad i=0,\dots,n. \equation(valueopt) $$ Next, the general term $\tilde g$ of $Tf$ is given by $\tilde g=T\rho-\tilde\rho+Tg$, and we use \equ(boundonT) to estimate $$ \|\tilde g\|_\beal\leq\|T\rho-\tilde\rho\|_\beal+\|g\|_\albe. \equation(defgenopt) $$ In order to bound the first term on the RHS of \equ(defgenopt), we use again \equ(boundonT) together with the linearity of $T$ and the fact that it is an involution. This leads to $$\eqalignno{ \|T\rho-\tilde\rho\|_\beal&=\|\rho-T\tilde\rho\|_\albe&\cr &\leq\sum_{i=1}^{n}\sup_{x\in I_i}w_\albe(x) \int_{I_i}\bigl|(\rho-T\tilde\rho)(x)\bigr|\,dx.&\equat(taa)\cr }$$ Finally, an explicit bound on the integral appearing in the previous expression follows from a direct calculation and is given in the next lemma. \claim Lemma(gentermopt) Let $\rho\in\AA^u$, and $\tilde\rho$ be defined as in \equ(defaffopt). With $\pi(\rho)=(\{x_i\}_{i=0}^n,\{\rho_i\}_{i=0}^n)$, $\varepsilon=x_1-x_0$ and $I_i=[x_{i-1},x_i]$, one has $$\displaylines{ \qquad \int_{I_i}|(\rho-T\tilde\rho)(x)|\,dx \leq{\veps\over4} \Bigl( \log\Bigl(1+{\varepsilon\over x_{i-1}}\Bigr)|\rho_{i}-\rho_{i-1}| \hfill\cr \hfill +\veps\Bigl|{\rho_{i}\over x_{i-1}}-{\rho_{i-1}\over x_{i}}\Bigr|+ \veps(x_{i}- \veps/2)\Bigl|{\rho_{i}\over x_{i-1}^2}-{\rho_{i-1}\over x_{i}^2}\Bigr|\Bigr). \qquad\equat(intgeneralopt)\cr }$$ \remark A bound on $T:\BB_\albe\rightarrow\BB_\beal$ is implemented as described here in the procedure \t{ft}. \subsection The Convolution For $\alpha,\beta>0$ and $\gamma\in[\alpha,4\alpha]$, we consider in this section the operator $$\eqalign{ C:\BB_\albe\times\BB_\albe&\rightarrow\BB_{\gamma\beta} \cr (f,h)& \mapsto f*h.\cr }\equation(opconvolution) $$ As mentioned before, we specifically introduced standard sets of piecewise affine functions defined on uniform partitions in order to simplify the construction of a bound on the convolution. Hence, our bound will act from $\std{\BB_\albe}^u\times\std{\BB_\albe}^u$ to $\std{\BB_\gabe}^u$. To simplify further the explicit expressions which we shall derive below, we restrict its domain to pairs $(F_1,F_2)$ for which the standard sets $(A_1,E_1)$ and $(A_2,E_2)$ containing the partitions associated with the affine functions in $F_1$ and $F_2$ satisfy $E_1=E_2$ and both $E_1$ and $E_2$ are singletons. This ensures that all affine functions in $F_1$ and $F_2$ are defined on (uniform) partitions with identical mesh size. Let $f=\rho+g_f$ and $h=\sigma+g_h$, with $\rho,\sigma\in\AA^u$ and $g_f, g_h\in\BB_\albe$. Then, one has $$ f*h=\rho*\sigma+\rho*g_h+g_f*h. \equation(convolution) $$ The relevant information is carried by the term $\rho*\sigma$. Since it does not belong to $\AA$, we will proceed as in the previous section and approximate it by a function $\tilde\rho\in\AA^u$. The general term of $f*h$ will be given by $$ \tilde g=(\rho*\sigma-\tilde\rho)+\rho*g_h+g_f*h. \equation(gentermconv) $$ One can estimate the last two terms on the RHS of the previous expression using the bound \equ(boundonconv). However, estimating the norm of the first term requires an explicit expression for $(\rho*\sigma)(x)$, $x>0$. We now derive this expression, which will be used also to specify $\tilde\rho$. We first state an intermediate result whose proof can be found in the appendix. \claim Lemma(rhoprhop) If $\rho,\sigma\in\AA^u$ have uniform partitions with identical mesh size $\varepsilon$, then $(\rho*\sigma)''\in\AA^u$. Furthermore, assume $n(\rho)=n(\sigma)\equiv n$, denote $\pi(\rho)=(\{x_i\},\{\rho_i\})_{i=0}^n$, $\pi(\sigma)=(\{y_i\},\{\sigma_i\})_{i=0}^n$, and define $\{S_k\}_{k=0}^{2n}$ to be the discrete convolution of $\{\rho_i\}_{i=0}^n$ and $\{\sigma_i\}_{i=0}^n$, i.e., $$ S_k=\sum_{i+j=k}\rho_i\sigma_j. \equation(discrconv) $$ Then $n((\rho*\sigma)'')=2n$, and $(\rho*\sigma)''=\TT_1\bigl(\{z_k\}_{k=0}^{2n},\{v_k\}_{k=0}^{2n}\bigr)$ where $$\eqalignno{ z_k&=x_0+y_0+k\varepsilon,&\equat(partz)\cr v_k&={1\over\varepsilon}\bigl(S_{k+1}-2S_k+S_{k-1}\bigr),&\equat(valz)\cr } $$ with the convention $S_{-1}=S_{2n+1}=0$. We now specify the nature of $\rho*\sigma$. \claim Lemma(rhorho) Let $\rho,\sigma\in\AA^u$ as in \clm(rhoprhop) and define $p$ to be the partition $\{z_k\}_{k=0}^{2n}$ where $z_k$ is given by \equ(partz). Then $\rho*\sigma$ is in $\CC^2(\real^+)$, has support in $[z_0,z_{2n}]$ and is there identical to its natural cubic spline approximation $\varphi$ at the nodes of $p$, i.e., with the boundary conditions $\varphi''(z_0)=\varphi''(z_{2n})=0$. \proof First, it follows from $\supp(\rho)\subseteq(x_0,x_n)$ and $\supp(\sigma)\subseteq(y_0,y_n)$ that the support of $\rho*\sigma$ is in $(x_0+y_0,x_n+y_n)$. Then, the regularity properties of the convolution imply $\rho*\sigma\in\CC^2(\real^+)$. To see that $\rho*\sigma$ is equal to its natural cubic spline approximation at the nodes $a\equiv z_00$, the interval $(x_0,x_n)$ is chosen to be the smallest interval such that $|\rho(x)|<\tau$ for $x\not\in(x_0,x_n)$. This interval might be fairly different from $\supp(\rho)$, leading to a mesh $\varepsilon$ smaller than $|\supp(\rho)|/n$ and hence a better approximation of $\rho$ on regions where the information is more relevant. The cutoff $\tau$ may vary from place to place in the proof and has been determined empirically. \remark Given an integer $n\geq2$, and two positive representable numbers $x_0$ and $s$, the procedure \t{fidentity} constructs a standard set in $\std{\PP_n^u}$ containing the uniform partition $p$ with $\supp(p)=(x_0,x_0+s)$, and computes a bound on the identity map as described above. The representable numbers $x_0$ and $s$ which describe the support adapted to a given function are determined in the procedure \t{rsupport}. %====================================================== % Section 5 %====================================================== \section The Maps $\NN_\lambda$ The goal of this section is to explain how bound \equ(precf0lp) of \clm(hypapcformlk) is computed and how inequality \equ(inequalityc1) of \clm(thmfornl) is checked. A major step is to compute $\NN_\lamp$ and $\NN_\lamm$ on various functions of $\BB_\mn$, with $\mu,\nu,\lamm$ and $\lamp$ as in \clm(hypapcformlk). We will see in Section~5.2 that these maps must be estimated from $\BB_\mn$ to $\BB_{\mu(\lamp\nu/\lamm)}$, a space slightly smaller than $\BB_\mn$, since $\lamp/\lamm\approx1+7\cdot10^{-7}$. In the sequel we denote $\delta=\lamm/\lamp$, and begin in the next section by describing the construction of a bound on the maps $\NN_\lambda:\BB_\mn\rightarrow\BB_{\mu(\nu/\delta)}$. \subsection A Bound on $\NN_\lambda$ We recall that for $\lambda>0$, $\NN_\lambda:\BB_\mn/\HH\rightarrow\BB_{\mu(\nu/\delta)}$ is well defined provided $1/\delta\leq\lambda\leq4$, cf. \clm(existenceofnl), and is given by \goodbreak $$ \NN_\lambda(f)=c_\lambda(f)\NN^1_\lambda(f)+c_2\NN^2_\lambda(f), \equation(nlch5) $$ where $$\eqalignno{ &\NN^1_\lambda(f)=S_\lambda(f*f),&\equat(nl1ch5)\cr &\NN^2_\lambda(f)=T\bigl(T\NN^1_\lambda(f)*T\NN^1_\lambda(f)\bigr), &\equat(nl2ch5)\cr &c_\lambda(f)={\lambda\over2}{1-c_2E\bigl(\NN^2_\lambda(f)\bigr)\over M(f)E(f)}.&\equat(formulec1ch5)\cr }$$ The expression \equ(formulec1ch5) is more convenient for our present purpose than \equ(formulec1). In principle, one readily gets a bound on $\NN_\lambda$ by composing the bounds constructed in the previous section. However, one can without too much effort improve this bound in two ways. First, the distributivity and commutativity properties of the operators involved in \equ(nlch5) give us the freedom to choose the order in which the bounds are composed. The order can affect the estimates, since in general these properties are not shared by the bounds. Regarding $\NN^1_\lambda$, the fact that $S_\lambda$ preserves $\AA^u$ yields slightly better estimates by using $$ \NN^1_\lambda(f)=S_\lambda f*S_\lambda f, \equation(nl1ch52nd) $$ instead of \equ(nl1ch5). Furthermore, in order to get as much contraction as possible from the scaling operator, cf. Section~4.2, one chooses the sequence of spaces $$ \NN^1_\lambda: \BB_\mn\buildrel{\hbox{$S_\lambda$}}\over{\hbox to 10mm{\rightarrowfill}} \BB_{{\mu\over4}(\nu/\delta)} \buildrel\hbox{$*$}\over{\hbox to 10mm{\rightarrowfill}} \BB_{\mu(\nu/\delta)}. \equation(suitedesp) $$ A bound on \equ(suitedesp) follows by composing the bounds of Section~4. We now turn to $\NN^2_\lambda$ and, as above, let $S_\lambda$ act first, considering \equ(nl2ch5) with $\NN^1_\lambda$ as in \equ(nl1ch52nd). Regarding the choice of spaces, we note that one could exploit the operators $T$ and the outer convolution to estimate $\NN^2_\lambda$ in the smaller space $\BB_{\mu(\nu/\delta)}$, as needed. That would permit us to consider $\nu$ instead of $\nu/\delta$ in \equ(suitedesp) for which $S_\lambda$ is a better contraction. Nevertheless, $\delta$ is so close to one that it does not lead to any significant improvement, and for convenience one simply constructs a bound on $\NN^2_\lambda$ by composing the previous bound on $\NN^1_\lambda$ and a bound on the map $$ \BB_{\mu(\nu/\delta)} \buildrel{\hbox{$T$}}\over{\hbox to 10mm{\rightarrowfill}} \BB_{(\nu/\delta)\mu} \buildrel\hbox{$*$}\over{\hbox to 10mm{\rightarrowfill}} \BB_{(\nu/\delta)\mu} \buildrel{\hbox{$T$}}\over{\hbox to 10mm{\rightarrowfill}} \BB_{\mu(\nu/\delta)}. \equation(suitedesp2) $$ The target space of the convolution above is chosen in order to minimize the norm of the general term arising from the convolution of the piecewise affine part. The second improvement concerns the computation of the coefficient $c_\lambda(f)$. An estimation of the quantity $E(\NN^2_\lambda(f))$ entering \equ(formulec1ch5) is of poor quality if obtained by composing the bound on $\NN^2_\lambda$ described above and the bound on the expectation as given in the previous section. Exploiting the structure of $\NN^2_\lambda$ and the fact that it maps into a smaller space, due to the outer convolution, leads to a substantial improvement. Defining $$\eqalign{ {\cal E}:\BB_{\alpha\beta}\times\BB_{\alpha\beta}&\rightarrow\real \cr (f,h)& \mapsto E(T(f*h)),\cr }\equation(mapexp) $$ one has $E(\NN^2_\lambda(f))=\EE(T\NN^1_\lambda(f),T\NN^1_\lambda(f))$. In order to construct a bound on $\EE$ acting from $\std{\BB_{\alpha\beta}}^u\times\std{\BB_{\alpha\beta}}^u$ to $\std{\real}$, we first observe that for $g\in\BB_{\zeta\eta}$, $$ |E(Tg)|\leq\int_0^\infty{1\over x}|g(x)|\,dx \leq\sup_{x>0}{x\over w_{\eta\zeta}(x)}\|g\|_{\zeta\eta}. \equation(et2tgen) $$ \goodbreak\noindent Next, for $f=\rho+g_f$ and $h=\sigma+g_h$, with $\rho,\sigma\in\AA^u$ and $g_f,g_h\in\BB_{\alpha\beta}$, one has $$ \EE(f,h)=E\bigl(T(\rho*\sigma)\bigr)+E\bigl(T(\rho*g_h+g_f*h)\bigr), \equation(mapexpfinal) $$ and, since $\rho*g_h+g_f*h\in\BB_{(4\alpha)\beta}$, one obtains from \equ(et2tgen) the following estimate on the second term in the RHS of \equ(mapexpfinal), $$ |E\bigl(T(\rho*g_h+g_f*h)\bigr)|\leq\sup_{x>0} {x\over w_{\beta(4\alpha)}(x)} (\|\rho\|_{\alpha\beta} \|g_h\|_{\alpha\beta}+\|g_f\|_{\alpha\beta}\|h\|_{\alpha\beta}). \equation(expconvt2t) $$ Finally, the first term on the RHS of \equ(mapexpfinal) can be computed explicitly. We use the same notation as in Section~4.4. Then, $\rho*\sigma$ is given on each interval $[z_k,z_{k+1}]$, $k=0,\dots,2n-1$, by the cubic polynomial $$\eqalignno{ &(\rho*\sigma)(z_k+\theta)= C_0(k)+ C_1(k)\theta+ C_2(k)\theta^2+ C_3(k)\theta^3,& \cr }$$ where the coefficients $C_i(k)$ are given by \equ(coeff). Hence, $$\eqalignno{ E\bigl(T(\rho*\sigma)\bigr)&=\int_0^\infty{1\over x}(\rho*\sigma)(x)\,dx&\cr &=\sum_{k=0}^{2n-1}\int_0^1{1\over\theta+z_k/\veps} \bigl( C_0+ \varepsilon\theta C_1(k)+ \varepsilon^2\theta^2C_2(k)+ \varepsilon^3\theta^3C_3(k) \bigr)\,d\theta,\qquad &\equat(exppwc1)\cr }$$ and using $$ \int_0^1{x^n\over x+1/a}dx=a^{-n}\int_0^a{x^n\over x+1}dx=\tflog_n(a), $$ where $\tflog_n$ is defined in \equ(logn), one can integrate each term in \equ(exppwc1) and gets finally $$ E\bigl(T(\rho*\sigma)\bigr)=\sum_{k=0}^{2n-1}\sum_{m=0}^3 \varepsilon^mC_m(k)\,\tflog_m\Bigl({\veps\over z_k}\Bigr). \equation(expconvt1t) $$ \remark A bound on \equ(mapexp) is implemented in \t{sexp\_of\_tconv}. Since the quantities entering \equ(expconvt1t) are computed during the estimation of the convolution, this subroutine is called in \t{fconvolute1} and \t{fconvolute2}. A bound on $\NN_\lambda$ is implemented in \t{fN}. This subroutine also returns a standard set containing the coefficient $c_\lambda(f)$ that will be used to check \equ(inequalityc1), treating separately the special case where the value of $E(f)$ is known exactly, cf. Section~5.3. \subsection Existence of the Family of Fixed Points: First Estimate We now explain how the quantity $\varepsilon$ entering inequality \equ(precf0lp) of \clm(hypapcformlk) is computed. Recall that it consists in an upper bound on $$ \|\MM_\lampka(f^0_\lamp)-f^0_\lamp\|_\mn, \equation(junkaa) $$ uniform in $\kappa\in[\delta,1]$, $\delta=\lamm/\lamp$, where $\lamm<\lamp,\mu$ and $\nu$ are given in \clm(hypapcformlk), and $f^0_{\lamp}$ is an approximate fixed point in $\AA^u$. From the definition of $\NN_\lampka$ and $\MM_\lampka$, cf. \equ(defnlk) and \equ(defmlk), it follows that $$ \|\MM_\lampka(f^0_\lamp)-f^0_\lamp\|_\mn \leq\|M\|\,\|\NN_\lampka(f^0_\lamp)-f^0_\lamp\|_\mn, \equation(precmlk) $$ and from \equ(boundongscal) one obtains $$\eqalignno{ \|\NN_\lampka(f^0_\lamp)-f^0_\lamp\|_\mn &\leq\|S_\kappa(\NN_\lamp(f^0_\lamp)-f^0_\lamp)\|_\mn +\|S_\kappa f^0_\lamp-f^0_\lamp\|_\mn&\cr &\leq\|\NN_\lamp(f^0_\lamp)-f^0_\lamp\|_{\mu(\nu/\delta)} +\|(S_\kappa-1)f^0_\lamp\|_\mn. \qquad&\equat(precnlk)\cr }$$ The last inequality is valid since $f^0_\lamp$ and $\NN_\lamp(f^0_\lamp)$ belong to $\BB_{\mu(\nu/\delta)}$. Indeed, $f^0_\lamp$ has compact support in $(0,\infty)$, and $\NN_\lamp$ preserves this property. Therefore, $\NN_\lamp(f^0_\lamp)\in\BB_\albe$ and $f^0_\lamp\in\BB_\albe$ for all $\alpha,\beta\geq0$. By composing the bounds constructed in the previous sections, one gets an estimate for the first term on the RHS of \equ(precnlk). At this point, the only dependence on the parameter $\kappa$ lies in the second term of \equ(precnlk), which one bounds uniformly using the following result. \claim Lemma(unifborne) Let $0<\kappa\leq1$ and $f\in W_1^1(\real^+,w_\albe(x)dx)$. If $f'\in\BB_{\alpha\gamma}$ for some $\gamma>\beta/\kappa$, then $$ \|(S_\kappa-1)f\|_\albe\leq(1-\kappa)\bigl(\|f\|_{\albe} +\|xf'(x)\|_{\alpha(\beta/\kappa)}\bigr). \equation(lbunif) $$ By definition, the function $f^0_\lamp\in\AA^u$ satisfies the hypothesis of \clm(unifborne) for all $\alpha,\beta\geq0$. Furthermore, the bound \equ(lbunif) is decreasing in $\kappa$. Hence, one has for all $\kappa\in[\delta,1]$, $$ \|(S_\kappa-1)f^0_\lamp\|_\mn \leq(1-\delta)\bigl(\|f^0_\lamp\|_\mn+ \|x(f^0_\lamp)'(x)\|_{\mu(\nu/\delta)}\bigr). \equation(precnlk2t) $$ Collecting the inequalities \equ(precmlk), \equ(precnlk) and \equ(precnlk2t) yields the desired bound $\varepsilon$ on \equ(junkaa), uniform in $\kappa\in[\delta,1]$. The only missing information is the norm of the operator $M$ appearing in \equ(precmlk), which will be given in Section 7.2. We end this section with the \noindent{\bf Proof of \clm(unifborne).} For $f\in W_1^1(\real^+,dx)$, one can rewrite $$ S_\kappa f(x)-f(x)=(\kappa-1)f(x)+\kappa\int_x^{\kappa x}f'(y)dy. $$ Hence, $$ \|(S_\kappa-1)f\|_\albe \leq(1-\kappa)\|f\|_\albe+ \kappa\int_0^\infty dx\,w_\albe(x)\int_{\kappa x}^x|f'(y)|dy. \equation(jaa) $$ \goodbreak\noindent Furthermore, $$\eqalign{ \int_0^\infty dx\,w_\albe(x)\int_{\kappa x}^x|f'(y)|dy &=\int_0^\infty dy\,|f'(y)|\int_y^{y/\kappa}w_\albe(x)dx\cr &\leq{1-\kappa\over\kappa}\int_0^\infty y|f'(y)| \max\{w_\albe(y),w_\albe(y/\kappa)\}dy\cr &\leq{1-\kappa\over\kappa}\sup_{y\geq0} {\max\{w_\albe(y),w_\albe(y/\kappa)\}\over w_{\alpha(\beta/\kappa)}(y)} \|yf'(y)\|_{\alpha(\beta/\kappa)}. }$$ Finally, one checks that the supremum in the previous expression is bounded by one for $\kappa\leq1$, which leads to \equ(lbunif). \qed \remark The quantity $\varepsilon$ is computed in the procedure \t{compute\_residual}. This procedure also returns a bound on the first term in the RHS of \equ(precnlk), a quantity that will be used in Section~5.3. The bound \equ(lbunif) is implemented in the procedure \t{snorm\_of\_Skappam1}, where the second term on the RHS of \equ(lbunif) is estimated using, for $\rho\in\AA$ with $\pi(\rho)=(\{x_i\},\{\rho_i\})_{i=0}^n$, $$ \|x\rho'(x)\|_{\alpha(\beta/\kappa)} \leq{1\over2}\sum_{i=1}^n\sup_{x\in I_i}w_{\alpha(\beta/\kappa)}(x) |\rho_i-\rho_{i-1}|(x_i+x_{i-1}). \equation(normxrhop) $$ Finally, the (positive) representable numbers that describe the approximate fixed point $f^0_\lamp\in\AA^u$ are contained in the file \t{fpoint.lp}. The first two numbers in this file are the boundary points of $\supp(f^0_\lamp)$. They determine the partition $p\in\PP^n_u$, $n=2^{17}+1$, satisfying $\pi(f^0_\lamp)=(p,\cdot)$. The last $2^{17}$ numbers are the (nonzero) entries of the vector $v$, where $\pi(f^0_\lamp)=(\cdot,v)$. Given a nonnegative $G\in\SS$, the subroutine \t{read\_fp} reads the file \t{fpoint.lp} and constructs a standard set $(P,V,G)$ with $(P,V)\in\std{\AA^u}$ containing $f^0_\lamp$. \subsection Existence of the Fixed Point $f_{\lambda^*}$ Recall that once the existence of the continuous family $\{f_\lambda\}$ of fixed points of $\NN_\lambda$ is established for $\lambda\in[\lamm,\lamp]$, our main result, namely the existence of a $\lambda^*\in[\lamm,\lamp]$ and a function $f^*$ satisfying $S_{\lambda^*}\DD(f^*)=f^*$, follows from $$ c_\lamm(f_\lamm)0$, one easily checks that the estimates of \clm(existenceofnl) imply that whenever $\NN_\lambda:\BB_\albe/\HH\rightarrow\BB_{\sigma\tau}$ is well defined, i.e., $\sigma\leq4\alpha/\lambda$, $\tau\leq\lambda\beta$, it is continuously differentiable on $\BB_\albe/\HH$. We will need later to estimate $D\NN_\lambda(f)$ for $\NN_\lambda:\BB_\albe\rightarrow\BB_{\alpha\gamma}$ with $\gamma$ slightly larger than $\beta$. In this case, the conditions for $D\NN_\lambda(f)$ to be bounded become $\gamma/\beta\leq\lambda\leq4$. We will see below that under the stronger conditions $\gamma/\beta<\lambda<4$, the tangent map $D\NN_\lambda(f)$ is actually compact provided $f$ is sufficiently regular, i.e., there is a sequence of subspaces of finite codimension on which $D\NN_\lambda(f)$ converges to zero. These subspaces are defined as follows. \claim Definition(remaindermasse0) For $p=\{x_0,\dots,x_n\}$ a partition in $\PP_n$ and $a,b>0$, we define the following subspaces of $\BB_\albe$, $$\eqalignno{ \LL_\alpha^a&=\{h\in\bigcup_{\eta\geq0}\BB_{\alpha\eta}\,|\; \supp(h)\subseteq(0,a)\},&\cr \CC_{p}&=\{h\in L^1(\real_+)|\,\supp(h)\subseteq(x_0,x_n),\ \int_{x_{i-1}}^{x_i}h(x)\,dx=0\ {\rm for}\ i=1,\dots,n\},&\cr \RR_\beta^b&=\{h\in\bigcup_{\zeta\geq0}\BB_{\zeta\beta}\,|\; \supp(h)\subseteq(b,\infty)\}.&\cr }$$ Furthermore, we denote by $\BB^p_\albe$ the following subspace of $\BB_\albe$, $$ \BB^p_\albe=\LL^{x_0}_\alpha\oplus\CC_p\oplus\RR^{x_n}_\beta. \equation(defbbp) $$ For $a$ small enough and $b$ large enough, it turns out that when restricted to $\LL_\alpha^a$, $\RR_\beta^b$, and $\CC_p$ respectively, the tangent map $D\NN_\lambda(f)$ has norms of value $\OO(e^{-1/a}||f||_\albe)$, $\OO(e^{-b}||f||_\albe)$, and $\OO(|p|\,||f'||_\albe)$, where $|p|$ denotes the mesh size of the partition $p$. More generally, if $\CC_p$ consists of functions whose $n-1$ first moments vanish on every interval of the partition $p$, the norm of the tangent map restricted to $\CC_p$ is $\OO(|p|^n||f^{(n)}||_\albe)$. For our purpose, it is sufficient to consider $n=1$. Before deriving explicitly the contraction factors, we remark that we will need to evaluate later $D\NN_\lambda(f)$ on the complement of $\BB^p_\albe$ in $\BB_\albe$. It is easily seen that for a given partition $p$, every $h\in\BB_\albe$ can be uniquely decomposed into a sum $h=g+\tau$ where $g\in\BB^p_\albe$ and where $\tau$ is constant on each interval of the partition $p$ and satisfies $\supp(\tau)=\supp(p)$. More precisely, with $p=\{x_0,\dots,x_n\}$ and $\chi_I$ the characteristic function of the interval $I$, one has $$ \BB_\albe=\BB^p_\albe\oplus\VV^p, \equation(firstdecomp) $$ where $\VV^p$ is the $n$-dimensional vector space defined by $$ \VV^p=\{\tau\,|\tau=\sum_{i=1}^n\lambda_i\chi_{[x_{i-1},x_i]}, \lambda_i\in\real\}. \equation(defvvp) $$ Section~6.4 is devoted to the construction of a bound on the map $D\NN_\lambda(f):\VV^p\rightarrow\BB_{\alpha\gamma}$. \remark In \equ(dnl2), there are factors that depend only on the base function $f$. These factors, namely $\NN^1_\lambda(f)$, $T\NN^1_\lambda(f)$ and their norms, together with $c_\lambda(f)$, $E(f)$ and $M(f)$, are computed once and for all in the subroutine \t{compute\_constant\_terms} using the bounds of Section~4. According to \clm(hypapcformlk), $f$ is represented by the standard set in $\std{\BB_\mn}^u$ whose affine part is the singleton $\{f^0_\lamp\}$ and whose general term $g$ satisfies $||g||_{\mu\nu}\leq9\cdot10^{-4}$. Finally, for given standard sets containing $M(h),E(h)$ and $E(D\NN^2_\lambda(f)h)$, a bound on $\delta_\lambda(f,h)$ is computed in the procedure \t{sdelta1} using \equ(formuledelta1). \subsection Oscillatory Functions We derive now an upper bound on the norm of the operator $D\NN_\lambda(f):\CC_p \rightarrow\BB_{\alpha\gamma}$, with $\CC_p$ as in \clm(remaindermasse0) and with $f=\rho+g\in\BB_\albe/\HH$, $\rho\in\AA^u$. For the first two terms in \equ(dnl2), and for $||g||_\albe$ small, the contraction factor will come from the convolution in $D\NN^1_\lambda$. Hence, we first use the bounds obtained in \clm(existenceofnl) and get in full generality $$ \|D\NN_\lambda(f)h\|_{\alpha\gamma} \leq(\,|c_\lambda(f)|+2c_2\|\NN^1_\lambda(f)\|_{\alpha\gamma})\, \|D\NN^1_\lambda(f)h\|_{\alpha\gamma}+|\delta_\lambda(f,h)|\, \|\NN^1_\lambda(f)\|_{\alpha\gamma}. \equation(boundonDnlav0) $$ In the previous expression, only the quantities that depend on $h$ remain to be estimated. Let us begin with $\delta_\lambda(f,h)$. For $h\in\CC_p$, one has $M(h)=0$ and the first term in \equ(formuledelta1) vanishes. Next, $E(h)$ is expressed in term of the largest interval in the partition $p$. Denoting $p=\{x_0,\dots,x_n\}$ and $I_i=[x_{i-1},x_i]$, $i=1,\dots,n$, the identity $\int_{I_i}h(x)dx=0$ implies $$ \Bigl|\int_{I_i}xh(x)\,dx\Bigr|=\Bigl|\int_{I_i}\bigl(x-{x_i+x_{i-1}\over2} \bigr)h(x)\,dx\Bigr| \leq{1\over2}(x_i-x_{i-1})\int_{I_i}|h(x)|\,dx, $$ which in turn yields $$ |E(h)| \leq{1\over2}\max_{i=1,\dots,n}\{x_i-x_{i-1}\} \sup_{x>0}\Bigl({1\over w_{\alpha\beta}(x)}\Bigr) \|h\|_{\alpha\beta}. \equation(eofav0) $$ Finally, since $D\NN^2_\lambda(f)h\in\BB_{\alpha(4\gamma)}$, it follows from \equ(boundonmete2) that $$ |E\bigl(D\NN^2_\lambda(f)h\bigr)|\leq2 \sup_{x>0}\Bigl({x\over w_{\alpha(4\gamma)}(x)}\Bigr) \|\NN^1_\lambda(f)\|_{\alpha\gamma}\|D\NN^1_\lambda(f)h\|_{\alpha\gamma}. \equation(boundone2Dn2lav0) $$ Inserting \equ(eofav0) and \equ(boundone2Dn2lav0) into \equ(formuledelta1) leads to an estimate for the second term on the RHS of \equ(boundonDnlav0). In order to bound the RHS of \equ(boundone2Dn2lav0) and the first term on the RHS of \equ(boundonDnlav0), it then remains to estimate $\|D\NN^1_\lambda(f)h\|_{\alpha\gamma}$. In order to treat $D\NN^1_\lambda(f)$, one has the possibility to exploit, as in the previous section, the distributivity of the scaling operator $S_\lambda$ with respect to the convolution. It turns out that the order is not crucial and we consider for simplicity $$ D\NN^1_\lambda: \BB_\albe\times\CC_p \buildrel\hbox{$*$}\over{\hbox to 10mm{\rightarrowfill}} \BB_{(4\alpha)\beta} \buildrel\hbox{$2S_\lambda$}\over{\hbox to 10mm{\rightarrowfill}} \BB_{\alpha\gamma}. \equation(suitedespav0) $$ We begin with the convolution and use the following result. \claim Lemma(lconvrhoav0) Let $f\in W^1_1(\real_+,w_\albe(x)\,dx)$ and $h\in\CC_p$ with $p=\{x_0,\dots,x_n\}$. Then, $$ \|f*h\|_{(4\alpha)\beta}\leq {1\over2}c_{\alpha\beta}(p) \|f'\|_{\alpha\beta}\|h\|_{\alpha\beta}, \equation(boundonconvfav0) $$ where, denoting $I_i=[x_{i-1},x_i]$, $$ c_{\alpha\beta}(p)= \max_{i=1,\dots,n}\Bigl\{ \sup_{x\in I_i}{1\over w_{\alpha\beta}(x)}\int_{I_i}w_{\alpha\beta}(x)\,dx \Bigr\}.\equation(defcp) $$ Since for $\rho\in\AA$ one has by definition $\rho\in W^1_1(\real_+,w_\albe(x)dx)$, the previous lemma together with \clm(existenceofnl) imply, with $f=\rho+g$, $\rho\in\AA^u$, $g\in\BB_\albe$, and $h\in\CC_p\,$, $$\eqalignno{ \|f*h\|_{(4\alpha)\beta} &\leq\|\rho*h\|_{(4\alpha)\beta}+ \|g\|_{\alpha\beta}\|h\|_{\alpha\beta}&\cr &\leq\Bigl({1\over2}c_\albe(p)\|\rho'\|_\albe+\|g\|_\albe\Bigr)\|h\|_\albe. &\equat(boundonCp)\cr }$$ Next, since $\gamma/\beta\leq\lambda\leq4$, inequality \equ(boundgscal) applies (with $\alpha$ replaced by $4\alpha$), and we finally obtain $$ \|D\NN^1_\lambda(f)h\|_{\alpha\gamma} \leq 2e^{-A} \Bigl({1\over2}c_{\alpha\beta}(p)\|\rho'\|_{\alpha\beta}+ \|g\|_\albe\Bigr)\|h\|_\albe, \equation(boundondnl1center) $$ where $A=2\sqrt{\alpha(4-\lambda)(\beta-\gamma/\lambda)}$. A few comments are in order. In \equ(boundondnl1center), the contraction factor is not only given by $c_\albe(p)$ but also by how close in $\BB_\albe$ the base function $f$ is to a regular function together with the norm of that function in $W^1_1(\real_+,w_\albe(x)dx)$. The fact that the fixed point whose existence we want to prove is smooth plays an important role here. To make a connection with \clm(hypapcformlk), the quantity $\|g\|_\albe$ in \equ(boundondnl1center) is the radius of the ball on which the tangent maps $D\MM_{\lamp,\kappa}$ need to be contractions. All the other terms can be made as small as we wish by letting the size of the largest interval in $p$ go to zero, cf. \equ(eofav0) and \equ(defcp). Note that $c_\albe(p)$ depends sensitively on $\alpha$ and $\beta$, and optimizing this factor requires to consider a partition $p$ with smaller intervals where the weight $w_\albe$ varies strongly. We will encounter later other optimization criteria for $p$. We shall denote by $p_r$ the partition $p$ which we will eventually choose, cf. Section~7.1. We end this section with the \noindent{\bf Proof of \clm(lconvrhoav0).} Define the function $h_1$ by $$ h_1(x)=\int_{x_0}^x h(\xi)\,d\xi, $$ for $x\in(x_0,x_n)$, and $h_1(x)=0$ otherwise. Note that $h_1'=h$ and, by definition of $h$, $h_1(x_i)=0$ for $i=0,\dots,n$. Hence, integration by parts leads to $$ \|f*h\|_{(4\alpha)\beta}=\|f'*h_1\|_{(4\alpha)\beta} \leq\|f'\|_{\alpha\beta}\|h_1\|_{\alpha\beta}. $$ It remains to estimate the norm of $h_1$ in term of $h$. For $i=1,\dots,n$ and $x\in[x_{i-1},x_i]$, one has $$ |h_1(x)| ={1\over2}\Bigl( \Bigl|\int_{x_{i-1}}^xh(\xi)\,d\xi\Bigr|+\Bigl|\int_x^{x_i}h(\xi)\,d\xi \Bigr|\Bigr) \leq{1\over2}\int_{I_i}|h(\xi)|\,d\xi, $$ which in turn yields $$\eqalign{ \|h_1\|_{\alpha\beta}&=\sum_{i=1}^n\int_{I_i} w_{\alpha\beta}(x)\,|h_1(x)|\,dx,\cr &\leq{1\over2}\sum_{i=1}^{n}\int_{I_i} w_{\alpha\beta}(x)\int_{I_i}|h(\xi)|\,d\xi\,dx\cr &\leq{1\over2}\max_{i=1,\dots,n}\Bigl\{\sup_{x\in I_i}{1\over w_{\alpha\beta}(x)}\int_{I_i}w_{\alpha\beta}(x)\,dx\Bigr\} \|h\|_{\alpha\beta}.\cr }$$ \qed \goodbreak \remark The quantity $c_\albe(p_r)$ is computed in the subroutine \t{swsupint}. The estimates \equ(eofav0), \equ(boundone2Dn2lav0) and \equ(boundondnl1center) are implemented in \t{fDN\_center} to compute \equ(boundonDnlav0), with a call to the subroutine \t{sdelta1} to get $\delta_\lambda(f,h)$. The quantity $\|\rho'\|_\albe$ entering \equ(boundondnl1center) is bounded in \t{snorm\_of\_der\_pl} by $$ \|\rho'\|_{\alpha\beta}\leq\sum_{j=1}^{m}\sup_{x\in I_j} w_\albe(x)|\rho_j-\rho_{j-1}|, \equation(boundonderrho) $$ where $\pi(\rho)=(\{y_j\},\{\rho_j\})_{j=0}^m$ and $I_j=[y_{j-1},y_j]$. \subsection Functions with Support Near the Origin In this section, we consider $D\NN_\lambda(f)$ acting on functions $h\in\LL^a_\alpha$ for $a$ small enough. As in the previous section, but for different reasons, the contractivity properties of $D\NN_\lambda(f)$ are entirely due to the term $D\NN^1_\lambda(f)$. Indeed, since the functions $f$ which will be considered have in general a support given by $\real_+$, the support of $D\NN^1_\lambda(f)\,h$ for $h\in\LL^a_\alpha$ is also equal to $\real_+$ due to the convolution. Hence, the size of $D\NN^2_\lambda(f)h$ is essentially given by the size of $D\NN^1_\lambda(f)\,h$, and we proceed as before starting with the bound \equ(boundonDnlav0) on $\|D\NN_\lambda(f)h\|_{\alpha\gamma}$. The last term in the expression \equ(formuledelta1) for $\delta_\lambda(f,h)$ is again bounded using \equ(boundone2Dn2lav0). The $h$-dependent coefficients of the first two terms in \equ(formuledelta1) are given by $M(h)$ and $E(h)$, which are bounded using $$\eqalignno{ |M(h)|&\leq{1\over w_{\alpha\beta}(a)}\|h\|_{\alpha\beta}, &\equat(boundonmlel1)\cr |E(h)|&\leq{a\over w_{\alpha\beta}(a)}\|h\|_{\alpha\beta}, &\equat(boundonmlel2)\cr }$$ provided $a\leq\sqrt{\alpha/\beta}$ for the first inequality, and $a\leq\bigl(1+\sqrt{1+4\alpha\beta}\bigr)/2\beta$ for the second inequality, cf. the discussion of \equ(supofinvw) and \equ(xcforsupofxonvw). It remains to bound $D\NN^1_\lambda(f)h$ in $\BB_{\alpha\gamma}$. We consider $$ D\NN^1_\lambda: \BB_\albe\times\LL^a_\alpha \buildrel\hbox{$*$}\over{\hbox to 10mm{\rightarrowfill}} \BB_{(\eta\alpha)\beta} \buildrel\hbox{$2S_\lambda$}\over{\hbox to 10mm{\rightarrowfill}} \BB_{\alpha\gamma}, \equation(suitedespleft) $$ with $\eta\in[\lambda,4]$ a parameter to be chosen later. For the convolution, we use the \claim Lemma(lboundonconvll) Let $f\in\banach\alpha\beta$ and $h\in\LL^a_{\alpha}$. Then, for $1\leq\eta\leq4$, $$ \|f*h\|_{(\eta\alpha)\beta}\leq\exp\Bigl(-{\alpha\sqrt{\eta} (2-\sqrt{\eta})\over a}\Bigr)\|f\|_{\alpha\beta}\|h\|_{\alpha\beta}. \equation(fboundonconvll) $$ \proof Exploiting $\supp(h)\subseteq(0,a)$, we proceed as in \clm(existenceofnl) and get $$ \|f*h\|_{(\eta\alpha)\beta}\leq \sup_{x>0\atop a>y>0}\exp\bigl(-\alpha g(x,y)\bigr) \|f\|_\albe\|h\|_\albe, $$ where $$ g(x,y)={x+y\over xy}-{\eta\over x+y}. $$ Since $g(x,y)\geq0$ for $\eta\leq4$, one has $\sup\exp(-\alpha g)= \exp(-\alpha\inf g)$ and, using $\eta\geq1$, we compute $$ \inf_{x>0\atop a>y>0}g(x,y)=\inf_{a>y>0}{\sqrt{\eta}(2-\sqrt{\eta}) \over y} ={\sqrt{\eta}(2-\sqrt{\eta})\over a}. $$ \qed We now turn to the scaling operator. Since $\supp(f*h)=\real_+$ for functions $f$ that will be considered, the following general bound is optimal, $$ \|S_\lambda g\|_{\alpha\gamma}\leq \exp\bigl(-2\sqrt{\alpha(1-\lambda/\eta)(\beta-\gamma/\lambda)}\,\bigr) \|g\|_{(\eta\alpha)\beta}, \equation(boundonscalll) $$ which is valid provided $\gamma/\beta\leq\lambda\leq\eta$. From \equ(boundonscalll) and \equ(fboundonconvll), we get a bound on $\|D\NN^1_\lambda(f)h\|_{\alpha\gamma}$. We now optimize the parameter $\eta$. Since ultimately we will get the needed contraction factor by choosing $a$ small enough, and since \equ(boundonscalll) does not depend on $a$, we consider \equ(fboundonconvll) only. For $\lambda\geq1$, the maximum of $\sqrt{\eta}(2-\sqrt{\eta})$ on $[\lambda,4]$ is taken at $\eta=\lambda$, and one gets finally $$ \|D\NN^1_\lambda(f)h\|_{\alpha\gamma}\leq2\exp\Bigl(-{\alpha\sqrt{\lambda} (2-\sqrt{\lambda})\over a}\Bigr)\|f\|_\albe\|h\|_\albe. \equation(boundondn1ll2) $$ Recall that \equ(boundondn1ll2) is valid provided $\gamma/\beta\leq\lambda\leq4$. Furthermore, it leads for $a$ small enough to a strict contraction only if $\lambda<4\,$: this is the first compactness condition. Before ending this section, let us comment on the optimization of the contraction factor. Instead of \equ(suitedespleft), one can consider $D\NN^1_\lambda(f)h=2(S_\lambda f*S_\lambda h)$ with $S_\lambda:\BB_\albe\rightarrow\BB_{(\alpha/\eta)\gamma}$ and $\eta\in[\lambda,4]$ a parameter to be optimized. Since $S_\lambda h\in\LL^{a/\lambda}_{(\alpha/\eta)\gamma}$ is of order $\OO(e^{-1/a})$ if $\eta>\lambda$, one gets a second $a$-dependent contraction factor from the convolution. However, optimizing $\eta$ leads to the same bound as \equ(boundondn1ll2), and we use \equ(suitedespleft) for convenience of implementation. \remark The bounds \equ(boundonmlel1), \equ(boundonmlel2) and \equ(boundondn1ll2) are implemented in the procedure \t{fDN\_left} to compute \equ(boundonDnlav0). The conditions on $a$ under which \equ(boundonmlel1) and \equ(boundonmlel2) are valid are first checked, namely $a\leq\sqrt{\alpha/\beta}$ and $a\leq\bigl(1+\sqrt{1+4\alpha\beta}\bigr)/2\beta$. An explicit check of $\gamma/\beta\leq\lambda\leq4$ is also necessary. Up to now, this inequality was implicitly verified when bounds were computed, as in \equ(boundondnl1center) for instance. \subsection Functions with Support Near Infinity We now consider functions $h\in\RR^b_\beta$ with $b$ large enough. Here, the situation differs from the previous cases in the sense that the term $D\NN^2_\lambda(f)h$ is small independently of the size of $D\NN^1_\lambda(f)h$. Indeed, the property of $h$ to have support away from the origin is preserved by $D\NN^1_\lambda(f)$. After applying the transformation $T$, one obtains a function whose support is near the origin, and the result from the previous section related to the convolution yields a second exponentially small factor. Hence, we simply start with the triangle inequality to get from \equ(dnl2) $$ \|D\NN_\lambda(f)h\|_{\alpha\gamma} \leq|c_\lambda(f)|\|D\NN^1_\lambda(f)h\|_{\alpha\gamma} +c_2\|D\NN^2_\lambda(f)h\|_{\alpha\gamma} +|\delta_\lambda(f,h)|\,\|\NN^1_\lambda(f)\|_{\alpha\gamma}. \equation(dnl2bound) $$ Let us begin with the first term. The main contraction factor is here entirely due to the scaling operator acting on $\RR^b_\beta$. Furthermore, for $f\in\BB_\albe$, the map $h\mapsto f*h$ preserves $\RR^b_\beta$. Hence, one has the choice of the order in which the scaling and the convolution are composed. By letting the scaling act first, one gains a ($b$-independent) contraction factor when applying this operator to the function $f$. Recall that $S_\lambda:\BB_\albe\rightarrow\BB_{(\alpha/4)\gamma}$ is a strict contraction for $\gamma/\beta<\lambda<4$. One can improve this factor by considering $\BB_{0\gamma}$ for the target space of $S_\lambda$. Hence, we consider finally $$ D\NN^1_\lambda/2: \BB_\albe\times\RR^b_\beta \buildrel\hbox{$S_\lambda$}\over{\hbox to 10mm{\rightarrowfill}} \BB_{0\gamma}\times\RR^{b/\lambda}_\gamma \buildrel\hbox{$*$}\over{\hbox to 10mm{\rightarrowfill}} \BB_{\alpha\gamma}. \equation(suitedespright) $$ Provided $\gamma/\beta\leq\lambda$, the scaling operator in \equ(suitedespright) is bounded, and, since $S_\lambda h$ has again support away from the origin, the convolution above is well defined even for $\alpha>0$. For $f\in\BB_\albe$, one estimates as usual $$\eqalignno{ \|S_\lambda f\|_{0\gamma}&\leq\sup_{x>0}{w_{0\gamma} (x/\lambda)\over w_\albe(x)}\,\|f\|_\albe&\cr &=\exp\Bigl(-2\sqrt{\alpha(\beta-\gamma/\lambda)}\,\Bigr)\,\|f\|_\albe,& \equat(boundgscal2)\cr }$$ and for $h\in\RR^b_\beta$, one uses the knowledge about the support of $h$ to get $$\eqalignno{ \|S_\lambda h\|_{0\gamma}&\leq \sup_{x>b}{w_{0\beta} (x/\lambda)\over w_{\alpha\gamma}(x)}\|h\|_{\alpha\beta}&\cr &=\exp\bigl(-\alpha/b-b(\beta-\gamma/\lambda)\bigr)\|h\|_{\alpha\beta}, &\equat(scalonh0beta)\cr }$$ the last equality being valid if $b\geq\sqrt{\alpha/(\beta-\gamma/\lambda)}$. Next, we consider the convolution in \equ(suitedespright). For $f\in\BB_{0\gamma}$ and $h\in\RR^{b/\lambda}_\gamma$, we proceed as in \clm(existenceofnl) and get $$\eqalignno{ \|f*h\|_{\alpha\gamma}&\leq \sup_{x>0\atop y>b/\lambda}\Bigl({w_{\alpha\gamma}(x+y)\over w_{0\gamma}(x)w_{0\gamma}(y)}\Bigr)\|f\|_{0\gamma} \|h\|_{0\gamma}&\cr &=\sup_{x>0\atop y>b/\lambda}\exp\Bigl({\alpha\over x+y}\Bigr) \|f\|_{0\gamma}\|h\|_{0\gamma}&\cr &=\exp(\alpha\lambda/b)\|f\|_{0\gamma} \|h\|_{0\gamma}. &\equat(conv0alpha)\cr }$$ Finally, \equ(boundgscal2), \equ(scalonh0beta) and \equ(conv0alpha) lead to $$ \|D\NN^1_\lambda(f)h\|_{\alpha\gamma}\leq2e^{-A} \exp\bigl(-b(\beta-\gamma/\lambda)\bigr) \|h\|_{\alpha\beta}\|f\|_{\alpha\beta}, \equation(dn1lonrr) $$ where $A=2\sqrt{\alpha(\beta-\gamma/\lambda)}-\alpha(\lambda-1)/b$. Although the convolution deteriorates the $b$-independent factor given by the scaling, \equ(suitedespright) is still a good choice due to the large values of $b$ that will be considered. Proceeding in this way is not crucial, but allows to take smaller values for $b$, thereby saving about 10 percent of the computation time devoted to the evaluation of $D\NN_\lambda(f)$ on $\VV^p$, the space of piecewise constant functions. We conclude by observing that \equ(dn1lonrr) yields a bound which is exponentially small in $b$ only if $\gamma/\beta<\lambda\,$: this is the second compactness condition. Next, we consider the second term in \equ(dnl2bound). One has $$ \|D\NN^2_\lambda(f)h\|_{\alpha\gamma} \leq2\|T\NN^1_\lambda(f)*TD\NN^1_\lambda(f)h\|_{\gamma\alpha}. $$ From $D\NN^1_\lambda(f)h\in\RR^{b/\lambda}_{\gamma}$ it follows that $TD\NN^1_\lambda(f)h\in\LL^{\lambda/b}_{\gamma}$, and applying \clm(lboundonconvll) with $\eta=1$ leads to $$ \|D\NN^2_\lambda(f)h\|_{\alpha\gamma}\leq2 \exp\Bigl(-{\gamma b\over\lambda}\Bigr) \|\NN^1_\lambda(f)\|_{\alpha\gamma}\|D\NN^1_\lambda(f)h\|_{\alpha\gamma}. \equation(dn2lonrr) $$ It remains to estimate $\delta_\lambda(f,h)$. The expectation of $D\NN^2_\lambda(f)h$ is simply bounded by $$ |E(D\NN^2_\lambda(f)h)|\leq \sup_{x>0}\Bigl({x\over w_{\alpha\gamma}(x)}\Bigr) \|D\NN^2_\lambda(f)h\|_{\alpha\gamma}. \equation(boundone2Dn2lrr) $$ Note that in the previous cases, we used the properties of the convolution near the origin to bound this quantity according to \equ(boundone2Dn2lav0). Here, these properties have been used already in the bound \equ(dn2lonrr) to extract a second exponentially small factor in $b$. Therefore, inserting \equ(dn2lonrr) into \equ(boundone2Dn2lrr) leads to a better estimate than \equ(boundone2Dn2lav0). Finally, for $h\in\RR^b_\beta$ and $b$ large, one has the following bounds on $M(h)$ and $E(h)$ $$ |M(h)|\leq{1\over w_{\alpha\beta}(b)}\|h\|_{\alpha\beta}, \qquad |E(h)|\leq{b\over w_{\alpha\beta}(b)}\|h\|_{\alpha\beta}, \equation(boundonmrer) $$ provided $b\geq\sqrt{\alpha/\beta}$ for the first inequality, and $b\geq(1+\sqrt{1+4\alpha\beta})/2\beta$ for the second inequality. \remark The bounds \equ(dn1lonrr),\equ(dn2lonrr), \equ(boundone2Dn2lrr) and \equ(boundonmrer) are implemented in \t{fDN\_right} to estimate \equ(dnl2bound). The validity conditions of \equ(scalonh0beta) and \equ(boundonmrer), namely $b\geq\sqrt{\alpha/(\beta-\gamma/\lambda)}$ ($\geq\sqrt{\alpha/\beta}$) and $b\geq(1+\sqrt{1+4\alpha\beta})/2\beta$, are explicitly checked. \subsection Piecewise Constant Functions Finally, we consider the case of functions $h$ in $\VV^p$. On this space, the tangent map $D\NN_\lambda(f)$ is not a contraction and the relevant information is contained in the images $D\NN_\lambda(f)h$ of the basis vectors $h$ of $\VV^p$. Therefore, in order to keep track of this information, we need to construct a bound on the tangent map in the sense of Section~3. For $p=\{x_0,\dots,x_n\}$ and $I_i=(x_i,x_{i-1})$, a basis of $\VV^p$ is given by $\{\chi_{I_i}\}_{i=1}^n$. Hence, we introduce the following set $\XX$ of characteristic functions, $$ \XX=\{c\chi_{[a,a+\delta]}|\,c\in\real,a>0,\delta>0\}, $$ and we construct a bound on $D\NN_\lambda:\BB_\albe\times\XX\rightarrow\BB_{\alpha\gamma}$ acting from $\std{\BB_\albe}^u\times\std\XX$ to $\std{\BB_{\alpha\gamma}}$, where we define $\std\XX$ to be the collection of all sets of the form $$ (A,B,C)=\{h\in\XX\,|\;h=c\chi_{[a,a+\delta]}\ {\rm with}\ a\in A,\delta\in B,c\in C\} \equation(stdsetchi) $$ for $C\in\std\real$ and $A,B\in\std{\real^*_+}$. Note that, once a bound on $D\NN^1_\lambda:\BB_\albe\times\XX\rightarrow\BB_{\alpha\gamma}$ has been obtained, composing it with the bounds of Section~4 readily yields bounds on the first two terms of $D\NN_\lambda(f)h$, cf. \equ(dnl2) and \equ(dn2). To compute the coefficient $\delta_\lambda(f,h)$ in the third term of \equ(dnl2), the only missing quantities are the mass and the expectation of $h\in\XX$. Those are obtained from the equalities $$ M(\chi_{[a,a+\delta]})=\delta\ ,\qquad E(\chi_{[a,a+\delta]})=\delta(a+\delta/2). \equation(massespofchi) $$ It remains to construct a bound on $D\NN^1_\lambda$. We consider $$ D\NN^1_\lambda/2: \BB_\albe\times\XX \buildrel\hbox{$S_\lambda$}\over{\hbox to 10mm{\rightarrowfill}} \BB_{{\alpha\over4}\gamma}\times\XX \buildrel\hbox{$*$}\over{\hbox to 10mm{\rightarrowfill}} \BB_{\alpha\gamma}. \equation(suitedespxx) $$ The reason for this choice is as follows. Some of the functions $h$ will have support close to the origin or far away from the origin. In such cases, we know from the previous sections that the scaling in \equ(suitedespxx) is a very good contraction. Hence, considering \equ(suitedespxx) will automatically yield an extra contraction factor and improve the bound on the convolution between $S_\lambda h$ and the general term of $S_\lambda f$. A bound on $S_\lambda:\XX\rightarrow\XX$ is easily obtained from $$ S_\lambda\chi_{[a,a+\delta]}=\lambda\chi_{[a/\lambda,(a+\delta)/\lambda]}. $$ Next, we construct a bound on the convolution defined from $\std{\BB_{\zeta\eta}}^u\times\std\XX$ to $\std{\BB_{(\gamma\eta}}^u$, with $\gamma\in[\zeta,4\zeta]$. Let $f=\rho+g$, $\rho\in\AA^u$ and $g\in\BB_{\zeta\eta}$. One has $ f*h=\rho*h+g*h, $ and the second term of this equality will be a part of the general term $\tilde g$ of $f*h$ and will be treated as usual. The first term contains the relevant information and needs to be computed explicitly. In the sequel, we consider for simplicity $h=\chi_{[a,a+\delta]}$. Let $\pi(\rho)=(\{x_i\},\{\rho_i\})_{i=0}^n$ and denote by $\varepsilon$ the mesh of the uniform partition associated with $\rho$. If $\varepsilon\geq\delta$, the function $\rho*h$ takes a simpler form than in the case $\varepsilon<\delta$, and we restrict the domain of our bound to such cases in order to simplify the implementation. Define $$ y_k=a+x_0+k\veps,\qquad k=0,\dots,n+1, \equation(defpartcfchi) $$ and $I_k=[y_k,y_{k+1}]$, $k=0,\dots,n$. It is clear from the properties of the convolution that $\rho*h$ is continuous and has a support equal to $(y_0,y_n+\delta)$. Next, a short computation shows that provided $\varepsilon\geq\delta$, $\rho*h$ is given on the interval $I_k$ by $$ (\rho*h)(y_k+\theta)=\left\{\matrix{ \delta(\rho_k-\delta\rho_{k-1}'/2)+ \theta\delta\rho_{k-1}'+ \theta^2(\rho_k'-\rho_{k-1}')/2\,,&0\leq\theta\leq\delta,\cr \cr \delta(\rho_k-\delta\rho_k'/2)+ \theta\delta\rho_k'\,, \hfill&\delta\leq\theta\leq\varepsilon,\cr }\right. \equation(fconvrhochi) $$ with the convention that $\rho_{-1}=\rho_{n+1}=0$, and where $$ \rho_k'={\rho_{k+1}-\rho_k\over\veps}. $$ Indeed, one has $$ (\rho*h)(y_k+\theta)=\int_a^b\rho(y_k+\theta-x)\,dx =\int_0^\delta\rho(x_k+\theta-\xi)\,d\xi. \equation(fcrp1) $$ Two cases arise: if $\theta\geq\delta$, the function $\rho$ in the above integral is given by $$ \rho(x)=\rho_k+(x-x_k)\rho_k'. \equation(rhoo1) $$ Inserting \equ(rhoo1) into \equ(fcrp1) and integrating lead to the second part of \equ(fconvrhochi). For $\theta<\delta$, we rewrite \equ(fcrp1) as $$ (\rho*h)(y_k+\theta)=\int_0^\theta\rho(x_k+\theta-\xi)\,d\xi+ \int_\theta^\delta\rho(x_k+\theta-\xi)\,d\xi. \equation(fcrp2) $$ In the first term, $\rho$ is again given by \equ(rhoo1), whereas in the second term one has $$ \rho(x)=\rho_{k-1}+(x-x_{k-1})\rho_{k-1}'. \equation(rhoo2) $$ Inserting \equ(rhoo1) and \equ(rhoo2) into \equ(fcrp2) yields the first part of \equ(fconvrhochi). Next, we define the affine part $\tilde\rho$ of $f*h$ to be the linear interpolation of $\rho*h$ at the nodes $\{y_k\}$. More precisely, we consider $$ \tilde\rho=\TT_1\bigl(\{y_i\}_{i=0}^{n+1},\{\tilde\rho_i\}_{i=0}^{n+1}\bigr), \equation(defvalcfchi11) $$ where $$ \tilde\rho_k=\delta(\rho_k-\delta\rho_{k-1}'/2),\qquad k=0,\dots,n+1. \equation(defvalcfchi) $$ Note that $\tilde\rho\in\AA^u$. Finally, the general term of $f*h$ is given by $\tilde g=\rho*h-\tilde\rho+g*h$ and one gets $$ \|\tilde g\|_{\gamma\eta}\leq\|\rho*h-\tilde\rho\|_{\gamma\eta} +\|g\|_{\zeta\eta}\|h\|_{\zeta\eta}. \equation(boundongdn1lchi) $$ For $h=\chi_{[a,a+\delta]}$, one simply uses that $$ ||h\|_{\zeta\eta}\leq\delta\sup_{x\in[a,a+\delta]}w_{\zeta\eta}(x). $$ To bound the first term on the RHS of \equ(boundongdn1lchi), we first note that on the interval $I_k$, $k=0,\dots,n$, $$\eqalign{ \tilde\rho(y)&=\tilde\rho_k+(y-y_k){\tilde\rho_{k+1}-\tilde\rho_k\over\veps}\cr &=\delta(\rho_k-\delta\rho_{k-1}'/2)+\delta(y-y_k) (\rho_k'-\delta(\rho_k'-\rho_{k-1}')/2\veps).\cr }$$ From this formula and the expression \equ(fconvrhochi) for $\rho*h$, one computes for $\theta\in[0,\delta]$, $$ |(\rho*h-\tilde\rho)(y_k+\theta)|=\theta\Bigl(\delta -{\theta\over2}-{\delta^2\over2\veps}\Bigr)|\rho_k'-\rho_{k-1}'|, \equation(absrcmtr) $$ and for $\theta\in[\delta,\varepsilon]$, $$ |(\rho*h-\tilde\rho)(y_k+\theta)|={\delta^2\over2} \Bigl(1-{\theta\over\veps}\Bigr)|\rho_k'-\rho_{k-1}'|. \equation(absrcmtr2) $$ Therefore, integrating \equ(absrcmtr) and \equ(absrcmtr2) leads to $$\eqalignno{ \|\rho*h-\tilde\rho\|_{\gamma\eta} &\leq\sum_{k=0}^n\sup_{x\in I_k}w_{\gamma\eta}(x)\int_{I_k} |(\rho*\chi-\tilde\rho)(y)|\,dy&\cr &= \delta^2\Bigl({1\over4}-{\delta\over6\varepsilon}\Bigr) \sum_{k=0}^n\sup_{x\in I_k}w_\zeet(x) |\rho_{k+1}-2\rho_k+\rho_{k-1}|\ ,\qquad& \equat(convrhochigen)\cr }$$ with the convention $\rho_{-1}=\rho_{n+1}=0$. \remark A set $(A,B,C)\in\std{\XX}$ is represented on the computer by a vector, say \t{fb}, with \t{fb(1)}=$A$, \t{fb(2)}=$B$, and \t{fb(3)}=$C$. The scaling $S_\lambda:\XX\rightarrow\XX$ is implemented in \t{fscale\_chi}. A bound on the convolution in \equ(suitedespxx) is implemented in the procedure \t{fconv\_chi} from \equ(defvalcfchi11), \equ(boundongdn1lchi) and \equ(convrhochigen), where we first check the condition $\varepsilon\geq\delta$. We note that for the purpose of the proof of \clm(hypapcformlk), the base function $f$ is always represented by the same standard set in $\std{\BB_\albe}^u$. Hence, the only quantity in \equ(defvalcfchi) that may change from basis vector to basis vector is $\delta$. By choice of the partition $p_r$, see Section~7.1, most of the basis vectors have equal $\delta$, and the computation of the $\tilde\rho_k$'s is carried out only once for such basis vectors. Finally, the bounds on the scaling and on the convolution in \equ(suitedespxx) and the bounds from Section~4 are composed in the subroutine \t{fDN\_chi} to implement a bound on $D\NN_\lambda(f):\XX\rightarrow\BB_{\alpha\gamma}$, $f\in\BB_\albe$. %====================================================== % Section 7 %====================================================== \section The Tangent Maps $D\MM_\lamka$ In this section we explain how a uniform upper bound on the contraction rate of the operators $\MM_{\lamp,\kappa}$ in a neighborhood of the fixed point $f^*$ is obtained for all $\kappa\in[\lambda^-/\lambda^+,1]$. This will complete the proof of \clm(hypapcformlk). We recall that the operators $\MM_{\lamp,\kappa}$ are given in terms of the original maps $\NN_{\lamp,\kappa}=S_\kappa\NN_{\lamp}$ by $$ \MM_{\lamp,\kappa}=1+M(\NN_{\lamp,\kappa}-1), \equation(tgmapml) $$ where $M$ is some fixed invertible linear map close to the inverse of $1-D\NN_{\lambda^*}(f_{\lambda^*})$. Since $\NN_{\lamp,\kappa}$ is already a good contraction on the subspaces $\BB^p_\albe$ for certain partitions $p$, we need $M$ to be different from the identity only on the finite dimensional subspace $\VV^p$, cf. \equ(firstdecomp). In Section~7.1, we introduce some notation and express the norm of a linear map in $\BB_\albe$ in terms of its norms when restricted to $\BB^p_\albe$ and $\VV^p$. The description of $M$ is given in Section~7.2. The last section is devoted to the final estimate needed to prove \clm(hypapcformlk). \subsection Decomposition of the Operator Norm Let $p=\{x_0,\dots,x_n\}$ be a partition in $\PP_n$. In order to express the projector on $\VV^p$, we introduce two maps associated with $p\,$: the finite rank operator $\II_p:\BB_\albe\rightarrow\real^n$ defined by $$ \II_p f=\biggr\{{1\over|I_i|}\int_{I_i}f(x) dx\biggl\}_{i=1,\dots,n}\,, \equation(projrn) $$ and $\JJ_p:\real^n\rightarrow\VV^p$ defined by $$ \JJ_p\{f_i\}_{i=1}^n=\sum_{i=1}^n f_i\chi_{I_i}, \equation(projba) $$ where $I_i=(x_{i-1},x_i]$ and $|I|$ is the Lebesgue measure of $I\subset\real$. With this notation, the projector $\QQ_p$ on $\VV^p$ may be written as $$ \QQ_p=\JJ_p\II_p. \equation(proj) $$ Let $A$ be a bounded linear map in $\BB_\albe$. One has $$ \|Af\|_\albe\leq\|f\|_p\max\bigl\{\|A\restr{\VV^p}\|, \|A\restr{\BB^p_\albe}\|\bigr\}, \equation(norma1st) $$ where $\|\cdot\|_p$ is the norm in $\BB_\albe$ given by $$ \|f\|_p=\|\QQ_p f\|_\albe+\|(1-\QQ_p)f\|_\albe. \equation(norm1) $$ The norms $\|\cdot\|_p$ and $\|\cdot\|_\albe$ are equivalent, with $$ \|f\|_\albe\leq\|f\|_p\leq K^\albe_p\|f\|_\albe \equation(equivalence) $$ for some constant $K^\albe_p$. From the definition of $\BB^p_\albe$ and its subspaces $\LL^a_\alpha$, $\CC_p$, $\RR^b_\beta$, it follows that $$ \|A\restr{\BB^p_\albe}\|=\max\bigl\{ \|A\restr{\LL^{x_0}_\alpha}\|, \|A\restr{\CC_p}\|, \|A\restr{\RR^{x_n}_\beta}\|\bigr\}. \equation(normabbp) $$ Furthermore, one has $$ \|A\restr{\VV^p}\|=\max_{i=1,\dots,n}\|A\eta_i\|_\albe, \equation(normavvp) $$ with $\eta_i$ the characteristic function of $I_i$ normalized in $\BB_\albe$, i.e., $$ \eta_i=\Bigl(\int_{I_i}w_\albe(x)\,dx\Bigr)^{-1}\chi_{I_i}. \equation(fcarnorm) $$ Inserting \equ(equivalence), \equ(normabbp) and \equ(normavvp) into \equ(norma1st), one gets $$ \|A\|\leq K^\albe_p\max\bigl\{ \{\|A\eta_i\|_\albe\}_{i=1}^n, \|A\restr{\LL^{x_0}_\alpha}\|, \|A\restr{\CC_p}\|, \|A\restr{\RR^{x_n}_\beta}\|\bigr\}. \equation(norma) $$ For $A=D\MM_{\lamp,\kappa}(f)$, evaluating the quantities in the RHS of this expression will yield the desired bound on the norm of the tangent map of $\MM_{\lamp,\kappa}$. The bounds obtained in the previous section will allow us to estimate each of the last three quantities in one step, by evaluating in turn $\|D\MM_{\lamp,\kappa}(f)h\|_\albe$ for all $h$ in the unit balls of $\LL^{x_0}_\alpha$, $\CC_p$ and $\RR^{x_n}_\beta$. In contrast, the contractivity of $\MM_{\lamp,\kappa}$ on $\VV^p$ follows from the specific choice of the operator $M$, and an explicit computation of the $n$ quantities $\|A\eta_i\|_\albe$ is required. This accounts for most of the computation time of the proof. This leads us to the problem of optimizing the partition $p$ in \equ(norma) with respect to $A=D\MM_{\lamp,\kappa}(f)$. Roughly speaking, the size of the intervals in $p=(x_0,\dots,x_n)$ is determined by the contraction rate of $A$ on $\CC_p$ that we need to obtain. Hence, the number of intervals $n$ is fixed by $x_0$ and $x_n$. In order to minimize $n$, we want to maximize $x_0$ and minimize $x_n$. These two parameters determine the contraction rate of $A$ on $\LL^{x_0}_\alpha$ and $\RR^{x_n}_\beta$. Increasing $\alpha$ and $\beta$ improves the contraction and allows to consider larger $x_0$ and smaller $x_n$. However, large values of $\alpha$ and $\beta$ deteriorate the estimate \equ(precf0lp) of \clm(hypapcformlk), i.e., the precision of the approximate fixed point. Good values for $\alpha$ and $\beta$ have been found empirically to be $\alpha=0.5$ and $\beta=0.9$, for which $x_0=0.065$, $x_n=11.83$ and a (non--uniform) partition of 5050 intervals give the desired bound \equ(normdmlpk). In the sequel, we will refer to this partition as $p_r$ and denote $n_r=5050$. We end this section with the computation of the equivalence constant $K^\albe_p$. First, we estimate $\|\QQ_p f\|_\albe\,$. From $$ |(\II_p f)_i|=\Bigl|{1\over|I_i|}\int_{I_i}f(x)\,dx\Bigr| \leq{1\over|I_i|}\sup_{x\in I_i}\Bigl({1\over w_\albe(x)}\Bigr) \int_{I_i}w_\albe(x)\,|f(x)|\,dx, \equation(boundonipfi) $$ it follows $$\eqalign{ \|\QQ_p f\|_\albe&=\sum_{i=1}^n|(\II_p f)_i|\int_{I_i}w_\albe(x)\,dx\cr &\leq\max_{i=1,\dots,n}\Bigl({1\over|I_i|} \sup_{x\in I_i}{1\over w_\albe(x)}\int_{I_i}w_\albe(x)\,dx\Bigr) \|f\|_\albe.\cr }$$ Hence, the following inequality $$ \|f\|_p=\|\QQ_p f\|_\albe+\|(1-\QQ_p)f\|_\albe \leq\|f\|_\albe+2\|\QQ_p f\|_\albe, $$ implies $$ K^\albe_p\leq1+2\max_{i=1,\dots,n}\Bigl({1\over|I_i|} \sup_{x\in I_i}{1\over w_\albe(x)}\int_{I_i}w_\albe(x)\,dx\Bigr). \equation(equivcte) $$ Note that the previous upper bound tends to $3$ from above when $n$ increases and when the size of each interval goes to zero. Also, the weight contributes to this bound by its largest variation on the intervals $\{I_i\}$. We have already encountered a similar situation, cf. \equ(defcp), and we chose to consider a non--uniform partition with a higher density of nodes where the weight varies strongly. For the partition $p_r$ introduced above, one has $K^{\mu\nu}_{p_r}<3.15$. \remark An upper bound on the equivalence constant $K^{\mu\nu}_{p_r}$ is computed in the procedure \t{compute\_equiv\_const}, using \t{swsupint} to estimate the second term in the RHS of \equ(equivcte). The first and last points in the partition $p_r$ are $x_0=0.065$ and $x_{n_r}=11.83$, respectively. The first $100$ (\t{npr1}) intervals are uniform with mesh $\varepsilon_{r_1}=(x_{n_r}-x_0)10^{-4}$ (\t{sepspr1}), whereas the remaining $4950$ (\t{npr2}) intervals are uniform with mesh $\varepsilon_{r_2}=2\,\varepsilon_{r_1}$ (\t{sepspr2}). \subsection The Operator $M$ As mentioned earlier, $M$ should be a good approximation to the inverse of $1-D\NN_{\lambda^*}(f_{\lambda^*})$, and needs to be different from the identity on the finite dimensional space $\VV^{p_r}$ only. Hence, for a certain partition $p\in\PP_m$ to be chosen later, we write $$ M=\bigl(1-\QQ_p D\NN_\lamp(f^0_\lamp)\QQ_p\bigr)^{-1}, \equation(opm1) $$ where $f^0_\lamp$ is the explicit approximate fixed point of $\NN_\lamp$ entering the statement of \clm(hypapcformlk). The previous expression involves the $m\times m$ matrix $$ A=\II_p D\NN_\lamp(f^0_\lamp)\JJ_p, \equation(matriceai) $$ and can be rewritten as $$ M=(1-\JJ_p A\II_p)^{-1}=1+\JJ_p A(1-A)^{-1}\II_p. \equation(opm2) $$ Since we look only for an approximation, the operations involved in the computation of the matrices $A$ and $A(1-A)^{-1}$ need not to be exact. Hence, the use of interval analysis is not required here and we will rely on numerics only. The result of this operation will be denoted by $B$, i.e., $$ B\approx A(1-A)^{-1}. \equation(matrixb) $$ With the notation $$ C_p=\JJ_p C\II_p, $$ for $C$ an $m\times m$ real matrix, $M$ is finally defined by $$ M=1+B_p. \equation(opm) $$ We note that the numerical invertibility of $1-A$ does not imply the invertibility of $M$. Since this property is required in order for the fixed points of $\NN_{\lamp,\kappa}$ and of $1+M(\NN_{\lamp,\kappa}-1)$ to be in correspondence, we must check that $M$ is indeed invertible. We exhibit a matrix $C$ for which $(1+B)C$ is invertible. This implies that the matrix $1+B$ is invertible, which in turn ensures the invertibility of $M$. For $C$, we consider the matrix $1-A$ that has been previously numerically determined. Then, we check rigorously with interval analysis that the matrix $X$ given by $$ X=(1+B)C-1, \equation(inverse1) $$ satisfies $$ \|X\|<1, \equation(niverse2) $$ for some norm on $\real^m$. From this inequality, it then follows that $1+X$ is invertible. The norm on $\real^m$ we use in the program is $\|x\|=\max_{i=1,\dots,m}|x_i|$, that is, for $C$ a real matrix with coefficients $\{c_{ij}\}$, $$ \|C\|=\max_{i=1,\dots,m}\sum_{j=1}^n|c_{ij}|. \equation(nmaxmatrice) $$ We now discuss the choice of the partition $p$ used in the definition of $M$. This partition will be denoted by $p_s$. Since the decomposition $\BB^{p_r}_\albe\oplus\VV^{p_r}$ has been introduced in order to isolate the subspace $\BB^{p_r}_\albe$ on which $\NN_{\lamp,\kappa}$ is a contraction and since the non trivial action of $M$ should turn $\MM_{\lamp,\kappa}$ into a contraction on $\VV^{p_r}$, it is natural to require $$ \BB^{p_r}_\albe\subseteq\Ker(B_{p_s}). $$ This is in particular true if $p_s$ is a subpartition of $p_r$, i.e., $$ p_s\subseteq p_r. \equation(condp1) $$ There is no need for $p_s$ to be equal to $p_r$. In particular, $p_s$ could have fewer nodes than $p_r$, which would improve performance with respect to memory and computation time. By trial and error, we have determined a small partition which satisfies \equ(condp1) and leads to a contraction on $\VV^{p_r}$. This (uniform) partition contains $m_s=500$ intervals. Hence, $B$ is a $500\times500$ matrix with entries in $\SS$, the set of (safe) representable real numbers. For technical reason, the matrix $A$ is not computed according to \equ(matriceai) with $p=p_s$. This would amount to computing the matrix elements $ a_{ij}=(\II_{p_s}D\NN_\lamp(f^0_\lamp)\JJ_{p_s}\hat x_j)_i, $ where $\{\hat x_j\}$ is the canonical base of $\real^{m_s}$. To avoid the writing of special procedures, we want to use our bound on $D\NN_\lamp$ acting on $\XX$, even though interval analysis is not required. Hence, the support of the characteristic function $\JJ_{p_s}\hat x_j$ must be small enough in order for $\JJ_{p_s}\hat x_j$ to be in the domain of that bound. (Recall that the convolution between a characteristic and a piecewise linear function was restricted to such cases.) For the partition $p_s$ introduced above, the bound does not apply for any of the $\JJ_{p_s}\hat x_j$. Hence, we proceed to the evaluation of a different matrix $A$ close to \equ(matriceai). First, we choose a partition $p_t\in\PP_{n_t}$ of which $p_s$ is a subpartition, all the intervals in $p_t$ being small enough. Then we compute $$ a_{ij}=(\II_{p_s}D\NN_\lamp(f^0_\lamp)\tilde\JJ_{p_t}\hat y_{k(j)})_i, \equation(matricea) $$ where $\{\hat y_k\}$ is the canonical basis of $\real^{n_t}$ and $\tilde\JJ_{p_t}:\real^{n_t}\rightarrow\VV^{p_t}$ satisfies $ \II_{p_s}\tilde\JJ_{p_t}\hat y_{k(j)}=\hat x_j. $ We recall that in \equ(matricea), the function $D\NN_\lamp(f^0_\lamp)\tilde\JJ_{p_t}\hat y_{k(j)}$ is given by our bound as a sum $\rho+g$, with $\rho\in\AA$ and $g$ a general term. For the purpose of computing $A$, $g$ is discarded and it remains to discuss the map $\II_{p_s}:\AA\rightarrow\real^{m_s}$. We will need to evaluate later this map rigorously and we now describe how to bound it. Let $p\in\PP_m$ and $\pi(\rho)=(p_\rho,\cdot)$. Define $\tilde p\equiv p\cup p_\rho=\{y_j\}_{j=0}^N$ and $\tilde\rho_j=\rho(y_j)$. Then, writing $\tilde I_j=(y_{j-1},y_j)$, one has for $i=1,\dots,m$, $$ \Bigl(\II_p\rho\Bigr)_i={1\over|I_i|}\sum_{{j\atop\tilde I_j\subseteq I_i}} \int_{\tilde I_j}\tilde\rho(x)\,dx= {1\over|I_i|}\sum_{{j\atop\tilde I_j\subseteq I_i}} |\tilde I_j|{\tilde\rho_j+\tilde\rho_{j-1}\over2}. \equation(projforrho) $$ We restrict the domain of this bound to those $\rho$'s for which the support of the partition $p$ contains the support of $p_\rho$. By proceeding so, we ensure that no information is lost when projecting on $\VV^p$. We end this section by deriving an expression for the operator norm of $M$ in $\BB_\albe$. Recall that this quantity was needed in Section~5, cf. \equ(precmlk), \equ(prec11) and \equ(prec22). We start with the trivial estimate $$ \|M\|\leq 1+\|B_{p_s}\|, \equation(normofm) $$ and express the norm of the finite rank operator $B_{p_s}$ in terms of the partition $p_s$ and the matrix elements of $B$. Let $p=\{x_0,\dots,x_n\}\in\PP_n$, $I_i=(x_{i-1},x_i)$ and let $C$ be an $n\times n$ matrix with real entries $\{c_{ij}\}$. For $f\in\BB_\albe$, one estimates $$\eqalign{ \|\JJ_p C\II_pf\|_\albe &=\sum_{i=1}^n\bigl|\sum_{j=1}^nc_{ij}(\II_pf)_j\bigr| \int_{I_i}w_\albe(x)\,dx\cr &\leq\sum_{j=1}^n|(\II_pf)_j|\sum_{i=1}^n|c_{ij}| \int_{I_i}w_\albe(x)\,dx,\cr }$$ and, using our previous bound \equ(boundonipfi) on $|(\II_pf)_j|$, one gets $$ \|C_p\|\leq N_p^\albe(C), \equation(boundonjjpciip) $$ where $$ N_p^\albe(C)= \max_{j=1,\dots,n}\Bigl({1\over|I_j|}\sup_{x\in I_j} \Bigl({1\over w_{\albe}(x)}\Bigr)\sum_{i=1}^n|c_{ij}| \int_{I_i}w_\albe(x)\,dx\Bigr). \equation(normofb) $$ \remark A bound on the map $\II_p:\AA\rightarrow\real^n$ is implemented in the procedure \t{projection}, checking first the condition on its domain of definition and using \t{fadd} to compute $\tilde p$ and $\{\tilde\rho_j\}$. In the procedure \t{compute\_matrix}, the matrix $B$ (\t{bm}) is computed using \equ(matrixb) and \equ(matricea). A call to \t{show\_invertibility} verifies that $1+B$ is invertible. For the numerical inversion of $1-A$, we use the standard algorithm of Gauss elimination, implemented in the subroutine \t{gaussj}. The operator norms of $M$ and $B_{p_s}$ are computed in \t{compute\_matrix\_norm}. Finally, the partition $p_s$ satisfies $\supp(p_s)=\supp(p_r)$ and is uniform with mesh $\varepsilon_s=10\,\varepsilon_{r_2}$ (\t{sepsps}). Hence, it contains $500$ (\t{mps}) intervals. \subsection Existence of the Family of Fixed Points: Second Estimate In this section, we derive a uniform bound on the norm of the tangent maps $D\MM_{\lamp,\kappa}(f)$ for $\kappa$ in $[\lamm/\lamp,1]$, and $f\in B_r(f^0_\lamp)\subset\BB_\mn$ with $\mu=0.5$, $\nu=0.9$ and $r=9\cdot10^{-4}$. In the sequel, we set $\delta=\lamm/\lamp$. By definition, one has $$D\MM_{\lamp,\kappa}(f) =1+M\bigl(S_\kappa D\NN_\lamp(f)-1\bigr). \equation(dmmlpk) $$ For $f\in\BB_\albe/\HH$, $D\NN_\lambda(f)$ is bounded from $\BB_\albe$ to $\BB_{\alpha\gamma}$ provided $\gamma/\beta\leq\lambda\leq4$. Hence, with $\beta=\nu$ and $\gamma=\nu/\kappa$, $S_\kappa D\NN_\lamp(f)$ is bounded as a map from $\BB_\mn$ to $\BB_\mn$ provided $1/\kappa\leq\lamp\leq4$. One concludes that for all $\kappa\in[\delta,1]$, $D\MM_{\lamp,\kappa}(f)$ is bounded as a map from $\BB_\mn$ to $\BB_\mn$ provided $1/\delta\leq\lamp\leq4$. For the values of $\lamp$ and $\lamm$ as given in the statement of \clm(hypapcformlk), the previous condition is satisfied. To estimate the norm of $D\MM_{\lamp,\kappa}(f)$ in $\BB_\mn$, we proceed as outlined in Section~7.1 and bound each term on the RHS of \equ(norma). We start with the simple case $h\in\BB^{p_r}_\mn$. The property $p_s\subseteq p_r$ implies $B_{p_s}h=0$, so that $Mh=h$. Hence, $$\eqalignno{ \|D\MM_{\lamp,\kappa}(f)h\|_\mn&\leq\|M\|\,\|S_\kappa D\NN_\lamp(f)h\|_\mn&\cr &\leq\|M\|\,\|D\NN_\lamp(f)h\|_{(\kappa\mu)(\nu/\kappa)}&\cr &\leq\|M\|\,\|D\NN_\lamp(f)h\|_{\mu(\nu/\delta)},&\equat(jjaa)\cr }$$ for all $\kappa\in[\delta,1]$. An upper bound on $\|M\|$ was described in the previous section. Representing $f$ by the standard set in $\std{\BB_\mn}^u$ whose affine part is the singleton $\{f^0_\lamp\}$ and whose general term has norm $r$, the bounds of Section~6 yield $0.85$ as an upper bound on the RHS of \equ(jjaa) for all $h$ in the unit ball of $\LL^{x_0}_\mu$, $\CC_{p_r}$, and $\RR^{x_n}_\nu$. Next we consider the more delicate case of $h\in\VV^{p_r}$. According to \equ(norma), one has to estimate the $n_r$ quantities $\|D\MM_{\lamp,\kappa}(f)\eta_i\|_\mn$ where $\eta_i$ is the normalized characteristic function of the $i^{\rm th}$ interval $I_i$ in the partition $p_r$. Recalling that $M=1+B_{p_s}$, we get from \equ(dmmlpk) $$ D\MM_\lampka(f)\eta_i =S_\kappa D\NN_\lamp(f)\eta_i +B_{p_s}(S_\kappa D\NN_\lamp(f)-1)\eta_i. \equation(dmmlpketa1) $$ The bound on the map $D\NN_\lambda(f):\XX\rightarrow\BB_{\alpha\gamma}$ previously constructed yields the function $D\NN_\lamp(f)\eta_i$ represented by a standard set in $\std{\AA}$ and a general term. We denote the former by $\rho_i$ and the latter by $g_i$, i.e., $$ D\NN_\lamp(f)\eta_i=\rho_i+g_i, $$ and rewrite \equ(dmmlpketa1) as $$\eqalignno{ D\MM_\lampka(f)\eta_i &=S_\kappa(\rho_i+g_i) +B_{p_s}\bigl(S_\kappa(\rho_i+g_i)-\eta_i\bigr)&\cr &=MS_\kappa g_i+S_\kappa\rho_i +B_{p_s}(S_\kappa\rho_i-\eta_i).&\equat(dmmlpketa2)\cr }$$ The norm of the first term is bounded as before for all $\kappa\in[\delta,1]$ by $$ \|MS_\kappa g_i\|_\mn \leq\|M\|\,\|g_i\|_{\mu(\nu/\delta)}. \equation(term1) $$ To treat the remaining terms in \equ(dmmlpketa2), we express them as $$\eqalign{ S_\kappa\rho_i+B_{p_s}(S_\kappa\rho_i-\eta_i)&= S_\kappa\bigl(\rho_i+B_{p_s}(\rho_i-\eta_i)\bigr)\cr &\quad+B_{p_s}(S_\kappa-1)\rho_i\cr &{\quad}+(1-S_\kappa)B_{p_s}(\rho_i-\eta_i).\cr }\equation(jjcc)$$ As we shall see below, the last two terms are of order $1-\kappa$, and the first term is small due to the cancellations arising by construction from the action of $M$. This term, therefore, needs to be computed explicitly. Let us begin with this term first. One starts by factorizing the action of $S_\kappa$ by using again $$ \|S_\kappa\bigl(\rho_i+B_{p_s}(\rho_i-\eta_i)\bigr)\|_\mn \leq\|\rho_i+B_{p_s}(\rho_i-\eta_i)\|_{\mu(\nu/\delta)}, \equation(term2) $$ which is valid for all $\kappa\in[\delta,1]$. Next, since $p_s\subseteq p_r$, it is enough to construct a bound on the map $$ \AA\times\XX\ni(\rho,\chi_I)\mapsto \|\rho+\JJ_p C\II_p(\rho-\chi_I)\|_{\alpha\gamma}, \equation(maprmbprmc) $$ for $p\in\PP_n$ and $C$ an $n\times n$ matrix, restricted to intervals $I$ satisfying $I\subseteq I_i$ for some interval $I_i$ in the partition $p$. A bound on $\rho\mapsto\II_p\rho$ has already been discussed in the previous section. Denoting by $I_i$ the $i^{\rm th}$ interval in the partition $p$, one has $(\II_p\chi_I)_i=0$ if $I\cap I_i=\emptyset$, and otherwise $$ (\II_p\chi_I)_i={|I|/|I_i|}. \equation(projonchi) $$ A bound on the map $C:\real^n\rightarrow\real^n$ is readily implemented with interval analysis, and it only remains to consider the map $(\rho,v)\mapsto\|\rho+\JJ_pv\|_{\alpha\gamma}$, $(\rho,v)\in\AA\times\real^n$. Let us denote $\pi(\rho)=(p_\rho,\cdot)$, $\tilde p=p\cup p_\rho=\{y_j\}_{j=0}^N$ and $\tilde\rho_j=\rho(y_j)$. Imposing the restriction $\supp(p_\rho)\subseteq\supp(p)$, one obtains $$ \|\rho+\JJ_pv\|_{\alpha\gamma}\leq\sum_{i=1}^n \sum_{{j\atop\tilde I_j\subseteq I_i}} \sup_{x\in\tilde I_j}w_{\alpha\gamma}(x) |\tilde I_j| {|\tilde\rho_j+v_i|+|\tilde\rho_{j-1}+v_i|\over2}, \equation(boundonndiff) $$ where $\tilde I_j$ stands for the $j^{\rm th}$ interval in $\tilde p$. This finishes the construction of a bound on the map \equ(maprmbprmc), which, given standard sets containing $\rho_i$ and $\eta_i$, provides an estimate on the RHS of \equ(term2). Next, the second term in \equ(jjcc) is simply bounded by $$\|B_{p_s}(S_\kappa-1)\rho_i\|_\mn \leq\|B_{p_s}\|\,\|(S_\kappa-1)\rho_i\|_\mn. \equation(term3) $$ The operator norm of $B_{p_s}$ has been determined in the previous section, and \clm(unifborne) provides a bound uniform in $\kappa$ for the second factor, namely $$ \|(S_\kappa-1)\rho_i\|_\mn\leq(1-\delta)\bigl(\|\rho_i\|_\mn+ \|x\rho_i'\|_{\mu(\nu/\delta)}\bigr). $$ To treat the last term in \equ(jjcc), we use the \claim Lemma(bbbbb) Let $p=\{x_0,\dots,x_n\}\in\PP_n$, $C$ an $n\times n$ matrix with coefficients $\{c_{ij}\}$, and $0<\kappa\leq1$. Then the operator norm of $(1-S_\kappa)C_p$ in $\BB_\albe$ satisfies $$ \|(1-S_\kappa)C_p\|\leq R^\albe_p(\kappa) N_p^\albe(C), \equation(normofb2) $$ where $N_p^\albe(C)$ is given by \equ(normofb) and $$ R_p^{\albe}(\kappa)=(1-\kappa)+ \max_{i=1,\dots,n} {1\over\int_{I_i}w_\albe(x)dx} \Bigl(\int_{x_{i-1}}^{x_{i-1}/\kappa}w_\albe(x)\,dx+ \int_{x_i}^{x_i/\kappa}w_\albe(x)\,dx\Bigr). $$ The only dependence on $\kappa$ in \equ(normofb2) is in the factor $R^\albe_p(\kappa)$. Furthermore, $R^\albe_p(\kappa)$ is decreasing in $\kappa$. Hence, one obtains $$ \|(1-S_\kappa)B_{p_s}(\rho_i-\eta_i)\|_\mn \leq R^\mn_{p_s}(\delta)N^\mn_{p_s}(B) (\|\rho_i\|_\mn+1), \equation(term4) $$ for all $\kappa\in[\delta,1]$. Finally, a bound on $\|D\MM_{\lamp,\kappa}(f)\eta_i\|_\mn$ follows from \equ(term1), \equ(term2), \equ(term3) and \equ(term4). As mentioned earlier, computing this bound for the $n_r=5050$ basis vectors $\eta_i$ of $\VV^{p_r}$ accounts for most of the computation time. In the terms involving explicitly $\eta_i$, one can, using the linearity, factorize the value of $\eta_i$, that is $(\int_{I_i}w_\mn)^{-1}$. Therefore, one only needs to compute an upper bound on this quantity. Furthermore, by proceeding like this one can take advantage of the fact that the value of $\chi_{I_i}$ can be represented by the standard set containing only the representable number one. This leads to a standard set containing $\rho_i+g_i$ which is more localized and improves the quality of the final bound. We end this section with the \noindent{\bf Proof of \clm(bbbbb).} For $f\in\BB_\albe$, one has $$\eqalignno{ \|(1-S_\kappa)C_p f\|_\albe &=\int_0^\infty w_\albe(x)\Bigl|(1-S_\kappa)\sum_{i=1}^n \sum_{j=1}^nc_{ij}(\II_pf)_j\chi_{I_i}(x)\Bigr|\,dx&\cr &\leq\sum_{j=1}^n|(\II_pf)_j|\sum_{i=1}^n|c_{ij}|\int_0^\infty w_\albe(x)|(1-S_\kappa)\chi_{I_i}(x)|\,dx.\qquad&\equat(jjdd)\cr }$$ Furthermore, one has $$ \int_0^\infty w_\albe|(1-S_\kappa)\chi_{I_i}| \leq\int_{x_{i-1}}^{x_{i-1}/\kappa}w_\albe+(1-\kappa) \int_{I_i}w_\albe+ \kappa\int_{x_i}^{x_i/\kappa}w_\albe. $$ Factorizing $\int_{I_i}w_\albe$ in the previous expression and inserting the bound \equ(boundonipfi) on $|(\II_p f)_i|$ into \equ(jjdd) finally leads to \equ(normofb2). \qed \remark A bound on the map $(\rho,v)\mapsto\|\rho+\JJ_{p_s}v\|_{\alpha\gamma}$ is implemented in the procedure \t{snorm\_add}, and the product $Cv$ is implemented in \t{linear\_app}. A uniform bound on the norm of $(1-S_\kappa)B_{p_s}$ is computed in \t{compute\_norm\_of\_1mSB}. Given $i\in\{1,\dots,n_r\}$, the subroutine \t{init\_chi} returns both a standard set in $\std{\XX}$ containing $\chi_{I_i}$ and the value of $\eta_i$, whereas the subroutine \t{fDM\_chi} computes a bound on $\|D\MM_{\lamp,\kappa}(f)\eta_i\|_\mn$. Finally, for all $f\in B_r(f^0_\lamp)$ and $\kappa\in[\delta,1]$, a uniform bound on the norm of the tangent maps $D\MM_{\lamp,\kappa}(f)$ is implemented according to \equ(norma) in \t{compute\_norm\_of\_DM}. \sectionnonr Acknowledgments J.W. would like to thank Jean--Pierre~Eckmann, Peter~Wittwer, and the Department of Theoreti\-cal Physics at the University of Geneva for their warm hospitality while part of this work was carried out. A.S. and P.W. would like to thank Jan~Wehr and the Department of Mathematics at the University of Arizona at Tucson for their warm hospitality and for providing us generously with computer resources while part of this work was carried out. %====================================================== % Appendix %====================================================== \sectionnonr Appendix %\sectionnonr Appendix \smallskip \noindent{\bf Proof of \clm(existenceofnl2).} \noindent If for some fixed $\lambda\in(1,4)$ and $\alpha,\beta>0$, $f_\lambda$ is a fixed point of $\NN_\lambda$ and belongs to $\BB_{\alpha\beta}$, then \clm(rembanach) and \clm(existenceofnl) imply that $f_\lambda\in\BB$. We now prove that, in addition, $f_\lambda$ is at least once differentiable, with $f_\lambda'\in\BB$. The regularization properties of the convolution imply then immediately that $f_\lambda$ is of class $\CC^\infty(\real_+)$. For $\zeta,\eta\geq0$, let $\BB^1_\zeet$ denote the Sobolev space of functions in $\BB_\zeet$ with one (distributional) derivative in $\BB_\zeet$, i.e., $$ \BB^1_\zeet=\bigl\{f\in\BB_\zeet\,|\;f'\in\BB_\zeet\bigr\}, $$ with the norm $$ \|f\|_\zeet^1=\|f\|_\zeet+\|f'\|_\zeet\,. $$ In the sequel, we adopt the shorter notation $\BB_\zeta=\BB_{\zeta\zeta}$ and $\BB^1_\zeta=\BB^1_{\zeta\zeta}$. One shows that the fixed point $f_\lambda$ belongs to $\BB^1_\zeta$ for all $\zeta>0$ by the following argument. One exhibits an $h\in\BB^1_\zeta$ and two sequences $\{f_n\}_{n\geq0}$ and $\{g_n\}_{n\geq0}$ satisfying $f_\lambda=h+f_n+g_n$ for all $n\geq0$, such that $\{f_n\}_{n\geq0}$ is Cauchy in $\BB^1_\zeta$ and $\{g_n\}_{n\geq0}$ converges to zero in $\BB_\zeta$. Hence, $f_\lambda$ is equal in $\BB_\zeta$ to a function belonging to $\BB^1_\zeta$. Since $\NN_\lambda$ preserves the regularity, this function is also a fixed point of $\NN_\lambda$. Therefore, it is equal to $f_\lambda$ in $\BB^1_\zeta$. We first construct recursively the sequences $\{f_n\}_{n\geq0}$ and $\{g_n\}_{n\geq0}$. Since $f_\lambda$ belongs to $\BB_\zeta$ for all $\zeta>0$, and since $\CC^\infty_0(\real_+)$ is dense in $\BB_\zeta$, there exist for every $\delta_0>0$ an $h\in\BB^1_\zeta$ and a $g_0\in\BB_\zeta$ satisfying $$ f_\lambda=h+g_0, \aequation(ap2) $$ with $$ \|g_0\|_\zeta\leq\delta_0. \aequation(ap1) $$ Moreover, one defines $$ f_0\equiv0. \aequation(apf0) $$ Denoting $c_\lambda(f_\lambda)=\bar c_\lambda$ and $\overline{\NN}_\lambda=\bar c_\lambda\NN^1_\lambda+c_2\NN^2_\lambda$, we now define for all $n\geq0$, $$\eqalign{ f_{n+1}&=\overline{\NN}_\lambda(h+f_n)-h+\CC_\lambda(f_n,g_n),\cr g_{n+1}&=\overline{\NN}_\lambda(g_n), }\aequation(ap2p)$$ where $$ \CC_\lambda(f,g)=\overline{\NN}_\lambda(h+f+g)-\overline{\NN}_\lambda(h+f)- \overline{\NN}_\lambda(g). $$ Note that $\CC_\lambda(f,g)$ contains only cross terms between $h+f$ and $g$. We now check that the sequences $\{f_n\}_{n\geq0}$ and $\{g_n\}_{n\geq0}$ have the desired properties, i.e., $f_\lambda=h+f_n+g_n$, $\{g_n\}_{n\geq0}$ converges to zero in $\BB_\zeta$, and $\{f_n\}_{n\geq0}$ is Cauchy in $\BB^1_\zeta$. Since $f_\lambda$ is a fixed point of $\overline{\NN}_\lambda$, it first follows from \equ(ap2) and \equ(ap2p) that $$ f_\lambda=h+f_n+g_n, $$ for all $n\geq0$. Furthermore, $\zeta>0$ and $\lambda\in(1,4)$ together with \clm(existenceofnl) imply that $\{g_n\}_{n\geq0}$ converges to zero in $\BB_\zeta$. Indeed, $\overline{\NN}_\lambda$ is well defined as a map from $\BB_\zeta$ to $\BB_\zeta$, and the bounds obtained in the proof of \clm(existenceofnl) lead to $$ \|g_n\|_\zeta\leq\bar c_\lambda\|g_{n-1}\|^2_\zeta+c_2\|g_{n-1}\|^4_\zeta. $$ Applying this inequality recursively and using $\|g_0\|_\zeta\leq\delta_0<1$, one gets for all $n\geq1$ $$ \|g_n\|_\zeta\leq\delta^{2^n}, \aequation(ap3) $$ where $$ \delta=(\bar c_\lambda+c_2)\delta_0<1 $$ for $\delta_0$ small enough. Note that \equ(ap2), \equ(ap1) and \equ(ap3) imply, for $\delta_0$ small enough, the uniform bound $$\eqalignno{ \|f_n\|_\zeta&\leq\|f_\lambda-h\|_\zeta+\|g_n\|_\zeta\cr &\leq2\delta_0. &\aequat(ap33)\cr }$$ Next, in order to show that $f_n\in\BB^1_\zeta$ for all $n\geq0$, one proceeds as in \clm(existenceofnl) and studies the maps which enter the definition of $\overline{\NN}_\lambda$ and $\CC_\lambda$, i.e., $S_\lambda$, $T$ and the convolution operator. From \equ(boundonconv) and $(f*g)'=f'*g$, it follows that $$ \|f*g\|^1_{(4\sigma)\tau}\leq\|f\|^1_{\sigma\tau}\|g\|_{\sigma\tau}, \aequation(ap5) $$ whereas \equ(boundongscal) together with $\lambda>1$ and $(S_\lambda f)'=\lambda S_\lambda f'$ leads to $$\eqalignno{ \|S_\lambda f\|^1_{(\sigma/\lambda)(\lambda\tau)}&\leq \|f\|_{\sigma\tau}+\lambda\|f'\|_{\sigma\tau}&\cr &\leq\lambda\,\|f\|^1_{\sigma\tau}. &\aequat(ap4)\cr }$$ \equ(ap5) and \equ(ap4) imply in particular $$ \|S_\lambda(f*g)\|^1_{(4\zeta/\lambda)(\lambda\zeta)} \leq\lambda\|f\|^1_\zeta\|g\|_\zeta. \aequation(ap6) $$ We now show that for all $\tau>\tau'$, $T$ is a bounded operator from $\BB^1_{\sigma\tau}$ to $\BB^1_{\tau'\sigma}$. One has $\|Tf\|_{\tau'\sigma}=\|f\|_{\sigma\tau'}\leq\|f\|_{\sigma\tau}$, and using $$ (Tf)'(x)=-{1\over x^2}\Bigl(2x(Tf)(x)+(Tf')(x)\Bigr), $$ one gets $$\eqalignno{ \|(Tf)'\|_{\tau'\sigma}&\leq2\int_0^\infty{1\over x}\,w_{\tau'\sigma}(x)\, |Tf|(x)\,dx +\int_0^\infty{1\over x^2}\,w_{\tau'\sigma}(x)\,|Tf'|(x)\,dx&\cr &=2\int_0^\infty x\,w_{\sigma\tau'}(x)\,|f|(x)\,dx +\int_0^\infty x^2\,w_{\sigma\tau'}(x)\,|f'|(x)\,dx&\cr &\leq2\sup_{x>0}{(1+x^2)w_{\sigma\tau'}(x)\over w_{\sigma\tau}(x)} \|f\|^1_{\sigma\tau}&\cr &\leq C_{\tau\tau'}\|f\|^1_{\sigma\tau}, &\aequat(ap66)\cr }$$ where $C_{\tau\tau'}$ is finite as long as $\tau>\tau'$. In particular, since $\lambda>1$, \equ(ap6) and \equ(ap66) imply $$ \|TS_\lambda(f*g)\|^1_{\zeta(4\zeta/\lambda)} \leq C\|f\|^1_\zeta\|g\|_\zeta, $$ which in turn, with $\lambda<4$, leads to $$ \|T\bigl(TS_\lambda(f*g)*TS_\lambda(\bar g*\bar{\bar g})\bigr)\|^1_{\zeta} \leq C\|f\|^1_\zeta\|g\|_\zeta\|\bar g\|_\zeta\|\bar{\bar g}\|_\zeta. \aequation(ap666) $$ Therefore, $\overline{\NN}_\lambda$ is well defined as a map from $\BB^1_\zeta$ to $\BB^1_\zeta$ for $\zeta>0$ and $\lambda\in(1,4)$. Assume now that $f_n\in\BB^1_\zeta$ and that $\delta_0$ is small enough. Then, \equ(ap33), \equ(ap6), and \equ(ap666) lead to $$\eqalignno{ \|\overline{\NN}_\lambda(h+f_n)\|^1_\zeta &\leq\|\overline{\NN}_\lambda(h)\|^1_\zeta+ C\|f_n\|_\zeta(\|h\|^1_\zeta+\|f_n\|^1_\zeta)&\cr &\leq C_1+{1\over2}\bar\delta\|f_n\|^1_\zeta, &\aequat(apfn1)\cr }$$ for some positive $\bar\delta<1$. Similarly, using \equ(ap3) and \equ(ap33), one gets $$ \|\CC_\lambda(f_n,g_n)\|^1_\zeta \leq C\|g_n\|_\zeta\bigl(\|f_n\|^1_\zeta+\|h\|^1_\zeta\bigr). \aequation(ap8) $$ Therefore, \equ(apfn1) and \equ(ap8) lead, together with \equ(ap3), to $$ \|f_{n+1}\|^1_\zeta\leq C_2+\bar\delta\|f_n\|^1_\zeta. $$ Using this bound recursively, one obtains $f_n\in\BB^1_\zeta$ for all $n\geq0$, together with the uniform estimate $$\|f_n\|^1_\zeta\leq C_2\sum_{k=0}^n\bar\delta^k \leq K. \aequation(ap9) $$ Finally, we check that the sequence $\{f_n\}_{n\geq0}$ is Cauchy in $\BB^1_\zeta$. Since \equ(ap3), \equ(ap8) and \equ(ap9) imply that $$ \lim_{n\rightarrow\infty}\|\CC_\lambda(f_n,g_n)\|^1_\zeta=0, $$ it only remains to show that $\{\overline{\NN}_\lambda(h+f_n)\}_{n\geq0}$ is Cauchy in $\BB^1_\zeta$. We first verify that the sequence $\{h_n\}_{n\geq0}$, with $$ h_n\equiv\NN^1_\lambda(h+f_n), $$ is Cauchy in $\BB^1_{(4\zeta/\lambda)(\lambda\zeta)}$. Since $\lambda\in(1,4)$, this implies in particular the convergence of $\{\NN^1_\lambda(h+f_n)\}_{n\geq0}$ in $\BB^1_\zeta$. Defining $$ \tilde\NN^1_\lambda(f,g)=S_\lambda(f*g), $$ one has $$\eqalign{ \|h_n-h_m\|^1_{\sigma\tau} &\leq 2\|\tilde\NN^1_\lambda(h,f_n)-\tilde\NN^1_\lambda(h,f_m)\|^1_{\sigma\tau} +\|\NN^1_\lambda(f_n)-\NN^1_\lambda(f_m)\|^1_{\sigma\tau}\cr &\leq 2\|\tilde\NN^1_\lambda(h,f_n-f_m)\|^1_{\sigma\tau} +\|\tilde\NN^1_\lambda(f_n,f_n-f_m)\|^1_{\sigma\tau} +\|\tilde\NN^1_\lambda(f_m,f_m-f_n)\|^1_{\sigma\tau},\cr }$$ which leads, with \equ(ap6) and \equ(ap9), to $$\eqalign{ \|h_n-h_m\|^1_{(4\zeta/\lambda)(\lambda\zeta)} &\leq C\bigl(2\|h\|^1_\zeta+\|f_n\|^1_\zeta+\|f_m\|^1_\zeta\bigr) \|f_n-f_m\|_\zeta\cr &\leq K\|g_n-g_m\|_\zeta.\cr }$$ Therefore, the convergence of $\{h_n\}$ in $\BB^1_{(4\zeta/\lambda)(\lambda\zeta)}$ follows from the convergence of $\{g_n\}$ in $\BB_\zeta$. Finally, in order to see that $\{\NN^2_\lambda(h+f_n)\}$ is Cauchy in $\BB^1_\zeta$, one observes that $$ \NN^2_\lambda(h+f_n)=F(h_n), $$ where $$ F(f)=T\bigl(Tf*Tf\bigr), $$ and that the bounds obtained above imply the continuity of $F$ as a map from $\BB^1_{(4\zeta/\lambda)(\lambda\zeta)}$ to $\BB^1_\zeta$ for $\zeta>0$ and $\lambda\in(1,4)$. Hence, the convergence of $\{\NN^2_\lambda(h+f_n)\}$ in $\BB^1_\zeta$ follows from the convergence of $\{h_n\}$ in $\BB^1_{(4\zeta/\lambda)(\lambda\zeta)}$. \qed \noindent{\bf Proof of \clm(gentermopt).} \noindent For $x\in I_i$, the functions $\rho$ and $T\tilde\rho$ are given by $$\eqalignno{ \rho(x)&={1\over\veps}\Bigl(\rho_{i}(x-x_{i-1})+\rho_{i-1}(x_{i}-x)\Bigr), &\aequat(tbb)\cr \tilde\rho(1/x)&=\Bigl({1\over x_{i-1}}-{1\over x_{i}}\Bigr)^{-1}\Bigl( x_{i-1}^2\rho_{i-1}\bigl({1\over x}-{1\over x_{i}}\bigr)+x_{i}^2\rho_{i} \bigl({1\over x_{i-1}}-{1\over x}\bigr)\Bigr)&\cr &={1\over\veps x}\Bigl(x_{i-1}^3\rho_{i-1}(x_{i}-x)+x_{i}^3\rho_{i} (x-x_{i-1})\Bigr),&\aequat(tcc)\cr }$$ and one computes $$\eqalignno{ |\rho(x)-T\tilde\rho(x)| &=|\rho(x)-\tilde\rho(1/x)/x^2|&\cr &={1\over\veps}\Bigl|\rho_{i-1}(x_{i}-x)\bigl(1-{x_{i-1}^3\over x^3}\bigr)- \rho_{i}(x-x_{i-1}) \bigl({x_{i}^3\over x^3}-1\bigr)\Bigr|&\cr &={(x-x_{i-1})(x_{i}-x)\over\veps x^3}\bigl|\rho_{i-1}(x-x_{i-1})^2-\rho_{i} (x_{i}-x)^2-3x(x_{i}\rho_{i}-x_{i-1}\rho_{i-1})\bigr|&\cr &\leq{\veps\over4}\Bigl({|\rho_{i}-\rho_{i-1}|\over x}+{|x_{i}\rho_{i}- x_{i-1}\rho_{i-1}|\over x^2}+{|x_{i}^2\rho_{i}-x_{i-1}^2\rho_{i-1}|\over x^3} \Bigr).&\aequat(tdd)\cr }$$ Integrating the expression on the RHS of \equ(tdd) leads to the stated result. \qed \goodbreak \noindent{\bf Proof of \clm(rhoprhop).} \noindent By definition of $\AA$, it is clear that $(\rho*\sigma)''\in\AA$. Furthermore, because $\rho$ and $\sigma$ have a uniform partition with identical mesh size $\varepsilon$, $(\rho*\sigma)''$ is also defined on a uniform partition. It is given by $\{z_k\}_{k=0}^{2n}$ where $z_k$ is defined in \equ(partz). It remains to compute $v_k\equiv(\rho*\sigma)''(z_k)$. With $\rho'(x)=\sum_{i=1}^n\rho'_i\chi_{I_i}(x)$ and $\sigma'(x)=\sum_{j=1}^n\sigma'_j\chi_{J_j}(x)$, where $\rho'_i=(\rho_i-\rho_{i-1})/\varepsilon$ and $\sigma'_j=(\sigma_j-\sigma_{j-1})/\varepsilon$, one gets $$ v_k=\varepsilon\sum_{i+j=k+1}\rho'_i\sigma'_j. $$ Expressing the RHS of the previous equality in terms of the coefficients \equ(discrconv) finally leads to the relation \equ(valz). \qed \noindent{\bf Proof of \clm(convgent).} \noindent For $k=2l$, $l=0,\dots,n-1$, and $\theta\in(0,2\varepsilon)$, one has $$ \tilde\rho(z_k+\theta) =C_0(k)+{\theta\over2\varepsilon}\Bigl(C_0(k+2)-C_0(k)\Bigr). $$ The continuity properties of $\rho*\sigma$ imply $$\eqalign{ C_0(k+1)&=C_0(k)+\varepsilon C_1(k)+\varepsilon^2 C_2(k) +\varepsilon^3 C_3(k),\cr C_1(k+1)&=C_1(k)+2\varepsilon C_2(k) +3\varepsilon^2 C_3(k),\cr C_2(k+1)&=C_2(k)+3\varepsilon C_3(k),\cr }$$ $k=0,\dots,2n-1$. For $\theta\in(0,\varepsilon)$, these relations allow us to write $$ \tilde\rho(z_k+\theta)= C_0(k)+\theta C_1(k)+\varepsilon\theta\Bigl(2C_2(k+1)- {5\varepsilon\over2}C_3(k)+{\varepsilon\over2}C_3(k+1)\Bigr), $$ and, from \equ(convpwl), $$ (\rho*\sigma)(z_k+\theta)=C_0(k)+\theta C_1(k)+ \theta^2\Bigl(C_2(k+1)-(3\varepsilon-\theta)C_3(k)\Bigr). $$ Hence, $$ (\tilde\rho-\rho*\sigma)(z_k+\theta)= \theta(2\varepsilon-\theta)C_2(k+1)+ \theta\Bigl(\theta(3\varepsilon-\theta)-{5\varepsilon^2\over2}\Bigr) C_3(k)+{\varepsilon^2\theta\over2}C_3(k+1), $$ which leads to the estimate $$ \int_{z_k}^{z_{k+1}}|(\tilde\rho-\rho*\sigma)(y)|dy\leq \varepsilon^3\Bigl({2\over3}|C_2(k+1)|+{\varepsilon\over2}|C_3(k)|+ {\varepsilon\over4}|C_3(k+1)|\Bigr). $$ For $\theta\in(\varepsilon,2\varepsilon)$, one proceeds similarly and obtains $$ \int_{z_{k+1}}^{z_{k+2}}|(\tilde\rho-\rho*\sigma)(y)|dy\leq \varepsilon^3\Bigl({2\over3}|C_2(k+1)|+{\varepsilon\over4}|C_3(k)|+ {\varepsilon\over2}|C_3(k+1)|\Bigr), $$ and the bound \equ(convgenpwl) follows immediately. \qed %====================================================== % References %====================================================== \sectionnonr References \ref \no ANW \by Ahlberg, J.H., E.N. 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Woo \paper A central limit theorem for nonlinear hierarchical sequences of random variables \jour Annals of Probability \toappear \endref \ref \no Z \by Ziman, J.M. \paper The localization of electrons in ordered and disordered systems: I.~Percolation of classical particles \jour J. Phys. C \vol 1 \pages 1532--1538 \yr 1968 \endref \message{*********************************************************************} \message{To extract SUPPLEMENTARY MATERIAL, see the beginning of this TeX file} \message{*********************************************************************} \bye % % Save the remainder of this file into supplement.uu %---------------------cut here---------------------- begin 664 Section.8.tar.gz M'XL("'S;(C@``U-E8W1I;VXN."YT87(`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`OSB3/?,+V9I&.'OYYOKLW1\O M=!N^?_=4:6]>*DE*7;YX__25U/"Y_T>7K^ZM(=R\8^G?V>?OGEY]?7'U[DR6 M3&HX_>_UO37$JQTH_ MOKJY-6[9)]+KRZ__^&#IXW^RX66RG]U0BGU^_>W35S?RMSFS;]Y__<>S]U>7 M,O+79[X=S_[+]>U:7GY[?RW_A1E[^N[LYH]/7[TZ>W9Q]N+RV\L7%R]NS_35 MRU^?_ZV<7UO=W5&3KV=A_ZG6KB1ZN9<_>Q M>NC)9[.>=V_>/7WUL=K6K)WV9#_*OSA[?7'S1]M)WV,$^]$__>Z[N[SOY[LKWQ](7S,Z;/EDQHFFSKM\MU=I1_].,=G-OOB M\O7%U8V>W+>/ME^$\^T7W$>/SU^^O=S_'7^NE'L^B,_?S>O+O_'QG]R\N'SY\M'-S;]? 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