% /u/lex2/agw/ibc/ibc.sty, Thu Jul 9 10:02:23 1987, edit by agw \documentstyle{amsppt} \font\twelverm=cmr12 \font\twelvebf=cmbx12 \def\makeatletter{\catcode`\@=11\relax} \def\makeatother{\catcode`\@=12\relax} \makeatletter \def\B@{B } \def\D@{D } \def\askstyle{\immediate\write\sixt@@n{Do you want B(ook style) or D(raft style)? (Type B or D, follow by :) }\loop \read\m@ne to\ans@ \xdef\next{\def\noexpand\Ans@{\ans@}}% \uppercase\expandafter{\next}% \ifx\Ans@\B@\badans@false\bookstyle\else \ifx\Ans@\D@\badans@false\draftstyle\else \badans@true\fi\fi \ifbadans@\immediate\write\sixt@@n{Type B or D, follow by : }% \repeat} \def\bookstyle{\hsize=28truepc\vsize=46truepc} \def\draftstyle{\magnification 1200\parindent=10pt\baselineskip=15pt} \askstyle \newif\iftitle % for title pages of chapters \titlefalse % in case we start in the middle of a chapter \def\chapterbeg#1#2{ \global\titletrue \vfill\eject \null\vskip 8 truepc \centerline{\twelverm Chapter #1} \line{\hrulefill} \centerline{\twelvebf #2} \global\def\chaptertitle{#1. \lowercase{#2}} \global\def\currentsectiontitle{\lowercase{#2}} \message{#1. #2: } \null\vskip 9 truepc } \def\unnumberedchapterbeg#1{ \global\titletrue \null\vskip 8 truepc \vskip 12 pt \line{\hrulefill} \centerline{\twelvebf #1} \global\def\chaptertitle{\lowercase{#1}} \global\def\currentsectiontitle{\lowercase{#1}} \message{#1} \null\vskip 9 truepc } \def\sectionbeg#1#2\medskip{ \vskip 2 truepc \centerline{\bf #1. #2} \nopagebreak\vskip 1truepc\nopagebreak \global\def\currentsectiontitle{#1. \lowercase{#2}} \message{#1. #2} } \newif\if@nobreak \@nobreakfalse \newif\ifignore \def\@makeother#1{\catcode`#112\relax} \def\newwrite{\alloc@7\write\chardef\sixt@@n} \newwrite\@unused \def\typeout#1{{\let\protect\string\immediate\write\@unused{#1}}} \newdimen\@savsk \newcount\@savsf \def\@bsphack{\@savsk\lastskip \ifhmode\@savsf\spacefactor\fi} \def\@esphack{\relax\ifhmode\spacefactor\@savsf {}\ifdim \@savsk >\z@ \global\ignoretrue \ignorespaces \fi \fi} \def\@sanitize{\@makeother\ \@makeother\\\@makeother\$\@makeother\&% \@makeother\#\@makeother\^\@makeother\^^K\@makeother\_\@makeother\^^A% \@makeother\%\@makeother\~} \def\makeindex{\newwrite\@indexfile \immediate\openout\@indexfile=\jobname.idx \def\index{\@bsphack\begingroup\@sanitize\@wrindex\@indexfile} \typeout{Writing index file \jobname.idx }} \def\thepage{\the\pageno} \def\@wrindex#1#2{\let\thepage\relax \xdef\gtempa{\write#1{\string \indexentry{#2}{\thepage}}}\endgroup\gtempa \if@nobreak \ifvmode\@nobreak\fi\fi\@esphack} \def\index{\@bsphack\begingroup\@sanitize\@index} \def\@index#1{\endgroup\@esphack} \def\@idxitem{\par\hangindent 40pt} \def\subitem{\par\hangindent 40pt \hskip 20pt} \def\subsubitem{\par\hangindent 40pt \hskip 30pt} \def\indexspace{\par \vskip 10pt plus 5pt minus 3pt\relax} \def\beginindex{\begingroup\let\item\@idxitem\begindoublecolumns} \def\endindex{\enddoublecolumns\endgroup} \newbox\partialpage \newdimen\pagewidth \pagewidth=\hsize \newdimen\pageheight \pageheight=\vsize \newdimen\colwidth \colwidth=\pagewidth \divide\colwidth by 2 \newdimen\twicecolheight \twicecolheight=\pageheight \multiply\twicecolheight by 2 \advance\twicecolheight by 1truepc % well, a little more than twice \def\begindoublecolumns{\begingroup \output={\global\setbox\partialpage=\vbox{\unvbox255\bigskip}}\eject \raggedright \output={\splitcolumns} \hsize=\colwidth \vsize=\twicecolheight \let\item\@idxitem} \def\enddoublecolumns{\output={\balancecolumns}\eject \endgroup \pagegoal=\vsize} \def\splitcolumns{\splittopskip=\topskip \splitmaxdepth=\maxdepth \dimen@=\pageheight \advance\dimen@ by-\ht\partialpage \setbox0=\vsplit255 to\dimen@ \setbox2=\vsplit255 to\dimen@ \unvbox255 \penalty\outputpenalty \doublecolumnout} \def\doublecolumnout{\shipout\vbox{% \hsize=\pagewidth \iftitle\else\makeheadline\fi% \unvbox\partialpage \wd0=\colwidth \wd2=\colwidth \hbox to\pagewidth{\box0\hfil\box2} \iftitle\makefootline\global\titlefalse\fi \hsize=\colwidth} \advancepageno \global\def\sectiontitle{\currentsectiontitle} \ifnum\outputpenalty>-\@MM \else\dosupereject\fi} \def\balancecolumns{\setbox0=\vbox{\unvbox255} \dimen@=\ht0 \advance\dimen@ by\topskip \advance\dimen@ by-\baselineskip \divide\dimen@ by2 \splittopskip=\topskip {\vbadness=10000 \loop \global\setbox3=\copy0 \global\setbox1=\vsplit3 to\dimen@ \ifdim\ht3>\dimen@ \global\advance\dimen@ by1pt \repeat} \setbox0=\vbox to\dimen@{\unvbox1} \setbox2=\vbox to\dimen@{\unvbox3} \doublecolumnout} \output={\ibcoutput} \def\leftheadline{\eightbf\folio\hfil\smc\chaptertitle} \def\rightheadline{\smc\sectiontitle\hfil\eightbf\folio} \headline={\ifodd\pageno\rightheadline\else\leftheadline\fi} \def\leftfootline{\eightbf\folio\hfil} \def\rightfootline{\eightbf\hfil\folio} \footline={\ifodd\pageno\rightfootline\else\leftfootline\fi} \def\ibcoutput{\shipout\vbox{% \iftitle\else\makeheadline\fi \pagebody \iftitle\makefootline\global\titlefalse\fi} \advancepageno \global\def\sectiontitle{\currentsectiontitle} \ifnum\outputpenalty>-\@MM \else\dosupereject\fi} \def\Y@{Y } \def\N@{N } \def\makingindex{\immediate\write\sixt@@n{Do you want to make an index? (Type Y or N, follow by :) }\loop \read\m@ne to\ans@ \xdef\next{\def\noexpand\Ans@{\ans@}}% \uppercase\expandafter{\next}% \ifx\Ans@\Y@\badans@false\makeindex\else \ifx\Ans@\N@\badans@false\else \badans@true\fi\fi \ifbadans@\immediate\write\sixt@@n{Type Y or N, follow by : }% \repeat} \makingindex \makeatother % various abbreviations ... form Greg and Henryk \define\a{\operatorname{a}} \define\avg{\operatorname{avg}} \define\card{\operatorname{card}} \define\codim{\operatorname{codim}} \define\comp{\operatorname{comp}} \define\cost{\operatorname{cost}} \define\trace{\operatorname{trace}} \define\diam{\operatorname{diam}} \define\dist{\operatorname{dist}} \define\e{\varepsilon} \define\E{\operatorname{E}} \define\error{\operatorname{error}} \define\non{\operatorname{non}} \define\nor{\operatorname{nor}} \define\sspan{\operatorname{span}} \define\var{\operatorname{var}} \define\ER{\operatorname{ER}} \define\HH{\operatorname{H}} \define\ter{\operatorname{ter}} \define\pre{\operatorname{pre}} \define\prob{\operatorname{prob}} \define\prerel{\operatorname{pre-rel}} \define\rad{\operatorname{rad}} \define\ran{\operatorname{ran}} \define\wor{\operatorname{wor}} \define\rel{\operatorname{rel}} \define\REL{\operatorname{REL}} \define\per{\operatorname{per}} \define\sign{\operatorname{sign}} \define\infp{\operatornamewithlimits{inf\vphantom{p}}} \define\maxp{\operatornamewithlimits{max\vphantom{p}}} \define\Prob{\operatorname{Prob}} \def\reals{\operatorname{I\negthinspace\bold R}} \def\Re{\reals} \def\complexes{\operatorname{\bold C}} \define\ofi{\overline \phi } \define\ofig{\overline {\phi^*}} \define\ofis{\overline {\phi^s}} \define\on{\overline N } \define\norm{||Sf-\phi_n(N_n(f))||} \define\norg{||Sf-\phi_n^*\,(N_n(f))||} \define\nors{||Sf-\phi_n^s\,(N_n(f))||} \define\ong{\overline {N^*}} \define\ongg{\overline {N^{**}}} \define\nnn{N^{\non}} \define\kr{\operatorname{kr}} \define\mr{\operatorname{mr}} \define\gmr{\operatorname{gmr}} \def\bullet{\raise.5ex\hbox{\bf .}} \def\liminf{\mathop{\underline{\text{lim}}}} % various abbreviations ... from Art \def\epsilon{\varepsilon} \def\COMP{\operatorname{\hbox{\smc comp}}} \def\esssup{\operatornamewithlimits{ess\ sup}} \def\FEM{\operatorname{\hbox{\smc fem}}} \def\FEMQ{\operatorname{\hbox{\smc femq}}} \def\interior{\operatorname{int}} \def\lin{\operatorname{lin}} \def\range{\operatorname{ran}} \def\SPLINE{\operatorname{\hbox{\smc spline}}} \def\ter{\operatorname{ter}} \def\volume{\operatorname{vol}} \define\Lstd{\Lambda^{\operatorname{std}}} \define\MFEM{\operatorname{\hbox{\smc mfem}}} %\def\reals{\Bbb R} %\def\complexes{\Bbb C} \def\reals{\operatorname{I\negthinspace\bold R}} %---For remarks and the like \define\demonp#1{\demo\nofrills{#1\usualspace}} %---Start a new section \define\subsection#1{\par\vskip 1pc\noindent{\bf #1} \nopagebreak\medskip\nopagebreak} %---Start and end the Notes & Remarks/Exercises (smaller type, single space) \define\startNRE{\bgroup\eightpoint\parindent=5pt} \define\endNRE{\egroup} \define\NR #1 {\noindent{\bf NR #1: }} % Notes and Remarks \define\Ex #1 {\noindent{\bf E #1: }} % Exercises % angle brackets for inner products \define\<{\langle} \define\>{\rangle} % Script letters and other useful control sequences \define\Sscr{{\Cal S}} \define\Tscr{{\Cal T}} \define\Sn{{\Cal S}_{n,k}} \define\Tn{{\Cal T}_n} \define\Snset{\{\Sn\}_{n=1}^\infty} \define\Tnset{\{\Tn\}_{n=1}^\infty} \define\phisn{\phi^{\operatorname{s}}_{n,k}} % Needed for References \define\BabAz{Babu\u ska and Aziz~[72]} \define\Bakhvalov{Bakhvalov~[77]} \define\Carasso{Carasso and Stone~[75]} \define\Ciarlet{Ciarlet~[78]} \define\CiarletRaviart{Ciarlet and Raviart~[72]} \define\Dunford{Dunford and Schwartz~[63]} \define\Dunn{Dunn~[67]} \define\Emel{Emelyanov and Ilin~[67]} \define\Fried{Fried~[73]} \define\Gehatia{Gehatia and Wiff~[70]} \define\Gelfand{Gel'fand and Vilenkin~[64]} \define\Hadamard{Hadamard~[52]} \define\Hammerlin{H\"ammerlin and Hoffman~[83]} \define\Kuo{Kuo~[75]} \define\Lattes{Lattes and Lions~[69]} \define\Morozov{Morozov~[85]} \define\OdRed{Oden and Reddy~[76]} \define\Pinkus{Pinkus~[85]} \define\Poggio{Poggio, Torre, and Koch~[85]} \define\Pour{Pour-El and Richards~[83]} \define\Rokhlin{Rokhlin~[86]} \define\Shilov{Shilov and Fan Dyk Tin~[67]} \define\Skorohod{Skorohod~[74]} \define\Tikhonov{Tikhonov~[63]} \define\TikhonovArsenin{Tikhonov and Arsenin~[77]} \define\GTOA{GOTA} \define\Twomey{Twomey~[77]} \define\GregVarying{Wasilkowski~[86]} %needed? \define\WWAvg{Wasilkowski and Wo\'zniakowski~[84]} %needed? \define\SecondOrder{Werschulz~[82]} \define\Fredholm{Werschulz~[85]} \define\Indef{Werschulz~[86a]} \define\IllPosed{Werschulz~[86b]} \define\Fembad{Werschulz~[87a]} \define\AprilAGW{Werschulz~[87b]} \define\Resid{Werschulz~[87c]} \define\Zen{Zeni\v sek~[72]}