% m211oh61.tex (prev.: oh59) - Function Spaces and Fourier Approximation 
%
\newcommand{\titlesize}{30pt}  % for title font
\input{m211oh.tex}
%
\newcommand{\SC}{\mbox{$\cal C$}}
%
\begin{titlepage}
\vspace*{-25pt}
\begin{center}\tcr{\rmH Fourier Approximation} \end{center} %\vspace{0pt}
%\LARGE              
\huge

\begin{description}  

\item[\tcr{Problem  C:}] 
\tcg{Given:} 1) A function \tcb{$g \in \SC[a,b]$}, and \\
\hspace*{2.05cm} 2) ``simple'' functions \tcb{$f_{\scr1}, \ldots, f_{\scr k} \in \SC[a,b]$}.

\tcg{Find:} a linear combination
$$
\tcb{f_{\scr0}\; = \; a_{\scr1}f_{\scr1} + \ldots + a_{\scr k}f_{\scr k} }
$$
which \tcr{best approximates} %(or \tcr{best fits}) 
the function \tcb{$g$} \tcr{in the $L^{\scr2}$-norm}; in other words, find  
\tcb{$f_{\scr0}$} as above 
\vspace{-10pt}
such that
$$
\tcb{\int_a^b (f_{\scr0} - g)^2 dx \; \le \; \int_a^b (f - g)^2,}\vspace{-5pt}
$$
for all functions \tcb{$f = a_{\scr1}'f_{\scr1} + \ldots + a_{\scr k}'f_{\scr k}\in 
V:= \lb f_{\scr1}, \ldots, f_{\scr k}\rb$}.   

\item[\tcr{Solution:}] \tcg{Step 1:} Apply the \tcr{Gram-Schmidt Method} to the functions \tcb{$f_{\scr1}, \ldots, f_{\scr k}$} by defining the \tcg{dot product} of two functions \tcb{$f, g \in\SC[a,b]$} to be
\vspace{-10pt}
$$
\tcb{(f\cdot g) = \int_a^b f(x)g(x)dx.}\vspace{-5pt}
$$
This gives us \tcr{orthogonal functions} \tcb{$h_{\scr1}, \ldots, h_{\scr k}$} which are certain linear combinations of \tcb{$f_{\scr1}, \ldots, f_{\scr k}$}.

\tcg{Step 2:} Use the \tcr{projection formula} of \tcg{Theorem 9:}
\vspace{-10pt}
$$
\tcb{f_{\scr0} = P_V(g) = \sum_{\scr i=1}^{\scr k} 
{\textstyle \frac{(g\cdot h_{\scr i})}{(h_{\scr i}\cdot h_{\scr i})}}h_{\scr i}}
%\vspace{-5pt}
$$    
 
\end{description}
\end{titlepage}
\end{document}                          
