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\markboth{\hfil A wavelet Galerkin method \hfil EJDE/Conf/10}
{EJDE/Conf/10 \hfil  J. R. L. de Mattos \& E. P. Lopes  \hfil}

\begin{document}
\setcounter{page}{211}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Fifth Mississippi State Conference on Differential Equations and
Computational Simulations, \newline
Electronic Journal of Differential Equations,
Conference 10, 2003, pp 211--225. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
  A wavelet Galerkin method applied to partial differential
  equations with variable coefficients
%
\thanks{ {\em Mathematics Subject Classifications:} 65T60.
\hfil\break\indent
{\em Key words:} Wavelet, multi-resolution analysis.
\hfil\break\indent
\copyright 2003 Southwest Texas State University. \hfil\break\indent
Published February 28, 2003. } }

\date{}
\author{Jos\'e Roberto Linhares de Mattos \&  Ernesto Prado Lopes}
\maketitle

\begin{abstract}
 We consider the problem $K(x)u_{xx}=u_{t}$ , $0<x<1$, $t\geq 0$,
 where $K(x)$ is bounded below by a positive constant.
 The solution on the boundary $x=0$ is a known function $g$
 and $u_{x}(0,t)=0$. This is an ill-posed problem in the sense
 that a small disturbance on the boundary specification $g$,
 can produce a big alteration on its solution, if it exists.
 We consider the existence of a solution $u(x,\cdot)\in L^{2}(R)$
 and we use a wavelet Galerkin method with the Meyer multi-resolution analysis,
 to filter away the high-frequencies and to obtain well-posed approximating
 problems in the scaling spaces $V_{j}$. We also derive an estimate for
 the difference between the exact solution of the problem and the orthogonal
 projection, onto $V_{j}$, of the solution of the approximating problem
 defined in $V_{j-1}$.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\numberwithin{equation}{section}

\section{Introduction}

In this paper, we consider the following problem, for
$0<\alpha \leq K( x) <+\infty$,
\begin{equation}
\begin{gathered}
K(x)u_{xx}(x,t)=u_{t}(x,t),\quad t\geq 0,\; 0<x <1 \\
u(0,\cdot) =g,\quad u_{x}(0,\cdot)=0
\end{gathered}\label{1}
\end{equation}
We assume  that this problem has a solution $u(x,\cdot
)\in L^{2}({R})$, for $K$ continuous, and we extend $u( x,t)$ and $g$ to $R$
assuming that both vanish for $t<0$.

Problem (\ref{1}) is  ill-posed in the sense
that a small disturbance on the boundary specification $g$, can produce a big
alteration on its solution, if it exists. This means that if the
solution exists, it does not depend continuously on $g$ (see note 1 below).


We consider the Meyer multi-resolution analysis. The advantage of using this
method is that it has good localization in the frequency domain,
since its Fourier transform has compact support. The orthogonal projection
onto Meyer scaling spaces, can be considered as a low pass filter, cutting
off the high frequencies. We get a version of the Gronwall
inequality that we use to obtain an estimate for the frequency of the
solution of the problem (\ref{1}).

>From the variational formulation of the approximating problem on the scaling space $V_{j}$, we get an infinite-dimensional system of second order ordinary
differential equations with variable coefficients. An
estimate obtained for the solution of this evolution problem, is used to
get the stability of the wavelet Galerkin method. Using an
estimate obtained for the difference between the exact solution of the
problem (\ref{1}) and its orthogonal projection onto $V_{j}$, we get an
 estimate for the difference between the exact solution of the problem (\ref{1}) and the orthogonal projection, onto $V_{j}$, of the solution of the approximating problem defined on the scaling space $V_{j-1}$.

Our approach is similar to the one used in \cite{r2} for the sideway heat equation.
The problem considered in \cite{r2} is an inverse problem for the heat equation
with constant coefficient. There the variational formulation,
on the scaling space $V_{j}$, of the approximating problem, produces an
infinite-dimensional system of second order ordinary differential
equations with constant coefficients, for which the solution is known.
Stability and convergence of the method follows from form of this solution.

In section 2, we construct the Meyer multi-resolution analysis. In
section 3, we get the estimates of the numerical stability and the
convergence of the wavelet Galerkin method.

For a function
$h \in L^{1}({R}) \bigcap L^{2}({R})$ its Fourier Transform is given by
$\widehat{h} ({\xi}):=\int_{\mathbb{R}} h( x) e^{-ix\xi } dx$.

\section{Meyer multi-resolution analysis}

To construct a wavelet basis from a mother wavelet,
we need an structure in $L^{2}(\mathbb{R})$ which allows us to decompose
$L^{2}( \mathbb{R}) $ in a direct sum of
mutually orthogonal spaces.

\paragraph{Definition} A \emph{multi-resolution analysis} is a sequence
of closed subspaces $V_{j}$ in $L^2(\mathbb{R})$, called
\emph{scaling spaces}, satisfying:
\begin{itemize}
\item[(M1)] $V_{j}\subseteq V_{j-1}$ for all $j\in \mathbb{Z}$

\item[(M2)] $\bigcup_{j\in \mathbb{Z}}V_{j}$ is dense in $L^{2}(\mathbb{R})$

\item[(M3)] $\bigcap_{j\in \mathbb{Z}}V_{j}=\{ 0\}$

\item[(M4)] $f\in V_{j}$ if and only if $f( 2^{j}\cdot ) \in V_{0}$

\item[(M5)] $f\in V_{0}$ if and only if $f( \cdot -k) \in V_{0}$ for all
$k\in \mathbb{Z}$

\item[(M6)] There exists $\phi \in V_{0}$ such that
$\{ \phi_{\stackrel{}{0,k}}:k\in \mathbb{Z}\}$ is an orthonormal basis in
$V_{0}$, where
$\phi_{j,k}(x)=2^{-j/2}\phi (2^{-j}x-k)$
for all  $j,k\in \mathbb{Z}$.
The function $\phi $ is called the \emph{scaling function} of the
multi-resolution analysis.
\end{itemize}

\paragraph{Remarks} 1) M4 and M6 imply
$\{ \phi_{j,k}:k\in \mathbb{Z}\}$ being an orthonormal basis for the
space $V_{j}$.
\\
2) Let $\phi \in L^{2}(\mathbb{R})$ and
$V_{j}=\overline{{\rm span}\{ \phi_{jk}\}}_{k\in \mathbb{Z}}$
 where  $\phi_{jk}( t):=2^{-j/2}\phi ( 2^{-j}t-k)$  and
$j\in \mathbb{Z}$.  Thus,
$\ V_{0}=\overline{{\rm span}\{ \phi ( \cdot -k)\}}_{k\in \mathbb{Z}}$.
 We have that $ V_{j}$ satisfy M1 if only if $ \phi \in V_{-1}$, that is,
 if only if there exists a $2\pi$-periodic square integrable function
 $m_{0}$, such that
\[
\widehat{\phi }( \xi ) =m_{0}( \frac{\xi }{2})
\widehat{\phi }( \frac{\xi }{2}).
\]

The Meyer multi-resolution analysis is constructed in the following
way:
Let $\varphi $ be the scaling function defined by its Fourier transform by
\[
\widehat{\varphi }( \xi ) =\begin{cases}
1, &\mbox{if } | \xi| \leq 2\pi/3 \\
\cos [ \frac{\pi }{2}\nu (\frac{3}{2\pi }| \xi | -1)] &
\mbox{if } 2\pi/3 \leq | \xi | \leq 4\pi/3 \\
0, & \mbox{otherwise,}
\end{cases}
\]
where $\nu $ is a differentiable function satisfying
\begin{gather}
\nu ( x) =\begin{cases}
0  &\mbox{if }  x\leq 0\\
1  &\mbox{if }  x\geq 1
\end{cases} \label{3} \\
\nu ( x) +\nu ( 1-x) =1  \label{4}
\end{gather}
>From (\ref{4}), it follows  that  $\sum_{k\in \mathbb{Z}}| \widehat{%
\varphi }( \xi +2k\pi ) | ^{2}=1$,
which  is  equivalent  to the orthonormality  of  $\varphi ( \cdot -k)$,
$k\in \mathbb{Z}$. Then M6 is satisfied. Here $m_{0}$ can be constructed
on $ [ 0,2\pi]$, from $\widehat{\varphi }$, by
\[
m_{0}( \xi ) =\sum_{l\in \mathbb{Z}}\widehat{\varphi }( 2( \xi +2\pi l) )
\]
This function is $2\pi $-periodic, square integrable, and, for
$\xi \in [ 0,2\pi]$,
\[
m_{0}( \frac{\xi }{2}) \widehat{\varphi }( \frac{\xi }{2})
= \sum_{l\in \mathbb{Z}}\widehat{\varphi }
( \xi +4\pi l) \widehat{\varphi }( \frac{\xi }{2})
=  \widehat{\varphi }( \xi ) \widehat{\varphi }( \frac{\xi }{2})
= \widehat{\varphi }( \xi )
\]
The second equality above follows from
 $$
\widehat{\varphi }
( \xi +4\pi l) \widehat{\varphi }( \frac{\xi }{2})=0, \quad \forall l\neq 0
 $$
 and the third equality follows from
$\widehat{\varphi }( \xi /2) =1$ for all
$\xi \in \mathop{\rm supp} \widehat{\varphi }$. Then M1 is satisfied and
the other conditions of the definition can also be proved. The
associated mother wavelet is given by (see \cite{d1})
\begin{eqnarray*}
\widehat{\psi }( \xi ) & = & e^{i \xi /2} \overline{m_{0}(
\xi /2 + \pi ) } \widehat{\varphi }( \xi /2)  \\
& = & e^{i \xi /2}\sum_{l\in \mathbb{Z}}\widehat{\varphi }%
( \xi +2\pi ( 2l+1) ) \widehat{\varphi }( \xi/2) \\
& = & e^{i \xi /2} [ \widehat{\varphi }( \xi +2\pi
) +\widehat{\varphi }( \xi -2\pi ) ] \widehat{\varphi
}( \xi /2)
\end{eqnarray*}
or equivalently,
\[
\widehat{\psi }( \xi ) =\begin{cases}
e^{i\xi /2}\sin [ \frac{\pi }{2}\nu ( \frac{3}{2\pi }| \xi | -1) ],
&\mbox{if } \frac{2\pi }{3}\leq | \xi | \leq \frac{4\pi }{3} \\
e^{i\xi/2}\cos [ \frac{\pi }{2}\nu ( \frac{3}{4\pi }| \xi | -1) ],
&\mbox{if } \frac{4\pi }{3}\leq | \xi | \leq \frac{8\pi }{3}\\
0, &\mbox{otherwise}.
\end{cases}
\]
The function $\psi$ is the Meyer wavelet.

Now, we consider the Meyer multi-resolution analysis. We have
\begin{align*}
\widehat{\psi_{jk}}( \xi ) & =  \int_{\mathbb{R}}\psi_{jk}( x) e^{-ix\xi } dx\\
&=  \int_{\mathbb{R}}2^{-\frac{j}{2}}\psi ( 2^{-j}x-k)e^{-ix\xi } dx \\
& =  \int_{\mathbb{R}}2^{j/2}\psi ( y-k)e^{-i2^{j}y\xi } dy\\
&= 2^{j/2}\int_{\mathbb{R}}\psi ( t)e^{-i2^{j}(t+k)\xi } dt \\
& =  2^{j/2}\int_{\mathbb{R}}\psi ( t)e^{-i2^{j}t\xi -i2^{j}k\xi } dt
 =  2^{j/2}e^{-i2^{j}k\xi }\widehat{\psi }( 2^{j}\xi )
\end{align*}
Since $\mathop{\rm supp}( \widehat{\psi }) =\left\{ \xi :\frac{2}{3}%
\pi \leq | \xi | \leq \frac{8}{3}\pi \right\} $ we have that
\begin{equation}
\mathop{\rm supp}( \widehat{\psi_{jk}}) =\big\{ \xi  ; \frac{2}{3}\pi
2^{-j}\leq | \xi | \leq \frac{8}{3}\pi 2^{-j}\big\}\quad
\forall k\in \mathbb{Z}  \label{5}
\end{equation}
Furthermore,
\begin{equation}
\mathop{\rm supp}( \widehat{\varphi_{jk}}) =\big\{ \xi  ; | \xi
| \leq \frac{4}{3}\pi 2^{-j}\big\} \quad \forall k\in \mathbb{Z}
\label{6}
\end{equation}
Now we consider the orthogonal projection onto $V_{j}$,
$P_{j}:L^{2}({R}) \to V_{j}$,
\[
P_{j}f(t)=\sum_{k\in \mathbb{Z}}\langle f,\varphi_{jk}\rangle \varphi
_{jk}(t)
\]
The hypothesis M1 and M2 imply that $\lim_{j\to -\infty }P_{j}f=f$,
for all $f\in L^{2}({R})$. This means that from a representation
of $f$ in a given scale, we can get $f$ by adding details which are given
at higher frequencies.
>From (\ref{6}), we see that $P_{j}$ filters away the frequencies higher
than $\frac{4}{3}\pi 2^{-j}$ (low pass filter).

We have, for all $ f\in L^{2}({R}),$%
\begin{eqnarray*}
f & = & P_{j}f-P_{j}f+f \\
& = & P_{j}f+( I-P_{j}) f \\
& = & \sum_{k\in \mathbb{Z}}\langle f,\varphi_{jk}\rangle \varphi
_{jk}+\sum_{l\leq j}\sum_{k\in \mathbb{Z}}\langle f,\psi
_{lk}\rangle \psi_{lk}
\end{eqnarray*}
This implies
\begin{equation}
\widehat{P_{j}f}( \xi ) =\widehat{f}( \xi ) \quad \mbox{for }
| \xi | \leq \frac{2}{3}\pi 2^{-j}  \label{7}
\end{equation}
since, by (\ref{5}),  $\widehat{\psi }_{lk}( \xi ) =0$
for all $l\leq j$ and $| \xi | \leq \frac{2}{3}\pi 2^{-j}$.

Considering  the corresponding orthogonal projections in the frequency
space, $\widehat{P_{j}}:L^{2}({R}) \to
\widehat{V_{j}}=\overline{\mathop{\rm span}\{ \widehat{\varphi_{jk}}\} }_{k\in
\mathbb{Z}}$,
\[
\widehat{P_{j}}f=\sum_{k\in \mathbb{Z}}\frac{1}{2\pi }\langle f,\widehat{%
\varphi_{jk}}\rangle \widehat{\varphi_{jk}}
\]
we have
\[
\widehat{P_{j}}\widehat{f}=\sum_{k\in \mathbb{Z}}\frac{1}{2\pi }\langle
\widehat{f},\widehat{\varphi_{jk}}\rangle \widehat{\varphi_{jk}}=\sum_{k\in \mathbb{Z}} \langle f,\varphi_{jk}\rangle \widehat{%
\varphi_{jk}}=\widehat{P_{j}f}
\]
Then (\ref{7}) implies that
\begin{equation}
\begin{aligned}
\| ( I-P_{j}) f\| & =  \frac{1}{\sqrt{2\pi }}\| [ ( I-P_{j}) f] ^\wedge \|
=  \frac{1}{\sqrt{2\pi }}\| ( I-\widehat{P_{j}}) \widehat{f}\|  \\
& =  \frac{1}{\sqrt{2\pi }}\| ( I-\widehat{P_{j}}) \chi_{j}\widehat{f}\|
\leq \| \chi_{j} \widehat{f}\|
\end{aligned}  \label{8}
\end{equation}
where $\chi_{j}$ is the characteristic function in $(-\infty ,-\frac{2}{3}
\pi 2^{-j}]\cup [\frac{2}{3}\pi 2^{-j},+\infty )$.

\section{Results of Stability and Convergence}

Hereafter, the multi-resolution analysis considered corresponds to the Meyer
multi-resolution analysis with scaling function $\varphi $.
The next lemma is a version of the Gronwall
inequality.

\begin{lemma} \label{lm1}
Let $u$ and $v$ be positive continuous functions, $x\geq a$ and $c > 0$. If
$u(x)\leq c +\int_{a}^{x}\int_{a}^{s}v(\tau )u(\tau )\,d\tau  ds$
then
\[
u(x)\leq c  \exp \Big( \int_{a}^{x}\int_{a}^{s}v(\tau )\,d\tau ds\Big)\,.
\]
\end{lemma}

\paragraph{Proof.}
Let $w(x)=c +\int_{a}^{x}\int_{a}^{s}v(\tau )u(\tau )\,d\tau  ds$.
Then $w'(x)=\int_{a}^{x}v(\tau ) u(\tau )\,d\tau $. Therefore,
\[
w''(x)=v(x) u(x)\leq v(x)w(x)
\quad\mbox{and}\quad \frac{w''(x)}{w(x)}\leq v(x)
\]
Now
$$
\frac{w''(x)}{w(x)}=(\frac{w'}{w})'(x)+(\frac {w'(x)}{w(x)})^{2}
$$
Thus
$\big(\frac {w'}{w}\big)'(x)\leq v(x)$  which implies
\[
\frac{w'(x)}{w(x)}\leq \int_{a}^{x}v(\tau )\,d\tau \quad\mbox{and}\quad
 ( \ln  w(x)) '\leq \int_{a}^{x}v(\tau )\,d\tau;
\]
Therefore,
\[
\ln  w(x)-\ln  w(a)\leq \int_{a}^{x}\int_{a}^{s}v(\tau ) \,d\tau  ds
\]
Since $w(a)=c $,
$\ln w(x)-\ln \ c \leq \int_{a}^{x}\int_{a}^{s}v(\tau )\,d\tau ds$,
which implies
\[
\ln \frac{w(x)}{c}\leq \int_{a}^{x}\int_{a}^{s}v(\tau ) \,d\tau ds
\quad\mbox{and}\quad
w(x)\leq c \exp \Big(\int_{a}^{x}\int_{a}^{s}v(\tau ) \,d\tau  ds\Big).
\]
Since, by hypothesis, $u(x)\leq w(x)$, we have
\[
u(x)\leq c \exp \Big( \int_{a}^{x}\int_{a}^{s}v(\tau )\,d\tau
 ds\Big)
\]
which completes the proof.
{}\hfill$\square$\smallskip

Applying the Fourier Transform with respect to  time in
Problem (\ref{1}), we obtain the following problem in the frequency space:
\begin{gather*}
\widehat{u}_{xx}( x,\xi ) =\frac{i\xi }{K(x)}  \widehat{u}(x,\xi ),
\quad  0<x<1 , \; \xi \in {R}  \\
\widehat{u}( 0,\xi ) =\widehat{g}(\xi ), \quad \widehat{u}_{x}(0,\cdot )=0
\end{gather*}
whose solution satisfies
\[
\widehat{u}( x,\xi ) =\widehat{g}(\xi)+\int_{0}^{x}
\int_{0}^{s}\frac{i\xi }{K(\tau )}\widehat{u}( \tau ,\xi ) \,d\tau  ds
\]
Then, from lemma \ref{lm1}, for $\widehat{g}(\xi )\neq 0$, we have
\begin{equation}
| \widehat{u}( x,\xi ) | \leq | \widehat{g}(
\xi ) |  \exp \Big[ | \xi |
 \int_{0}^{x}\int_{0}^{s}\frac{1}{K(\tau )}\,d\tau  ds\Big]  \label{9}
\end{equation}
The next lemma corresponds to proposition 3.1 in \cite{r2}, when $K(x)$ is constant.

\begin{lemma} \label{lm2}
The operator $D_{j}(x)$ defined by
$$[ (
D_{j})_{lk}( x) ]_{l\in \mathbb{Z},\ k\in \mathbb{Z}} =
\Big[ \frac{1}{K( x) }\langle \varphi_{jl}',\varphi_{jk}
\rangle \Big]_{l\in \mathbb{Z}, k\in \mathbb{Z}}
$$
satisfies the following three conditions:
1) $( D_{j})_{lk}(x)=-( D_{j})_{kl}(x)$\
\newline
2) $( D_{j})_{lk}(x)=( D_{j})_{( l-k)0}( x)$. Hence,
$( D_{j})_{lk}( x)$ are equal along diagonals.
\newline
3) $\| D_{j}(x)\| \leq \frac{\pi 2^{-j}}{K(x)}$
\end{lemma}

\paragraph{Proof.}
This proof follows quite closely the proof in \cite{r2} for
$D_{j}(x)$ independent of $x$. \newline
1) As we already know,
$\widehat{\varphi_{j0}}( \xi ) =2^{j/2}\widehat{\varphi }( 2^{j}\xi )$,
 $\widehat{\varphi_{jk}}( \xi ) =2^{j/2}e^{-ik\xi 2^{j}}\widehat{\varphi }
 ( 2^{j}\xi )= e^{-ik\xi 2^{j}}\widehat{\varphi_{j0}}( \xi) $,
and the Fourier transform of the scaling function $\varphi $ is even.
Then $ \widehat{\varphi_{j0}}( \xi ) =\widehat{\varphi_{j0}}( -\xi )$ and
\begin{eqnarray*}
( D_{j})_{lk}(x) & = & \frac{1}{K(x)}\langle \varphi
_{jl}',\varphi_{jk}\rangle
\ \ = \ \ \frac{1}{K(x)}\frac{1}{2\pi }\langle \widehat{\varphi
_{jl}'},\widehat{\varphi_{jk}}\rangle  \\
& = & \frac{1}{K(x)}\frac{1}{2\pi }\int_{\mathbb{R}}\widehat{\varphi
_{jl}'}( \xi ) \overline{\widehat{\varphi_{jk}}}( \xi ) \,d\xi  \\
& = & \frac{i}{K(x)}\frac{1}{2\pi }\int_{\mathbb{R}}\xi \widehat{\varphi
_{jl}}( \xi ) \overline{\widehat{\varphi_{jk}}}( \xi) \,d\xi  \\
& = & \frac{i}{K(x)}\frac{1}{2\pi }\int_{\mathbb{R}}\xi e^{-i(l-k)\xi
2^{j}}| \widehat{\varphi_{j0}}( \xi ) | ^{2}d\xi
\end{eqnarray*}
Then
\begin{eqnarray*}
( D_{j})_{lk}(x) & = & \frac{i}{K(x)}\frac{1}{2\pi }\int_{\mathbb{R}}( -\omega )
e^{-i(k-l)\omega 2^{j}}| \widehat{\varphi_{j0}}( -\omega
) | ^{2}d\omega  \\
& = & -\frac{i}{K(x)}\frac{1}{2\pi }\int_{\mathbb{R}}\omega
e^{-i(k-l)\omega 2^{j}}| \widehat{\varphi_{j0}}( \omega )
| ^{2}d\omega  \\
& = & -\frac{i}{K(x)}\frac{1}{2\pi }\int_{\mathbb{R}}\omega  \widehat{
\varphi_{jk}}( \omega ) \overline{\widehat{\varphi_{jl}}}( \omega ) \,d\omega  \\
& = & -\frac{1}{K(x)}\frac{1}{2\pi }\int_{\mathbb{R}} \widehat{\varphi
_{jk}'}( \omega ) \overline{\widehat{\varphi_{jl}}}( \omega ) \,d\omega .
\end{eqnarray*}
Thus
\[
( D_{j})_{lk}(x) = -\frac{1}{K(x)}\frac{1}{2\pi }\langle \widehat{\varphi
_{jk}'},\widehat{\varphi_{jl}}\rangle  \\
=-\frac{1}{K(x)}\langle \varphi_{jk}',\varphi
_{jl}\rangle = -( D_{j})_{kl}(x)
\]
2) As proved above
\[
( D_{j})_{lk}(x) = \frac{i}{K(x)}\frac{1}{2\pi }
\int_{\mathbb{R}}\xi e^{-i(l-k)\xi 2^{j}}| \widehat{\varphi_{j0}}( \xi)
| ^{2}d\xi = ( D_{j})_{( l-k) 0}(x)
\]
Then $( D_{j})_{lk}( x)$ is equal along
diagonals. \newline
3) We have
\[
\| D_{j}(x)\| =\big\| \frac{1}{K(x)}B_{j}\big\| =\frac{1}{K(x)}\| B_{j}\|
\]
where $( B_{j})_{lk}=\langle \varphi_{jl}',\varphi_{jk}\rangle $.
>From results 1) and 2), we have
$( B_{j})_{lk}=-(B_{j})_{kl}$,
 $( B_{j})_{lk}=\frac{1}{2\pi }\int_{\mathbb{R}}\xi e^{-i( l-k) \xi  2^{j}}|
\widehat{\varphi_{j0}}( \xi ) | ^{2}d\xi
=(B_{j})_{(l-k) 0}$ and $( B_{j})_{lk}$ is constant
along diagonals. We will show that $\ \| B_{j}\| \leq \pi 2^{-j}$.
Thus, we will have
\[
\| D_{j}(x)\| \leq \frac{\pi }{K(x)}2^{-j}
\]
For \ $| t| \leq \pi 2^{-j}$,
\begin{align*}
\Gamma_{j}( t) =& i2^{-j}\big[ ( t-2^{-j+1}\pi )
| \widehat{\varphi_{j0}}( t-2^{-j+1}\pi ) |
^{2}+t| \widehat{\varphi_{j0}}( t) | ^{2}\\
&+( t+2^{-j+1}\pi ) | \widehat{\varphi_{j0}}( t+2^{-j+1}\pi ) | ^{2}\Big]
\end{align*}
Extend $\Gamma_{j}$ periodically to $\mathbb{R}$ and expand it in Fourier
series as
\[
\Gamma_{j}( t) =\sum_{k\in \mathbb{Z}}\gamma_{k}e^{ikt2^{j}}
\]
We have $\gamma_{k}=b_{k}$ for all $k$, where $b_{k}$ is the element in
diagonal $k$ of $B_{j}$. In fact, since $\widehat{\varphi_{j0}}(
t) =0$ \ for $| t| \geq \frac{4}{3}\pi 2^{-j}$, it follows that
\begin{eqnarray*}
\gamma_{k} & = & \frac{1}{2^{-j+1}\pi }\int_{-\pi 2^{-j}}^{\pi
2^{-j}}\Gamma_{j}( t) e^{-ikt2^{j}}dt  \\
& = & \frac{i}{2\pi }\int_{-\pi 2^{-j}}^{\pi 2^{-j}}(
t-2^{-j+1}\pi ) | \widehat{\varphi_{j0}}( t-2^{-j+1}\pi
) | ^{2}e^{-ikt2^{j}}dt  \\
&  & +\frac{i}{2\pi }\int_{-\pi 2^{-j}}^{\pi 2^{-j}}t|
\widehat{\varphi_{j0}}( t) | ^{2}e^{-ikt2^{j}}dt \\
&  & +\frac{i}{2\pi }\int_{-\pi 2^{-j}}^{\pi 2^{-j}}(
t+2^{-j+1}\pi ) | \widehat{\varphi_{j0}}( t+2^{-j+1}\pi
) | ^{2}e^{-ikt2^{j}}dt
\end{eqnarray*}
Making a change of variable, we obtain:
\begin{eqnarray*}
\gamma_{k} & = & \frac{i}{2\pi }\int_{-3\pi 2^{-j}}^{-\pi
2^{-j}}t| \widehat{\varphi_{j0}}( t) |
^{2}e^{-ikt2^{j}}dt+\frac{i}{2\pi }\int_{-\pi 2^{-j}}^{\pi
2^{-j}}t| \widehat{\varphi_{j0}}( t) |
^{2}e^{-ikt2^{j}}dt  \\
&  & +\frac{i}{2\pi }\int_{\pi 2^{-j}}^{3\pi 2^{-j}}t|
\widehat{\varphi_{j0}}( t) | ^{2}e^{-ikt2^{j}}dt
 \\
& = & \frac{i}{2\pi }\int_{-3\pi 2^{-j}}^{3\pi 2^{-j}}t|
\widehat{\varphi_{j0}}( t) | ^{2}e^{-ikt2^{j}}dt
 \\
& = & \frac{i}{2\pi }\int_{\mathbb{R}}t| \widehat{\varphi_{j0}}( t)
| ^{2}e^{-ikt2^{j}}dt
 = b_{k}
\end{eqnarray*}
Now, $\| B_{j}\| =\sup_{\| f\| =1}\|B_{j}f\| $ where
 $\| f\| ^{2}=\sum_{k\in \mathbb{Z}}| f_{k}| ^{2}$.
Let  $F( t) =\sum_{k\in \mathbb{Z}}f_{k}e^{ikt2^{j}}$
and define $W( t) =\Gamma_{j}(t) F( t)$.
We have
\[
W( t) =\sum_{k\in \mathbb{Z}}\omega
_{k}e^{ikt2^{j}}\quad \mbox{and}\quad
\omega_{k}=\sum_{l\in \mathbb{Z}}b_{k-l} f_{l}=( B_{j}f)_{k}
\]
Hence
\begin{align*}
\| \omega \| ^{2} & =  \sum_{k\in \mathbb{Z}}| \omega_{k}| ^{2}
 =  \frac{1}{2\pi 2^{-j}}\int_{-\pi 2^{-j}}^{\pi 2^{-j}}|
W( t) | ^{2}dt  \\
& =  \frac{1}{2\pi 2^{-j}}\int_{-\pi 2^{-j}}^{\pi 2^{-j}}|
\Gamma_{j}( t) F( t) | ^{2}dt\\
& \leq  \sup_{| t| \leq \pi 2^{-j}}| \Gamma
_{j}( t) | ^{2}\frac{1}{2\pi 2^{-j}}\int_{-\pi
2^{-j}}^{\pi 2^{-j}}| F( t) | ^{2}dt\\
& =  \sup_{| t| \leq \pi 2^{-j}}| \Gamma_{j}(
t) | ^{2}\| f\| ^{2}
\end{align*}
Then
\[
\| B_{j}\| \leq \sup_{| t| \leq \pi
2^{-j}}| \Gamma_{j}( t) | ^{2}
\]
On the other hand, $\Gamma_{j}$ is an odd function. Hence
\[
\sup_{| t| \leq \pi 2^{-j}}| \Gamma_{j}(
t) | ^{2}=\sup_{0\leq t\leq \pi 2^{-j}}| \Gamma
_{j}( t) | ^{2}
\]
But, for \ $0\leq t\leq \pi 2^{-j}$, we have $t+\pi 2^{-j+1}\geq \pi
2^{-j+1}$  and  $t-\pi 2^{-j+1}\leq 0$. Hence
\[
\widehat{\varphi_{j0}}( t+\pi 2^{-j+1}) =0 \quad\mbox{and}\quad
( t-\pi 2^{-j+1}) | \widehat{\varphi_{j0}}(
t-\pi 2^{-j+1}) | ^{2}\leq 0
\]
for $t\in [0,\pi 2^{-j}]$. Thus
\begin{eqnarray*}
\sup_{0\leq t\leq \pi 2^{-j}}| \Gamma_{j}( t)
| ^{2} & \leq & \pi 2^{-j+1}\sup_{0\leq t\leq \pi
2^{-j}}t| \widehat{\varphi_{j0}}( t) | ^{2}
 \\
& = & \pi 2^{-j+1}\sup_{0\leq t\leq \pi 2^{-j}}(t2^{j})|
\widehat{\varphi }( 2^{j}t) | ^{2}  \\
& = & \pi 2^{-j+1}\sup_{0\leq s\leq \pi }s| \widehat{\varphi }%
( s) | ^{2}
\end{eqnarray*}
By definition of $\widehat{\varphi }$ we have
$| \widehat{\varphi }( s) | ^{2}\leq \frac{1}{2\pi }$  and therefore
$s| \widehat{\varphi }( s) | ^{2}\leq \frac{\pi }{2\pi}=\frac{1}{2}$
 for  $0\leq s\leq \pi $. Then
\[
\sup_{0\leq t\leq \pi 2^{-j}}| \Gamma_{j}( t)
| ^{2}  \leq  \sup_{0\leq s\leq \pi }s| \widehat{\varphi
}( s) | ^{2}
 \leq  \frac{\pi 2^{-j+1}}{2} =  \pi 2^{-j}
\]
Thus
\[
\| D_{j}( x) \| =\frac{1}{K( x) }\|
B_{j}\| \leq \frac{1}{K( x) }\sup_{| t|
\leq \pi 2^{-j}}| \Gamma_{j}( t) | ^{2}\leq \frac{\pi
2^{-j} }{K( x) }
\]
which completes the proof of lemma \ref{lm2}. \hfill$\square$\smallskip

Let us now consider the following approximating problem in $V_{j}$,
where the projection in the first equation of (\ref{10})
is due to the fact that we can have $\varphi \in V_{j}$
with $\varphi'\notin V_{j}$ (see note 2 below),
\begin{equation}
\begin{gathered}
K(x)u_{xx}(x,t)=P_{j}u_{t}(x,t),\quad t\geq 0,\; 0<x<1  \\
u(0,\cdot) =P_{j}g  \quad u_{x}(0,\cdot)=0  \\
u(x,t) \in V_{j}
\end{gathered} \label{10}
\end{equation}
Its variational formulation is
\begin{gather*}
\langle K( x) u_{xx}-u_{t} , \varphi_{jk}\rangle =0  \\
\langle u(0,\cdot), \varphi_{jk}\rangle = \langle P_{j}g , \varphi_{jk}\rangle,
\quad \langle u_{x}(0,\cdot), \varphi_{jk}\rangle
= \langle 0,\varphi_{jk}\rangle, \quad k\in \mathbb{Z}
\end{gather*}
where $\varphi_{jk}$ is the orthonormal basis of $V_{j}$ given by the
scaling function $\varphi$.
Consider $u_{j}$ a solution  of  the  approximating  problem (\ref{10}),
given  by  $u_{j}( x,t) =\sum_{l\in \mathbb{Z}}w_{l}( x)\varphi_{jl}( t)$.
Then, we have
$( u_{j})_{t}( x,t) =\sum_{l\in\mathbb{Z}}w_{l}( x)\varphi_{jl}'( t)$ and
$( u_{j})_{xx}( x,t) =\sum_{l\in \mathbb{Z}}w_{l}''( x) \varphi_{jl}(t)$.
Therefore,
\[
K( x) ( u_{j})_{xx}( x,t)-( u_{j})_{t}( x,t) =K( x)
\sum_{l\in \mathbb{Z}}w_{l}''( x)  \varphi
_{jl}( t) -\sum_{l\in \mathbb{Z}}w_{l}( x)
 \varphi_{jl}'( t)
\]
Hence
\begin{align*}
&\langle K( x) ( u_{j})_{xx}-( u_{j})
_{t},\varphi_{jk}\rangle =0 \\
&\Longleftrightarrow \langle
\sum_{l\in \mathbb{Z}}K( x)  w_{l}'' \varphi_{jl}-\sum_{l\in \mathbb{Z}}w_{l} \varphi_{jl}',\varphi_{jk} \rangle =0 \\
&\Longleftrightarrow \sum_{l\in \mathbb{Z}}K( x)
 w_{l}'' \langle \varphi_{jl},\varphi
_{jk}\rangle =\sum_{l\in \mathbb{Z}}w_{l} \langle \varphi
_{jl}',\varphi_{jk}\rangle  \\
&\Longleftrightarrow K( x)  w_{k}''=\sum_{l\in \mathbb{Z}}w_{l}
\langle \varphi_{jl}',\varphi_{jk}\rangle \quad k\in \mathbb{Z}.
\end{align*}
Therefore,
\[
\frac{d^{2}}{dx^{2}} w_{k}=\sum_{l\in \mathbb{Z}
}w_{l} \frac{1}{K( x) } \langle \varphi_{jl}',\varphi_{jk}\rangle
\quad\mbox{and}\quad \frac{d^{2}}{dx^{2}} w_{k}
=\sum_{l\in \mathbb{Z}}w_{l}  ( D_{j})_{lk}( x)
\]
where, as defined before, $ ( D_{j})_{lk}( x) =\frac{1}{K(
x) } \langle \varphi_{jl}',\varphi
_{jk}\rangle $. Thus, we get an infinite-dimensional system of ordinary
differential equations
\begin{equation}
\begin{gathered}
\frac{d^{2}}{dx^{2}} w=-D_{j}( x) w \\
w( 0) =\gamma, \quad w'(0)=0
\end{gathered}\label{11}
\end{equation}
where $\gamma $ is given by
\[
P_{j}g=\sum_{z\in \mathbb{Z}}\gamma_{z}\varphi_{jz}=\sum_{z\in \mathbb{Z}%
}\langle g,\varphi_{jz}\rangle \varphi_{jz}
\]

\begin{lemma} \label{lm3}
If $w$ is a solution of the evolution problem of second order (\ref{11}),
then
\[
\| w( x) \| \leq \| \gamma\| \exp \Big( 2^{-j}\pi \int_{0}^{x}\int_{0}^{s}
\frac{1}{K( \tau ) }\,d\tau \, ds\Big)
\]
\end{lemma}

\paragraph{Proof} Since
$w( x) =\gamma +\int_{0}^{x}\int_{0}^{s}( -D_{j} )( \tau )
 w( \tau ) \,d\tau  ds$,
\[
\| w( x) \| \leq \| \gamma
\| +\int_{0}^{x}\int_{0}^{s}\| D_{j}(
\tau ) \| \| w( \tau ) \| \,d\tau\, ds
\]
By lemma \ref{lm2} this implies
\[
\| w( x) \| \leq \| \gamma \| +
\int_{0}^{x}\int_{0}^{s} \frac{2^{-j}\pi}{K(x)} \| w  ( \tau )\|
\,d\tau \, ds.
\]
Then by lemma \ref{lm1} we have
\[
\| w( x) \| \leq \| \gamma\| \exp \Big( 2^{-j}\pi \int_{0}^{x}\int_{0}^{s} %
\frac{1}{K( \tau ) }\,d\tau  ds\Big)
\]
which completes the proof. \hfill$\square$

\begin{theorem}[Stability of the wavelet Galerkin method] \label{thm4}
Let $u_{j}$ and $v_{j}$ be solutions in $V_{j}$ of the approximating problems
 (\ref{10}) for the boundary specifications $g$ and $\widetilde{g}$,
respectively. If $\| g-\widetilde{g}\| \leq \epsilon $ then
\[
\| u_{j}( x,\cdot ) -v_{j}( x,\cdot )
\| \leq \epsilon  \exp  \big( \frac{2^{-j-1}\pi }{\alpha } x^2\big)
\]
where $\alpha$ satisfies $0<\alpha\leq K(x)<+\infty$ as in the definition of
the problem (\ref{1}).
For $j$ such that $2^{-j}\leq \frac{2\alpha }{\pi } \log  \epsilon ^{-1}$ we
have
\[
\| u_{j}( x,\cdot ) -v_{j}( x,\cdot )\| \leq  \epsilon ^{1-x^{2}}
\]
\end{theorem}

\paragraph{Proof.}
$u_{j}( x,t) =\sum_{l\in \mathbb{Z}} w_{l}(x) \varphi_{jl}(t)$,
$v_{j}(x,t) =\sum_{l\in \mathbb{Z}} \widetilde{w}_{l}(x) \varphi_{jl}(t)$
where $w$ and $\widetilde{w}$ are solutions of the
Galerkin problem (\ref{11}) with conditions $w(0)=\gamma $
and  $\widetilde{w}(0)=\widetilde{\gamma }$, respectively. So, by lemma \ref{lm3}
and linearity of (\ref{11}) we have
\begin{eqnarray*}
\| u_{j}( x,\cdot ) -v_{j}(x,\cdot )\| & = & \| w(x)-\widetilde{w}(x)\|   \\
& \leq & \| \gamma -\widetilde{\gamma }\|  \exp ( 2^{-j}\pi
\int_{0}^{x}\int_{0}^{s}\frac{1}{K(\tau )}\,d\tau
 ds)   \\
& \leq & \epsilon \exp ( 2^{-j}\pi
\int_{0}^{x}\int_{0}^{s}\frac{1}{\alpha }\,d\tau
 ds)   \\
& = & \epsilon \exp ( 2^{-j-1}\frac{\pi }{\alpha } x^{2})
\end{eqnarray*}
For $j=j( \epsilon )$ such that
$2^{-j}\leq \frac{2\alpha }{\pi}\log  \epsilon ^{-1}$, we have
\[
\| u_{j}( x,\cdot ) -v_{j}(x,\cdot )\|
\leq  \epsilon \exp  ( x^{2}\log  \epsilon ^{-1})
 =  \epsilon ^{1-x^{2}}
\]
which completes the proof. \hfill$\square$

Now, we are interested in the solutions $u(x,\cdot
)\in L^{2}({R})$ of problem (\ref{1}), for the functions $ g\in L^{2}({R})$
such that $\widehat{g}( \cdot )\exp(|\cdot | /(2\alpha)) \in L^{2}({R})$,
 where $\widehat{g}$ is the Fourier Transform of $g$.
The Inverse Fourier Transform of $\exp ( -\frac{\xi ^{2}+| \xi | }{2\alpha })$,
for instance, satisfies this condition. Define
\begin{equation}
f:=\widehat{g}( \cdot )\exp \big( \frac{|\cdot|}{2\alpha }\big) \in L^{2}({R})   \label{2}
\end{equation}

\begin{proposition} \label{prop5}
If $u( x,t) $ is a solution of problem (\ref{1}), then
\[
\| u( x,\cdot ) -P_{j}u( x,\cdot ) \| \leq
\| f\|_{L^2(\mathbb{R}) } \exp ( -\frac{1}{3}
\frac{\pi }{\alpha }2^{-j}( 1-x^{2}) )
\]
where $f$ is given by (\ref{2}).
\end{proposition}

\paragraph{Proof}
>From (\ref{8}) and (\ref{9}), we have
\begin{eqnarray*}
\| ( I-P_{j}) u( x,\cdot ) \| & \leq &
\| {\chi }_{j} \widehat{u}( x,\cdot ) \|
\\
& = & [ \int_{| \xi | >\frac{2}{3}\pi
2^{-j}}| \widehat{u}( x,\xi ) | ^{2}\,d\xi ] ^{1/2} \\
& \leq & [ \int_{| \xi | >\frac{2}{3}\pi
2^{-j}}| \widehat{g}( \xi ) | ^{2}  \exp [ 2| \xi
| \int_{0}^{x}\int_{0}^{s}\frac{1}{K( \tau
) }\,d\tau \, ds] \,d\xi ] ^{1/2}
\end{eqnarray*}
Then
\begin{eqnarray*}
\| ( I-P_{j}) u( x,\cdot ) \| & \leq &
[ \int_{| \xi | >\frac{2}{3}\pi 2^{-j}}| \widehat{g}( \xi ) | ^{2}
 \exp ( | \xi |  \frac{x^{2}}{\alpha }) \,d\xi ] ^{1/2} \\
& \leq & [ \int_{| \xi | >\frac{2}{3}\pi
2^{-j}}| f( \xi ) | ^{2}  \exp ( -\frac{|
\xi | }{\alpha })  \exp ( \frac{| \xi | }{\alpha } x^{2}) d\xi ] ^{1/2} \\
& = & [ \int_{| \xi | >\frac{2}{3}\pi
2^{-j}}| f( \xi ) | ^{2}  \exp ( -\frac{|
\xi | }{\alpha } ( 1-x^{2}) ) \,d\xi ] ^{1/2}
\end{eqnarray*}
For $| x| <1$,
\begin{eqnarray*}
\| ( I-P_{j}) u( x,\cdot ) \| & \leq &
[ \int_{\mathbb{R}}| f( \xi ) | ^{2}\,d\xi
] ^{1/2}  \exp ( -\frac{( 2/3) \pi 2^{-j}}{%
2\alpha } ( 1-x^{2}) )  \\
& \leq & \| f\|_{L^2(\mathbb{R}) }  \exp ( -\frac{%
1}{3}\frac{\pi }{\alpha }2^{-j} ( 1-x^{2}) )
\end{eqnarray*}
which completes the proof. \hfill$\square$

\begin{proposition} \label{prop6}
If $u$ is a solution of problem (\ref{1}) and $u_{j-1}$ is a solution of
the approximating problem in $V_{j-1}$ then
\begin{equation}
\widehat{u}(x,\xi )=\widehat{u}_{j-1}(x,\xi ) \quad\mbox{for }
| \xi | \leq \frac{4}{3}\pi 2^{-j} \label{12}
\end{equation}
Consequently,
\begin{equation}
P_{j}u(x,\cdot )=P_{j}u_{j-1}(x,\cdot ) \label{13}
\end{equation}
\end{proposition}

\paragraph{Proof}
Let $\Lambda ( x,\xi ) =\widehat{u}( x,\xi ) -\widehat{u}_{j-1}( x,\xi )$.
We will show that $\Lambda (x,\xi ) =0$
 for $| \xi | \leq \frac{4}{3}\pi 2^{-j}$. Consider the approximating problem
 in $V_{j-1}$:
\begin{gather*}
K(x) ( u_{j-1})_{xx} = P_{j-1}(u_{j-1})_{t}\quad t\in \mathbb{R},\; 0<x<1 \\
u_{j-1}( 0,\cdot ) = P_{j-1}g,\quad
( u_{j-1})_{x}( 0,\cdot ) = 0 \\
u_{j-1}( x,\cdot ) \in V_{j-1}
\end{gather*}
Applying the Fourier transform with respect to time, we have
\[
K( x) ( \widehat{u}_{j-1})_{xx}( x,\xi )
= \widehat{P}_{j-1}[ ( u_{j-1})_{t}] \widehat{}\
( x,\xi )
 = \widehat{P}_{j-1}( i\xi \widehat{u}_{j-1}( x,\xi ))
\]
for $0\leq x<1$, $\xi \in \mathbb{R}$,
with the conditions:
$\widehat{u}_{j-1}( 0,\xi ) =\widehat{P}_{j-1}\widehat{g}( \xi )$
and $( \widehat{u}_{j-1})_{x}( 0,\cdot ) =0$.
Now, by (\ref{7}),
$$\widehat{P}_{j-1}( i\xi \widehat{u}_{j-1}(x,\xi ) )
=i\xi \widehat{u}_{j-1}( x,\xi ) \quad
\mbox{and}\quad \widehat{P}_{j-1}\widehat{u}( 0,\xi )
 =\widehat{u}(0,\xi )
 $$
for $| \xi | \leq \frac{4}{3}\pi 2^{-j}$.
Thus, for $| \xi | \leq \frac{4}{3}\pi 2^{-j}$, we have
\begin{align*}
&K( x) \Lambda_{xx}( x,\xi ) -i\xi \Lambda (x,\xi )\\
&=  K( x) \widehat{u}_{xx}( x,\xi )-K( x) ( \widehat{u}_{j-1})_{xx}( x,\xi )
-i\xi [ \widehat{u}( x,\xi ) -\widehat{u}_{j-1}(x,\xi ) ]
= 0
\end{align*}
\[
\Lambda ( 0,\xi )  =  \widehat{u}( 0,\xi ) -\widehat{u}_{j-1}( 0,\xi )
 =  \widehat{u}( 0,\xi ) -\widehat{P}_{j-1}\widehat{g}( \xi )
 =  \widehat{u}( 0,\xi ) -\widehat{P}_{j-1}\widehat{u}(0,\xi ) = 0
\]
\[
\Lambda_{x}( 0,\xi ) =  \widehat{u}_{x}( 0,\xi )
-( \widehat{u}_{j-1})_{x}( 0,\xi )  =  0
\]
Hence, for $| \xi | \leq \frac{4}{3}\pi 2^{-j}$, fixed,
$\Lambda ( x,\xi ) $ is solution on $0\leq x<1$ of the
problem
\begin{gather*}
K( x) \Lambda_{xx}( x,\xi ) -i\xi \Lambda (x,\xi ) = 0 , \quad 0<x<1  \\
\Lambda ( 0,\xi ) = 0,  \quad
\Lambda_{x}( 0,\xi ) = 0
\end{gather*}
This problem has an unique solution $\Lambda ( x,\xi )=0$, for all
$x \in [0,1)$. Thus,
\[
\widehat{u}( x,\xi ) =\widehat{u}_{j-1}( x,\xi ) \quad
\mbox{for }| \xi | \leq \frac{4}{3}\pi 2^{-j}
\]
Now, (\ref{13}) is consequence of (\ref{12}) and the definition of
$\widehat{P}_{j}$. \hfill$\square$

\begin{theorem} \label{thm7}
Let $u$ be a solution of  (\ref{1}) with the condition $u( 0,\cdot )=g$,
 and let $f$ be given by (\ref{2}). Let $v_{j-1}$ be a solution of
 (\ref {10}) in $V_{j-1}$ for the boundary specification $\widetilde{g}$
 such that $\| g-\widetilde{g}\| \leq \epsilon $. If $j=j( \epsilon )$ is
such that $2^{-j}= \frac{\alpha }{\pi } \log  \epsilon ^{-1}$,
then
\[
\| P_{j}v_{j-1}( x,\cdot ) -u( x,\cdot ) \|
\leq \epsilon ^{1-x^{2}} \ + \ \| f\|_{L^{2}({R})} \cdot \epsilon
^{\frac{1}{3}(1-x^{2})}
\]
\end{theorem}

\paragraph{Proof}
Note that
\begin{align*}
\| P_{j}v_{j-1}(x,\cdot )-u( x,\cdot ) \|
& \leq\| P_{j}v_{j-1}( x,\cdot ) -P_{j}u( x,\cdot )
+P_{j}u( x,\cdot ) -u( x,\cdot ) \|  \\
& \leq  \| P_{j}v_{j-1}( x,\cdot ) -P_{j}u( x,\cdot) \|
+ \| P_{j}u( x,\cdot ) -u( x,\cdot) \|\,.
\end{align*}
Let $u_{j-1}$ be a solution of (\ref {10}) in $V_{j-1}$ for the boundary specification $g$. By (\ref{13}), $P_{j}u(
x,\cdot ) =P_{j}u_{j-1}( x,\cdot ) $. Thus, by theorem \ref{thm4}, we have
\begin{align*}
\| P_{j}v_{j-1}( x,\cdot ) -P_{j}u( x,\cdot )\|
& =  \| P_{j}v_{j-1}( x,\cdot )-P_{j}u_{j-1}( x,\cdot ) \|  \\
& \leq  \| v_{j-1}( x,\cdot ) -u_{j-1}( x,\cdot) \|
\leq  \epsilon ^{1-x^{2}}
\end{align*}
Now, by proposition \ref{prop5},
\[
\| P_{j}u( x,\cdot ) -u( x,\cdot ) \|
\leq \| f\|_{L^2(\mathbb{R}) } \exp ( -\frac{1}{3}
\frac{\pi }{\alpha}2^{-j}( 1-x^{2}) )
 \leq  \| f\|_{L^2(\mathbb{R}) }\cdot \epsilon ^{\frac{1}{3}(1-x^{2})}
\]
Then
$\| P_{j}v_{j-1}( x,\cdot ) -u( x,\cdot ) \| \leq  \epsilon ^{1-x^{2}}
 +  \| f\|_{L^{2}({R})}  \epsilon ^{\frac {1}{3}(1-x^{2})}$
 \hfill$\square$

\subsection*{Conclusion}

We had considered solutions $u( x,\cdot)\in L^{2}(R)$ of the problem $
K(x)u_{xx}=u_{t}$, $0<x<1$ , $t\geq 0$ , with boundary specification $g$
and $u_{x}(0,\cdot)=0$, where $K(x)$ is bounded below by a positive constant.
The inequality (\ref {9}) implies that a solution of the problem above will
 be in $L^2(\mathbb{R}) $ if $\widehat{g}$ has a rapid decay at high
frequencies. Since the Meyer wavelet has compact
support in the frequency domain, it cuts the high frequencies.
Utilizing a wavelet Galerkin method with the Meyer multi-resolution analysis,
we regularize the ill-posedness of the problem, approaching it by well-posed
problems in the scaling spaces, as shown by theorem \ref{thm4}. We had shown the
convergence of the wavelet Galerkin method applied to our problem, with an
estimate error, in theorem \ref{thm7}. A more direct result would be to have a similar
estimate for the difference between the exact solution of the problem and the
solution of the approximating problem defined on the scaling space $V_{j}$.
We are working towards this goal at the moment.

\paragraph{Notes:} 1) Consider the problem
\begin{gather*}
u_{xx}(x,t)=u_{t}(x,t),\quad t\geq 0,\; 0<x <1  \\
u(0,\cdot) =g_{n},\quad u_{x}(0,\cdot)=0\,,
\end{gather*}
where
\[
g_{n}(t)=\begin{cases}
n^{-2}\cos 2n^{2}t, & \mbox{if } 0\leq t \leq t_{0}  \\
0,&  \mbox{if } t>t_{0}\,.
\end{cases}
\]
The solution of this problem is
\[
u_{n}(x,t)=\begin{cases}
\sum_{j=0}^{\infty }n^{-2}\cos (2n^{2}t+j \frac{\pi}{2})
\frac{(\sqrt{2}nx)^{2j}}{(2j)!},& \mbox{if }0\leq t \leq t_{0}  \\
0,& \mbox{if } t>t_{0}\,.
\end{cases}
\]
Note that  $g_{n}(t)$ converges uniformly to zero as $n$ tends to infinity,
while for $x>0$, the solution $u_{n}(x,t)$ does not tend to zero.
This example was inspired by \cite{c1}.

\noindent 2) Note that $({\varphi_{jl}})' \notin V_{j}$. In fact, if $({\varphi_{jl}})' \in V_{j}$ then $({\varphi_{jl}})'=\sum_{k\in Z}\alpha_{k} \varphi_{jk}$. Hence
$$\widehat{({\varphi_{jl}})'}=\sum_{k\in Z}\alpha_{k} \widehat{\varphi_{jk}}$$
So, we would have
$$
i{2}^{j/2}{e}^{-i{2}^{j}l \xi}\xi \widehat{\varphi}({2}^{j}\xi)
= \sum_{k\in Z}\alpha_{k}{2}^{j/2}{e}^{-i{2}^{j/2}\xi}
\widehat{\varphi}({2}^{j}\xi)
$$
This equality implies
$\xi = \sum_{k\in Z}\alpha_{k}{e}^{-i[{2}^{j}(k-l)\xi + \frac{\pi}{2}]}$.

\begin{thebibliography}{0} \frenchspacing

\bibitem{c1} J. R Cannon, \emph{The one dimensional heat equation},
(Reading) MA: Addison-Wesley, 1984.

\bibitem{d1} I. Daubechies, I., \emph{Ten Lectures on Wavelets},
CBMS - NSF 61 SIAM, Regional Conferences Series in \ Applied Mathematics, 1992.

\bibitem{r1} T. Reginska, \emph{Sideways heat equation and wavelets},
J. Comput. Appl. Math. 63 (1995) 209-214.

\bibitem{r2} T. Reginska and L. Eld\'{e}n, \emph{Solving the
sideways heat equation by a wavelet Galerkin method}, Inverse Problems 13
(1997) 1093-1106.

\bibitem{r3} T. Reginska,  \emph{Stability and Convergence of a
wavelet Galerkin method for the Sideways Heat Equation}, J. Inverse Ill
Posed Probl., 8, no 1, 2000.

\end{thebibliography}

\noindent\textsc{Jos\'{e} Roberto Linhares de Mattos}\\
Rural Federal University of Rio de Janeiro, \\
Exact Sciences Institute, Department of Mathematics, \\
BR465  Km 7,  Serop\'{e}dica  RJ,  CEP 23890-000,  Brazil \\
email: linhares@cos.ufrj.br  \medskip

\noindent\textsc{Ernesto Prado Lopes}\\
Federal University of Rio de Janeiro,\\
COPPE, Systems and Computing Engineering Program,\\
Tecnology Center, Bloco H \\
and \\
Institute of Mathematics,
Tecnology Center, Bloco C, Ilha do Fund\~{a}o,\\
Rio de Janeiro RJ, CEP 21945-970, Brazil \\
email: lopes@cos.ufrj.br 

\end{document}