\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \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 $K(x)u_{xx}=u_{t}$ , $0From the variational formulation of the approximating problem on the scaling space$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$\psiis 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 $$\mathop{\rm supp}( \widehat{\psi_{jk}}) =\big\{ \xi ; \frac{2}{3}\pi 2^{-j}\leq | \xi | \leq \frac{8}{3}\pi 2^{-j}\big\}\quad \forall k\in \mathbb{Z} \label{5}$$ Furthermore, $$\mathop{\rm supp}( \widehat{\varphi_{jk}}) =\big\{ \xi ; | \xi | \leq \frac{4}{3}\pi 2^{-j}\big\} \quad \forall k\in \mathbb{Z} \label{6}$$ 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 $$\widehat{P_{j}f}( \xi ) =\widehat{f}( \xi ) \quad \mbox{for } | \xi | \leq \frac{2}{3}\pi 2^{-j} \label{7}$$ 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{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} 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 0From 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{gathered} K(x)u_{xx}(x,t)=P_{j}u_{t}(x,t),\quad t\geq 0,\; 0From (\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 $$\widehat{u}(x,\xi )=\widehat{u}_{j-1}(x,\xi ) \quad\mbox{for } | \xi | \leq \frac{4}{3}\pi 2^{-j} \label{12}$$ Consequently, $$P_{j}u(x,\cdot )=P_{j}u_{j-1}(x,\cdot ) \label{13}$$ \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},\; 0t_{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}