\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 130, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/130\hfil Regularization of the backward heat equation]
{Regularization of the backward heat equation via heatlets}

\author[B. M. Campbell H., R. Hughes, E. McNabb, \hfil EJDE-2008/130\hfilneg]
{Beth Marie Campbell Hetrick, Rhonda Hughes, Emily McNabb}  % in alphabetical order

\address{Beth Marie Campbell Hetrick \newline
Gettysburg College, Gettysburg, PA 17325, USA}
\email{bcampbel@gettysburg.edu}

\address{Rhonda Hughes \newline
Bryn Mawr College, Bryn Mawr, PA 19010, USA}
\email{rhughes@brynmawr.edu}

\address{Emily McNabb \newline
Bryn Mawr College, Bryn Mawr, PA 19010, USA}
\email{emily.a.mcnabb@accenture.com}

\thanks{Submitted March 27, 2008. Published September 18, 2008.}
\subjclass[2000]{47A52, 42C40}
\keywords{Ill-posed problems; backward heat equation; wavelets;\hfill\break\indent
 quasireversibility}

\begin{abstract}
 Shen and Strang \cite{Shen2} introduced heatlets  in order to solve
 the heat equation using wavelet expansions of the initial data.
 The advantage of this approach is that heatlets, or the heat evolution
 of the wavelet basis functions, can be easily computed and stored.
 In this paper, we use heatlets to regularize the {\it backward}
 heat equation and, more generally, ill-posed Cauchy problems.
 Continuous dependence results obtained by Ames and Hughes \cite{AHb}
 are applied to approximate stabilized solutions to ill-posed problems
 that arise from the method of quasi-reversibility.
\end{abstract}

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

\section{Introduction} \label{Introduction}

Shen and Strang \cite{Shen2} introduced heatlets in order to solve
the heat equation using wavelet expansions of the initial data.
The advantage of this approach is that heatlets, or the heat
evolution of the wavelet basis functions, can be computed easily
and stored.  When the initial data is expanded in terms of the
wavelet basis, the solution to the heat equation is then obtained
from an expansion using the heatlets and the corresponding wavelet
coefficients of the data.  In this paper, we turn our attention to
ill-posed problems, using heatlets, and the method of
quasi-reversibility  \cite{LL}, to regularize  the {\it
backward} heat equation \cite{Miller1, Payne1,Showalter} as
well as more general ill-posed problems.

Given an ill-posed problem, it is often convenient to define an
approximate problem that is well-posed.  Generally, we seek to
ensure that a solution to the original problem, if it exists, will
be appropriately close to the solution to the approximate problem.
In our main results, we show that for a wide range of ill-posed
problems, heatlets may be used to obtain such approximate
solutions.  In addition, applying the results of \cite{AHb, AH2}, we
obtain H\"{o}lder-continuous dependence results for the difference
between solutions of the ill-posed and approximate well-posed
problems.    Previously, wavelets have been used by Liu et al. to
decompose the regularized solution of inverse heat conduction
problems using a sensitivity decomposition \cite{Liu}, but
heatlets do not play a role in that work.

 We consider the backward heat equation
\begin{equation} \label{e1}
\begin{gathered}
   \frac{\partial u}{\partial t}
= - \frac{\partial ^2 u}{\partial x^2}  \quad \text{where } 0<x<c, \; 0<t<T,  \\
  u(0, t) = u(c,t) = 0, \quad  0 < t < T,   \\
  u(x,0)  = k(x), \quad  0<x<c,
\end{gathered}
\end{equation}
 for suitable initial data $k(x)$.
The continuous dependence results in \cite{AHb, AH2} use semigroup theory
and the notion of $C$-semigroups  \cite{Mel,Miller,Showalter}.
   If we let
$A= - \Delta$ denote the self-adjoint Laplacian in
$L^2(\mathbb{R})$, then the backward heat equation can be written
as an \emph{abstract Cauchy problem} \cite{Fatt}:
\begin{equation} \label{e2}
\begin{gathered}
\frac{du}{dt}  =  Au,  \\
u(0)  =  f.
\end{gathered}
\end{equation}
 Following \cite{Ames,Miller1}, we define an approximate well-posed problem as follows:
\begin{equation} \label{ee}
\begin{gathered}
\frac{dv}{dt}   =  (A- \epsilon A ^{2}) v
=- \frac{\partial ^{2} v}{\partial x^{2}}
 - \epsilon \frac{\partial ^{4} v}{\partial x ^{4}},\\
v(0) =  f.
\end{gathered}
\end{equation}
This equation is well-posed, since  the spectrum of
$A- \epsilon A ^{2}$ is bounded above.  From the Spectral Theorem,
it follows that  solutions to the approximate well-posed problem
are of the form
\begin{equation} \label{e4}
v(t )  =  e^{t(A-\epsilon A^{2})}f.
\end{equation}

  Quasi-reversibility is
a regularization technique for ill-posed problems that is designed to
generate approximate solutions to the problem in
question.  The central idea of quasi-reversibility is to solve the
original problem backward, after first replacing $A$ by
an approximate operator whose spectrum is bounded above.
Miller \cite{Miller1,Miller} refines the
quasi-reversibility approach of Lattes and Lions, finding
sufficient conditions on the approximate operator to guarantee
  H\"{o}lder continuous dependence on the data when the method is
stabilized; he refers to his approach as an $SQR$-method.
 To implement the method of quasi-reversibility, we consider the
well-posed final value problem
\begin{equation} \label{e5}
\begin{gathered}
\frac{dw}{dt}  =  A w,\\
 w(T)  =  v(T)=e^{T(A-\epsilon A^{2})}f,
\end{gathered}
 \end{equation}
with solution
\begin{equation} \label{e6}
w(t)  =  e^{(t-T)A}e^{T(A-\epsilon A^{2})}f
  =  e^{tA}e^{-T\epsilon A^{2}}f.
\end{equation}
We then have the following regularization result from
\cite{AHb}.



\begin{theorem}[{\cite[Theorem 2]{AHb}}] \label{thme}
If $u(t)$ and $w(t)$ are solutions to \eqref{e2} and \eqref{e5} respectively,
and $\| u(T)\| \le k$, for some constant  $k$, then there exist constants
$C$ and $M$, independent of $\epsilon > 0$, such that for $0 \le t <T$,
\[
\| u(t)- w(t) \| \le C\epsilon^{1- \frac{t}{T}}M^{t/T}.
\]
\end{theorem}

In light of this result, we ask whether a heatlet decomposition of
the initial data can be used to determine the regularization $w(t)$.
First, we turn to the main result in \cite{Shen2}, which deals
with the well-posed {\it forward} heat equation
\begin{equation} \label{e7}
\begin{gathered}
\frac{du}{dt}   =  - Au,  \\
u(0)  =  f.
\end{gathered}
\end{equation}


\begin{theorem}[{\cite[Theorem 3.1]{Shen2}}] \label{thmb}
Let $f \in L^{2}(\mathbb{R})$, and $\{\psi_{j,n}\}$ be an orthonormal wavelet
basis.  Then the corresponding heat evolution in
$L^{2}(\mathbb{R})$  is given by
\[
u(x,t)= \sum_{j,n \in \mathbb{Z}} c_{j,n}\Psi^{h}_{j,n} (x,t),
\]
where $c_{j,n}$ is the wavelet coefficient of $f(x)$ attached to
$\psi_{j,n}=2^{j/2}\psi(2^{j}x-n)$, and
$\Psi^{h}_{j,n} (x,t)$ is the solution of \eqref{e5} with initial data
$\psi_{j,n}$.
Moreover, the infinite series converges in  $L^{2}(\mathbb{R})$
uniformly with respect to $t$ .
\end{theorem}

 Using quasi-reversibility, we determine that $w(t)$ can be obtained
by evaluating a heatlet at time $T-t$.  This will yield our main result,
the {\it heatlet decomposition for the backward heat equation}
(Theorem 3.3):



\begin{theorem} \label{thm1}
 Let $f \in L^{2}(\mathbb{R})$.   If
$u(t)$ is a stabilized solution of \eqref{e2}, so that
$\|u(T)\| \leq \tilde{M}$, we have
\[
\| u(t)-  \sum_{j,n \in \mathbb{Z}} c_{j,n}
 e^{T(A-\epsilon A^{2})}\Psi^{h}_{j,n} (x,T-t) \|
\le C\epsilon^{1- \frac{t}{T}}M^{t/T},
\]
 for constants $C$ and $M$ that are independent of
$\epsilon > 0$, and $c_{j,n}$ is the wavelet coefficient of $f(x)$
attached to $\psi_{j,n}=2^{j/2}\psi(2^{j}x-n)$.   Thus,
for small values of $\epsilon > 0$,
\[
  \sum_{j,n \in \mathbb{Z}}
c_{j,n} e^{T(A-\epsilon A^{2})}\Psi^{h}_{j,n} (x,T-t)
\]
is close to $u(t)$ in $L^{2}(\mathbb{R})$, for $0 \leq t < T$.
\end{theorem}

The value of the above theorem  lies in the fact that,
as in the case of the well-posed heat equation, the heatlets may be
computed and stored, and the approximation $w(t)$ will require
 evaluation of $e^{T(A-\epsilon A^{2})}\Psi^{h}_{j,n} (x,T-t)$,
rather than $e^{T(A- \epsilon A^{2})}e^{(t-T)A}f$. Finally, in
Section 4, we show that Theorem 1.3 may be framed in a  more general
setting, with other choices of the approximating operators.  To
pursue this generalization, we introduce the terminology of
\cite{AHb},
and define {\it generalized heatlets}, that is, solutions of the
{\it abstract} Cauchy problem with initial data consisting of
elements of a wavelet basis.  We then approximate the solution to
the ill-posed problem using the wavelet coefficients in a manner
analagous to that in Theorem 1.3 (Theorem 4.2).

\section{Wavelets and Heatlets}

 In $L^{2}(\mathbb{R})$ we define the \emph{mother
wavelet} of the Haar basis as
\[
\psi(x) = \begin{cases}
1 & 0 \le x < \frac{1}{2}\\
-1 & \frac{1}{2} \le x < 1\\
0 & \text{otherwise.}
\end{cases}
\]
For  positive integers $n,j$ define
$\psi_{n}^{j}(x)=2^{j/2}\psi(2^{j}x-n)$. Then according to a
theorem of Haar, $\{\psi_{n}^{j}\}$ is an orthonormal basis for
$L^{2}(\mathbb{R})$ (cf. \cite{Daub}).



\noindent \textbf{Definition.}
 A \emph {multiresolution analysis}
of $L^{2}( \mathbb{R})$ is a chain of approximate spaces $V_{j}$
such that $-\infty \le j \le \infty$.  These closed subspaces
satisfy the following properties:
\begin{itemize}
\item[(i)]  The $V_{j}$ spaces are nested:
$\dots V_{-1} \subset V_{0} \subset V_{1} \subset V_{2} \subset
\dots$

\item[(ii)] These spaces are complete; that is,
\begin{gather*}
\overline{\cup_{j \in \mathbb{Z}} V_{j}}
= L^{2}(\mathbb{R}) \quad
(\text{i.e. } \lim_{j \to \infty} V_{j} =  L^{2}(\mathbb{R}) ),\\
 \cap_{j \in \mathbb{Z}} V_{j}= {0}  \quad
(\text{i.e. } \lim_{j \to -\infty} V_{j} = {0}).
\end{gather*}

\item[(iii)]  $f (x) \in V_{j}$  if and only if $f(2x) \in V_{j+1}$.

\item[(iv)]  $f (x) \in V_{0}$  if and only if $f(x-k) \in V_{0}$.

\item[(v)]  There exists a scaling function $\phi (x) \in V_{0}$
such that $\{ \phi(x-k): k \in \mathbb{Z} \}$ is an orthonormal
basis of $V_{0}$ (cf. \cite{Daub}).

\end{itemize}

 To create a multiresolution, one needs to construct a
scaling function $\phi(x)$.  Then, using the  properties of a
multiresolution analysis, the entire chain can be constructed from
$\phi(x)$.  For example, we can let $V_{0}=\{\phi(x-n)|n \in \mathbb{Z}\}$.
Then
\begin{gather*}
V_{1}=\{\phi(2x-n):n \in \mathbb{Z}\}, \\
V_{2}=\{\phi(2^{2}x-n):n \in \mathbb{Z}\}, \\
V_{-1}=\{\phi(\frac{x}{2}-n):n \in \mathbb{Z}\}.
\end{gather*}
This chain of approximate spaces $V_{j}$ forms a multiresultion analysis of
$L^{2}(\mathbb{R})$ \cite{Daub}.


The multiresolution analysis  associated with the Haar basis is
provided by
\[
V_{j} = \{ f \in L^{2}( \mathbb{R}) :
 f | _ {[\frac{k}{2^{j}}, \frac{(k+1)}{2^{j}}]} = \text{constant, }
 k \in \mathbb{Z} \}.
\]

Next, we summarize the definitions and results from
\cite[Section 3]{Shen2}.



\noindent \textbf{Definition.}
Let $\phi(x)$ be the scaling function and $\psi(x)$ be the wavelet
associated to a multiresolution analysis.  Define the
\emph{heat evolutions} of $\phi(x)$ and $\psi(x)$ to be $\Phi^{h}(x,t)$
and $\Psi^{h}(x,t)$, where
\[
 \Phi^{h}_{t}= \Phi^{h}_{xx}, \quad \Phi^{h}(x,0)=\phi(x), \quad
 \text{for } t>0, \;  x \in \mathbb{R}.
\]
 Similarly,
\[
 \Psi^{h}_{t}= \Psi^{h}_{xx}, quad \Psi^{h}(x,0)=\psi(x),\quad
\text{for } t>0, \;  x \in \mathbb{R}.
 \]
The function $\Psi^{h}$ is called a \emph{heatlet} and
$\Phi^{h}$ is a \emph{refinable heat}.


\begin{proposition} \label{prop1}
Assume that $\phi(x)$ and $\psi(x)$ satisfy the equations
\begin{gather*}
 \phi(x)=2 \sum_{n \in Z} h_{n}\phi(2x-n), \\
 \psi(x)=2 \sum_{n \in Z} g_{n}\phi(2x-n),
\end{gather*}
where $(h_{n}),(g_{n}) \in l^{2}$.  Then, the refinable heat and
heatlet will satisfy
\begin{gather*}
\Phi^{h}(x,t)=2 \sum_{n \in Z} h_{n}\Phi^{h}(2x-n,4t), \\
\Psi^{h}(x,t)=2 \sum_{n \in Z} g_{n}\Phi^{h}(2x-n,4t).
\end{gather*}
\end{proposition}

\begin{proposition} \label{prop2}
Define $\Psi^{h}_{j,n} (x,t)$ to be the solution of \eqref{e5}
with initial data $\psi_{j,n}$.  Then
\[
\Psi^{h}_{j,n} (x,t)= 2^{j/2}\Psi^{h}(2^{j}x-n,4^{j}t).
 \]
\end{proposition}

The main theorem of Shen and Strang \cite{Shen2} is as follows.


\begin{theorem}[\cite{Shen2}] \label{thmc}
 Let $f \in L^{2}(\mathbb{R})$.
Then the corresponding heat evolution in  $L^{2}(\mathbb{R})$ is
given by
\[
u(x,t)= \sum_{j,n \in \mathbb{Z}} c_{j,n}\Psi^{h}_{j,n} (x,t),
 \]
where $c_{j,n}$ is the wavelet coefficient of $f(x)$ attached to
$\psi_{j,n}=2^{j/2}\psi(2^{j}x-n)$.  Moreover, the
infinite series converges in  $L^{2}(\mathbb{R})$ uniformly with
respect to $t$ .
\end{theorem}

\section{Regularization of  the Backward Heat Equation}
\label{Regularization of the Backward Heat Equation}

 Consider the \emph{final value} problem
\begin{gather*}
\frac{\partial u}{\partial t} = \frac {\partial ^{2} u} {\partial x^{2}}
\quad \text{for }  0<t<T,\;  x \in (0,l), \\
u(x,T)= \phi(x), \\
u(0,t)=u(l,t)=0.
\end{gather*}
This problem is ill-posed, and equivalent to \eqref{e1}.
Following \cite{Payne1}, we will stabilize the problem as follows.
Define $\mathcal{M}$ to be the collection of all continuous functions
$\phi(x,t)$ in $D  \times   [0,T)$ such that $\phi(x,t)$ is twice
differentiable in $x$ and continuously differentiable in
 $t$ for $t \in (0,T)$. Furthermore, assume
\[
\|\phi(T) \|^{2} \le k^{2}
 \]
for some prescribed constant $k$ which is a natural bound.
 The following stability result is well-known:


\begin{theorem}[\cite{Payne1}] \label{thmd}
If $u(x,t) \in \mathcal{M}$ is a solution to the backward heat equation
and $\| u(T) \|^{2} \le k^{2} $, then
\[
\|u(t)\|^{2} \le \|f\|^{2(1-\frac{t}{T})} k^{\frac{2t}{T}}.
 \]
\end{theorem}

In addition, we have the previously mentioned  H\"{o}lder-continuity result from
\cite{AHb} (Theorem 1.1).

Now, recall that for $f \in L^{2}(\mathbb{R})$, the
corresponding heat evolution in  $L^{2}(\mathbb{R})$ from $f$ (for
the well-posesd problem) is given by
\[
u(x,t)= \sum_{j,n \in \mathbb{Z}} c_{j,n}\Psi^{h}_{j,n} (x,t),
\]
where $c_{j,n}$ are the wavelet coefficients of $f(x)$ attached to
$\psi_{j,n}=2^{j/2}\psi(2^{j}x-n)$.  Using quasireversibility,
we find that $w(t)$ may be obtained by evaluating a heatlet at time $T-t$.
This will yield the {\it heatlet decomposition for the backward heat equation}.


\begin{theorem} \label{thm1b}
Let $f \in L^{2}(\mathbb{R})$, and let
$c_{j,n}$ denote the wavelet coefficient of $f(x)$ attached to
$\psi_{j,n}=2^{j/2}\psi(2^{j}x-n)$. Assume that $u(t)$  is
a stabilized solution of \eqref{e2}.  Then there exist constants $C$ and
$M$, independent of $\epsilon > 0$, such that
\[
\| u(t)-  \sum_{j,n \in \mathbb{Z}} c_{j,n}
e^{T(A-\epsilon A^{2})}\Psi^{h}_{j,n} (x,T-t) \|
\le C \epsilon^{1- \frac{t}{T}}M^{t/T}.
\]
 Thus, for small values of $\epsilon > 0$,
$\sum_{j,n \in \mathbb{Z}} c_{j,n} e^{T(A-\epsilon A^{2})}\Psi^{h}_{j,n}
(x,T-t)$ is close to $u(t)$ in $L^{2}(\mathbb{R})$, for
$0 \leq t < T$.
\end{theorem}


\begin{proof}
 Recall that the solution to \eqref{e5} is
\begin{align*}
w(t) & =   e^{(t-T)A}e^{T(A-\epsilon A^{2})}f \\
& =  \sum_{j,n \in \mathbb{Z}} c_{j,n}e^{(t-T)A} e^{T(A-\epsilon A^{2})}
 \psi_{j,n} \\
& =   \sum_{j,n \in \mathbb{Z}} c_{j,n}e^{T(A-\epsilon A^{2})}
 \Psi^{h}_{j,n} (x,T-t),
\end{align*}
where for each $j,n$, $e^{(t-T)A} \psi_{j,n}$ is the heatlet
$\Psi^{h}_{j,n} (x,T-t)$.
 We consider
\[
\|u(t) - w(t) \| = \|e^{tA} \chi - e^{tA} e^{-\epsilon T A^2} f\| =
\|(I - e^{-\epsilon TA^2}) e^{tA} f\|.
\]
In order to obtain a convexity result, we set
\[
\phi_n(\alpha) =  (e^{ {\alpha}^2}[e^{\alpha A} - e^{\alpha A}
e^{-\epsilon T A^2}]f_n,\,h),
 \]
where $f_n = E(e_n)$, $E(\cdot)$ is the resolution of the identity
 for $A$, $e_n$ is a bounded Borel function, and $h$ is an arbitrary
element of $\mathcal{H}$.  Then
\begin{align*}
|\phi_n(\alpha)|
 &\leq   e^{t^2 - \eta^2} \|e^{(t+i\eta)A} f_n -
e^{(t+i\eta)A}e^{-\epsilon TA^2} f_n\| \, \|h\|\\
 &\leq   e^{t^2 - \eta^2}\|(I - e^{-\epsilon TA^2})e^{tA} f_n\|\,\|h\|\\
 &\leq   C_1 \, e^{t^2 - \eta^2} \epsilon \|A^{2}e^{tA}f_n\|\,\|h\|.
\end{align*}
Thus $\phi_n(\alpha)$ is
bounded in the strip $0 \leq \Re \alpha \leq T$, and so by the Three
Lines Theorem, we obtain
\[
|\phi_n(t)| \leq M(0)^{1-t/T}M(T)^{t/T}  ,
\]
 where $M(t) = \max_{\eta \in \mathbb{R}}|\phi(t +i\eta)|$.
Since $M(0) \leq C_1 \epsilon \|A^{2} f_n\|\,\|h\|$,
and
\[
M(T) \leq e^{T^2} \|(I - e^{-\epsilon TA^2})
e^{TA}f_n\|\,\|h\| \leq C_2 \, e^{T^2}\|e^{TA}f_n\|\,\|h\|,
\]
 we obtain, taking the supremum over all $h \in \mathcal{H}$,
with $\|h\| \leq 1$,
\[
\|u(t) - w(t)\| \leq C\{\epsilon \|A^{2} f_n\|\}^{1-t/T} \{
\|e^{TA}f_n\|\}^{t/T}.
\]
for a suitable constant $C$.  If we take the limit as $n
\to \infty$, and assume in addition that
$\|e^{TA}f\| \leq
\tilde{M}$, from which it follows that
$\|A^{2}f\| \leq \tilde{M}$, for a possibly different
constant,  we have
\[
\|u(t) -  \sum_{j,n \in \mathbb{Z}} c_{j,n}e^{T(A-\epsilon A^{2})}
 \Psi^{h}_{j,n} (x,T-t)\|
 =  \|u(t) - w(t)\|
 \leq  C\epsilon^{1-t/T}M^{t/T}.
\]
 Thus, for small values of $\epsilon > 0$,
$\sum_{j,n \in \mathbb{Z}} c_{j,n} e^{T(A-\epsilon A^{2})}\Psi^{h}_{j,n}
(x,T-t)$ is close to $u(t)$ in $L^{2}(\mathbb{R})$, for
$0 \leq t <T$.
\end{proof}

\section{Applications to Ill-Posed Problems}


In this section, following \cite{Ames0,Ames,AmesLecture,AHb,AH2},
we consider ill-posed Cauchy
problems in $L^2(\mathbb{R})$, where $A$ is now  any positive
self-adjoint operator.  Let $\psi$ be associated with a
multiresolution analysis, and let  be the corresponding wavelet
basis be $\{\psi_{j,n}\}$.  We show that the choice of approximate
problem can be generalized.  


\noindent\textbf{Definition.}
 Let $f: [0, \infty) \to \mathbb{R}$ be a Borel function, and
assume that there exists $\omega \in \mathbb{R}$ such that
$f(\lambda ) \leq \omega$ for all $\lambda \in [0, \infty)$.
Then $f$ is said to satisfy Condition (A)
 if there exist positive constants $\beta$, $\delta$ with $0 < \beta < 1$,
for which  $\mathop{\rm Dom}(A^{1+ \delta}) \subseteq \mathop{\rm Dom}(f(A))$ and
\[
\| (-A + f(A)) \psi \| \leq \beta \| A^{1+ \delta} \psi \| .
\]
Set $g(A) = -A + f(A)$.

As in the previous section, we also obtain an approximation $w(t)$
through quasireversibility:
$w(t) = e^{(t-T)A}e^{Tf(A)}\chi$, where we replace the initial data
 $f$ by $\chi$, to avoid confusion.


\begin{theorem}[{\cite[Theorem 2]{AH2}}] \label{thmf}
 Let $A$ be a positive self-adjoint
operator acting on $\mathcal{H}$, let $f$ satisfy Condition
(A), and assume that there exists a constant
$\gamma$, independent of $\beta$, such that
$(g(A)\psi,\psi) \leq \gamma (\psi,\psi)$, for all
$\psi \in \mathcal{H}$.  If $u(t)$ and
$w(t)$ are solutions of \eqref{e2} and \eqref{e4}, respectively,
 and $\|u(T)\|\leq \tilde{M}$, then there exist constants $C$ and $M$,
independent of $\beta$,  such that for $0 \leq t < T$,
\[
\|u(t) - w(t)\| \leq C\beta^{1-t/T}M^{t/T}.
 \]
\end{theorem}

\noindent \textbf{Definition.}
  For a self-adjoint operator $A$, we define a  generalized heatlet
 to be the solution $\Psi_{n}^{j}$ of the abstract Cauchy problem
$\frac{du}{dt} = -Au$,  with initial data $\psi_{j,n}$.


 The next theorem follows in the same manner as
Theorem \ref{thm1b} in the previous section, using the realization
of $w(t)$ in terms of heatlets.

\begin{theorem} \label{thm2}
Let $\chi\in L^{2}(\mathbb{R})$, and
let  $c_{j,n}$ denote the wavelet coefficient of $\chi(x)$
attached to $\psi_{j,n}=2^{j/2}\psi(2^{j}x-n)$. Assume
that $u(t)$  is a stabilized solution of \eqref{e2}, where $A$ is a
positive self-adjoint operator on $ L^{2}(\mathbb{R})$, and that
$f$ satisfies Condition (A).  Then there exist constants $C$ and
$M$, independent of $\epsilon > 0$, such that
\[
 \| u(t)-  \sum_{j,n \in \mathbb{Z}} c_{j,n} e^{Tf(A)}\Psi^{h}_{j,n} (x,T-t)\|
\le C \epsilon^{1- \frac{t}{T}}M^{t/T}.
\]
 Thus, for small values of $\epsilon > 0$,
$\sum_{j,n \in \mathbb{Z}} c_{j,n} e^{Tf(A)}\Psi^{h}_{j,n} (x,T-t)$
is close to
$u(t)$ in $L^{2}(\mathbb{R})$, for $0 \leq t < T$.
\end{theorem}

\subsection*{Acknowledgements}
The authors gratefully
acknowledge the contributions of Professor Walter Huddell (Eastern
University) and Ayako Fukui (Bryn Mawr College) to this work.

\begin{thebibliography}{00}

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\end{thebibliography}

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