@techreport{Koster.Griebel:1998,
author = { F.~Koster and M.~Griebel},
title = { Orthogonal Wavelets on the Interval},
year = {1998},
institution = {SFB 256, Universit\"{a}t Bonn, Germany},
number = {No. 576},
annote = {report,unrefereed,CNRS},
ps = {http://wissrech.ins.uni-bonn.de/research/pub/koster/boundwave.ps.gz 1},
abstract = { In this paper we generalize the constructions
\cite{ChLi1}, \cite{CDV1} and \cite{MoPe1} of wavelets on
the interval. These schemes give boundary modifications of
compactly supported orthogonal wavelets $\psi \in L_2({\bf R})$ with $supp~\psi=[-N+1,N]$, where $N$ denotes the
number of vanishing moments of $\psi$. Our new scheme
overcomes this restrictive condition. Furthermore, the
constructions \cite{ChLi1}, \cite{CDV1}, \cite{MoPe1}
involve Gram matrices for explicit orthogonalization steps.
These Gram matrices tend to be very ill conditioned for
increasing $N$. It is shown that for the present scheme the
condition numbers of the resulting matrices are smaller by
orders of magnitude. Therefore our scheme is numerically
more stable. We also point out how wavelets can be obtained
satisfying homogeneous Dirichlet or Neumann conditions. In
addition, we deal with the requirement of the discrete
wavelet transform on the interval to find, e.g. from nodal
values of $u \in L_2([0,1])$, an approximation $\tilde{u}$
by a linear combination of dilated scaling functions. We
present and compare two methods. One method has already
been used in \cite{ChLi1} and \cite{MoPe1}. Our experiments
show that this scheme is not suited for data compression on
the interval. The other method however is designed for data
compression applications and leads to cheap and very well
conditioned approximation mapping}
}