Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation

  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 = { 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}