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   -> Volume 7, Issue 5

Preprint: Two preprints from I. Selesnics on multiwavelets.
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Ivan Selesnick (

PostPosted: Sat May 09, 1998 1:45 am    
Subject: Preprint: Two preprints from I. Selesnics on multiwavelets.
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#9 Preprint: Two preprints from I. Selesnics on multiwavelets.


Cardinal Multiwavelets and the Sampling Theorem

This paper considers the classical Shannon sampling theorem in
multiresolution spaces with scaling functions as interpolants. As
discussed by Xia and Zhang, for an orthogonal scaling function to
support such a sampling theorem, the scaling function must be
cardinal. They also showed that the only orthogonal scaling function
that is both cardinal and of compact support is the Haar function,
which has approximation order 1 only. This paper addresses the same
question, but in the multiwavelet context, where the situation is
different. This paper presents the construction of orthogonal
multiscaling functions that are simultaneously cardinal, of compact
support, and have approximation order K>1. The scaling functions
thereby support a Shannon-like sampling theorem.


Multiwavelets with Extra Approximation Properties

Because multiwavelet bases normally lack important properties
traditional wavelet bases (of equal approximation order)
possess, the discrete multiwavelet transform is less useful
for signal processing, unless preceded by a preprocessing step.
This paper examines properties of orthogonal multiwavelet bases,
with approximation order K>1, that possess those properties
normally absent. For these ``balanced" bases (after Lebrun and
Vetterli) prefiltering can be avoided. Using results regarding
M-band wavelet bases, it has been found that balanced multiwavelet
bases can be characterized in terms of the divisibility of certain
transfer functions by powers of (z^{-2r}-1)/(z^{-1}-1). The paper
also presents a balanced version of the DGHM basis --- the scaling
functions are simultaneously symmetric, orthogonal and of compact

Ivan Selesnick
Polytechnic University
Brooklyn, NY
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