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   -> Volume 1, Issue 9

Preprints available.
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Author Message
Wim Sweldens, Department of Mathematics, University of South Carolina

PostPosted: Tue Dec 01, 1992 12:00 am    
Subject: Preprints available.
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Preprints available.

The following two papers are available.

1. Quadrature Formulae for the Calculation of the Wavelet Decomposition

Wim Sweldens and Robert Piessens


In many applications concerning wavelets, inner products of functions
f(x) with wavelets and scaling functions have to be calculated.
This paper involves the calculation of these inner products from
function evaluations of f(x).
Firstly, one point quadrature formulae are presented. Their accuracy
is compared for different classes of wavelets. Therefore the relationship
between the scaling function phi(x), its values at the integers and
the scaling parameters h_k is investigated.
Secondly, multiple point quadrature formulae are constructed. A method
to solve the nonlinear system coming from this construction is presented.
Since the construction of multiple point formulae using monomials is
ill-conditioned, a modified, well-conditioned construction using
Chebyshev polynomials is presented.

Status: Preprint Department of Computer Science, K.U.Leuven, Belgium, submitted

Can be retrieved as a postscript file using anonymous ftp to, file /pub/wavelet/papers/

2. Asymptotic error expansions for wavelet approximations of smooth functions

Wim Sweldens and Robert Piessens


This paper deals with asymptotic error expansions of orthogonal wavelet
approximations of smooth functions. Two formulae are derived and compared.
As already known, the error decays as O(2^(-jN)) where j is the multi-
resolution level and N is the number of vanishing wavelet moments.
It is shown that the most significant term of the error expansion is
proportional to the N-th derivative of the function multiplied with
an oscillating function. This result is used to derive asymptotic
interpolating properties of the wavelet approximation. Also a numerical
extrapolation scheme based on the multiresolution analysis is presented.

Status: Report TW 164, Department of Computer Science, K.U.Leuven,
Belgium, submitted.

Can be retrieved as a postscript file using anonymous ftp to, file /pub/wavelet/papers/

Wim Sweldens
Department of Mathematics
University of South Carolina
Columbia SC 29208
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