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   -> Volume 6, Issue 12


Preprint: Approximation by multi-wavelets
 
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Thierry BLU (thierry.blu@cnet.francetelecom.fr)
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PostPosted: Mon Dec 01, 1997 11:24 am    
Subject: Preprint: Approximation by multi-wavelets
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#7 Preprint: Approximation by multi-wavelets

We would like to make the following paper available to people working
in the fields of wavelets, multi-wavelets and approximation theory. It has
been submitted to Applied and Computational Harmonic Analysis, and revised
in October 1997.

Please contact Thierry BLU (e-mail: thierry.blu@cnet.francetelecom.fr)
in order to receive a postscript version of this preprint.

TITLE
Approximation Error for Quasi-Interpolators
and (Multi-) Wavelet Expansions

AUTHORS
Thierry BLU (France Telecom) and
Michael UNSER (Swiss Federal Institute of Technology, Lausanne)

ABSTRACT
We investigate the approximation properties of general polynomial
preserving operators that approximate a function into some scaled
subspace of $L^2$ via an appropriate sequence of inner products. In
particular, we consider integer shift-invariant approximations such as
those provided by splines and wavelets, as well as finite elements and
multi-wavelets which use multiple generators. We estimate the
approximation error as a function of the scale parameter $T$ when the
function to approximate is sufficiently regular. We then present a
generalized sampling theorem, a result that is rich enough to provide
tight bounds as well as asymptotic expansions of the approximation
error as a function of the sampling step $T$. Another more
theoretical consequence is the proof of a conjecture by Strang and
Fix, stating the equivalence between the order of a multi-wavelet
space and the order of a particular subspace generated by a single
function. Finally, we consider refinable generating functions and use
the two-scale relation to obtain explicit formulae for the
coefficients of the asymptotic development of the error. The leading
constants are easily computable and can be the basis for the
comparison of the approximation power of wavelet and multi-wavelet
expansions of a given order $L$.
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