The Wavelet Digest Homepage
Return to the homepage
Search the complete Wavelet Digest database
Help about the Wavelet Digest mailing list
About the Wavelet Digest
The Digest The Community
 Latest Issue  Back Issues  Events  Gallery
The Wavelet Digest
   -> Volume 6, Issue 12

Preprint: Approximation by multi-wavelets
images/spacer.gifimages/spacer.gif Reply into Digest
Previous :: Next  
Author Message
Thierry BLU (

PostPosted: Mon Dec 01, 1997 11:24 am    
Subject: Preprint: Approximation by multi-wavelets
Reply with quote

#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:
in order to receive a postscript version of this preprint.

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

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

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 $L$.
All times are GMT + 1 Hour
Page 1 of 1

Jump to: 

disclaimer -
Powered by phpBB

This page was created in 0.024771 seconds : 18 queries executed : GZIP compression disabled