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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Approximation with wave packets generated by a refinable function


Authors: Lasse Borup and Morten Nielsen
Journal: Proc. Amer. Math. Soc. 133 (2005), 2409-2418
MSC (2000): Primary 41A46; Secondary 41A17, 42C40
Published electronically: February 25, 2005
MathSciNet review: 2138884
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Abstract: We consider best $m$-term approximation in $L_p(\mathbb{R} ^d)$ with wave packets generated by a single refinable function. The main examples of wave packets are orthonormal wavelets, or more generally wavelet frames based on a multiresolution analysis (so-called framelets). The approximation classes associated with best $m$-term approximation in $L_p(\mathbb{R} ^d)$ for a large class of wave packets are completely characterized in terms of Besov spaces.

As an application of the main result, we show that for $m$-term approximation in $L_p(\mathbb{R} ^d)$ with elements from an oversampled version of a framelet system with compactly supported generators, the associated approximation classes turn out to be (essentially) Besov spaces.


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Additional Information

Lasse Borup
Affiliation: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg East, Denmark
Email: lasse@math.auc.dk

Morten Nielsen
Affiliation: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg East, Denmark
Email: mnielsen@math.auc.dk

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07778-6
PII: S 0002-9939(05)07778-6
Keywords: Refinable functions, nonlinear approximation, framelet systems, Jackson inequality, Bernstein inequality, Besov spaces
Received by editor(s): July 15, 2003
Received by editor(s) in revised form: April 14, 2004
Published electronically: February 25, 2005
Additional Notes: This work was supported in part by the Danish Technical Science Foundation, Grant no. 9701481
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.