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SPARSE APPROXIMATION AND RECOVERY BY GREEDY ALGORITHMS IN BANACH SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices Approximations and expansions

Published online by Cambridge University Press:  27 May 2014

V. N. TEMLYAKOV
V. N. TEMLYAKOV*
Affiliation:
University of South Carolina, USA Steklov Institute of Mathematics, Russia; temlyakovusc@gmail.com

Abstract

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We study sparse approximation by greedy algorithms. We prove the Lebesgue-type inequalities for the weak Chebyshev greedy algorithm (WCGA), a generalization of the weak orthogonal matching pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. The results are proved for redundant dictionaries satisfying certain conditions. Then we apply these general results to the case of bases. In particular, we prove that the WCGA provides almost optimal sparse approximation for the trigonometric system in $L_p$, $2\le p<\infty $.

MSC classification

Secondary: 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 41A46: Approximation by arbitrary nonlinear expressions; widths and entropy 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e12
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author 2014

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