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Lebesgue-Type Inequalities for Greedy Approximation

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New Trends in Applied Harmonic Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

The estimation of a solution that satisfies a certain optimality criterion is the goal of many engineering applications. In many contemporary problems one would often like to obtain an approximation of the solution using a sparse linear combination of elements of a given system (dictionary). This paper is devoted to theoretical aspects of sparse approximation. The main motivation for the study of sparse approximation is that many real world signals can be well approximated by sparse ones. Sparse approximation automatically implies a need for nonlinear approximation, in particular, for greedy approximation. The paper is a survey on results in constructive sparse approximation. Two directions are discussed here: (1) Lebesgue-type inequalities for thresholding greedy algorithms with respect to bases, and (2) Lebesgue-type inequalities for Chebyshev Greedy Algorithms with respect to a special class of dictionaries. In particular, these algorithms provide constructive sparse approximation with respect to the trigonometric system. The technique used is based on fundamental results from the theory of greedy approximation. Results in the direction (2) are based on deep methods developed recently in compressed sensing. We present some of these results with detailed proofs.

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Acknowledgements

This research was supported by NSF grant DMS-1160841.

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Authors and Affiliations

  1. University of South Carolina and Steklov Institute of Mathematics, Columbia, USA

    Vladimir Temlyakov

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  1. Vladimir Temlyakov
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Correspondence to Vladimir Temlyakov .

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Editors and Affiliations

  1. Department of Mathematics, Vanderbilt University, Nashville, Tennessee, USA

    Akram Aldroubi

  2. Depto. de Matemática, Univ. de Buenos Ai, IMAS - UBA/CONICET, Buenos Aires, Argentina

    Carlos Cabrelli

  3. UFR de Sciences et Technologie, Universite Paris Est Creteil, Creteil Cedex, Paris, France

    Stephane Jaffard

  4. Depto. de Matemática, Univ. de Buenos Ai, IMAS, UBA/CONICET, Buenos Aires, Argentina

    Ursula Molter

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Temlyakov, V. (2016). Lebesgue-Type Inequalities for Greedy Approximation. In: Aldroubi, A., Cabrelli, C., Jaffard, S., Molter, U. (eds) New Trends in Applied Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-27873-5_4

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