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Simultaneous approximation by greedy algorithms

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Abstract

We study nonlinear m-term approximation with regard to a redundant dictionary \(\mathcal {D}\) in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each f∈H and any dictionary \(\mathcal {D}\) an expansion into a series

$$f=\sum_{j=1}^{\infty}c_{j}(f)\varphi_{j}(f),\quad\varphi_{j}(f)\in \mathcal {D},\ j=1,2,\ldots,$$

with the Parseval property: ‖f‖2=∑j|cj(f)|2. Following the paper of A. Lutoborski and the second author we study analogs of the above expansions for a given finite number of functions f1,. . .,fN with a requirement that the dictionary elements φj of these expansions are the same for all fi, i=1,. . .,N. We study convergence and rate of convergence of such expansions which we call simultaneous expansions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    D. Leviatan

  2. Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA

    V. N. Temlyakov

Authors
  1. D. Leviatan
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  2. V. N. Temlyakov
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Additional information

Communicated by Yuesheng Xu

Dedicated to our colleague and friend Dr. Charles Micchelli on his 60th birthday

Mathematics subject classifications (2000)

primary 41A65; secondary 41A25, 41A46, 46B20.

D. Leviatan: Part of this work was done while the first author visited the University of South Carolina in January 2003.

V.N. Temlyakov: This research was supported by the National Science Foundation Grant DMS 0200187 and by ONR Grant N00014-96-1-1003.

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Leviatan, D., Temlyakov, V.N. Simultaneous approximation by greedy algorithms. Adv Comput Math 25, 73–90 (2006). https://doi.org/10.1007/s10444-004-7613-4

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  • DOI: https://doi.org/10.1007/s10444-004-7613-4

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