Weak convergence of greedy algorithms in banach spaces

S. J. Dilworth; Denka Kutzarova; Karen L. Shuman; V. N. Temlyakov; P. Wojtaszczyk

doi:10.1007/s00041-008-9034-0

Weak convergence of greedy algorithms in banach spaces

S. J. Dilworth, Denka Kutzarova, Karen L. Shuman, V. N. Temlyakov, P. Wojtaszczyk

Research output: Contribution to journal › Article › peer-review

Abstract

We study the convergence of certain greedy algorithms in Banach spaces. We introduce the WN property for Banach spaces and prove that the algorithms converge in the weak topology for general dictionaries in uniformly smooth Banach spaces with the WN property. We show that reflexive spaces with the uniform Opial property have the WN property. We show that our results do not extend to algorithms which employ a 'dictionary dual' greedy step.

Original language	English (US)
Pages (from-to)	609-628
Number of pages	20
Journal	Journal of Fourier Analysis and Applications
Volume	14
Issue number	5-6
DOIs	https://doi.org/10.1007/s00041-008-9034-0
State	Published - Dec 2008
Externally published	Yes

Keywords

Greedy algorithms
Uniform Opial property
Uniformly smooth Banach spaces
Weak convergence

ASJC Scopus subject areas

Analysis
General Mathematics
Applied Mathematics

Online availability

10.1007/s00041-008-9034-0

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AU - Dilworth, S. J.

AU - Kutzarova, Denka

AU - Shuman, Karen L.

AU - Temlyakov, V. N.

AU - Wojtaszczyk, P.

PY - 2008/12

Y1 - 2008/12

N2 - We study the convergence of certain greedy algorithms in Banach spaces. We introduce the WN property for Banach spaces and prove that the algorithms converge in the weak topology for general dictionaries in uniformly smooth Banach spaces with the WN property. We show that reflexive spaces with the uniform Opial property have the WN property. We show that our results do not extend to algorithms which employ a 'dictionary dual' greedy step.

AB - We study the convergence of certain greedy algorithms in Banach spaces. We introduce the WN property for Banach spaces and prove that the algorithms converge in the weak topology for general dictionaries in uniformly smooth Banach spaces with the WN property. We show that reflexive spaces with the uniform Opial property have the WN property. We show that our results do not extend to algorithms which employ a 'dictionary dual' greedy step.

KW - Greedy algorithms

KW - Uniform Opial property

KW - Uniformly smooth Banach spaces

KW - Weak convergence

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ER -

Weak convergence of greedy algorithms in banach spaces

Abstract

Keywords

ASJC Scopus subject areas

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