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Weak Convergence of Greedy Algorithms in Banach Spaces

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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.

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Authors and Affiliations

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

    S. J. Dilworth & V. N. Temlyakov

  2. Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Denka Kutzarova

  3. Department of Mathematics, Grinnell College, Iowa, 50112, USA

    Karen L. Shuman

  4. Institute of Applied Mathematics, Warsaw University, Banacha 2, 02-097, Warsaw, Poland

    P. Wojtaszczyk

Authors
  1. S. J. Dilworth
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  2. Denka Kutzarova
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  3. Karen L. Shuman
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  4. V. N. Temlyakov
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  5. P. Wojtaszczyk
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Corresponding author

Correspondence to S. J. Dilworth.

Additional information

Communicated by Ronald A. DeVore.

The research of the first author was supported by NSF Grant DMS 0701552. The research of the third author was supported by the Grinnell College Committee for the Support of Faculty Scholarship. The research of the fourth author was supported by NSF Grant DMS 0554832 and by ONR Grant N00014-91-J1343. The research of the fifth author was supported by KBN grant 1 PO03A 038 27 and the Foundation for the Polish Science. The fifth author thanks the Industrial Mathematics Institute of the University of South Carolina and Prof. R. DeVore and Prof. V.N. Temlyakov personally for their hospitality. The first and second authors were supported by the Concentration Week on Frames, Banach Spaces, and Signals Processing and by the Workshop in Analysis and Probability at Texas A&M University in 2006.

Current address of D. Kutzarova: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

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Dilworth, S.J., Kutzarova, D., Shuman, K.L. et al. Weak Convergence of Greedy Algorithms in Banach Spaces. J Fourier Anal Appl 14, 609–628 (2008). https://doi.org/10.1007/s00041-008-9034-0

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  • DOI: https://doi.org/10.1007/s00041-008-9034-0

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