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Weak Convergence of a Greedy Algorithm and the WN-Property

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Abstract

We study the weak convergence of a greedy algorithm of approximation by a given set in a Banach space. It is proved that the greedy algorithm of approximation by a strongly norm-reducing set in a uniformly smooth Banach space with the WN-property weakly converges. In an arbitrary separable Banach space without the WN-property, we construct an example of a strongly norm-reducing set such that the greedy algorithm of approximation by this set does not weakly converge for some initial element.

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References

  1. S. J. Dilworth, D. Kutzarova, K. L. Shuman, V. N. Temlyakov, and P. Wojtaszczyk, “Weak convergence of greedy algorithms in Banach spaces,” J. Fourier Anal. Appl. 14 (5–6), 609–628 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  2. V. Temlyakov, Greedy Approximation (Cambridge Univ. Press, Cambridge, 2011).

    Book  MATH  Google Scholar 

  3. V. N. Temlyakov, “Nonlinear methods of approximation,” Found. Comput. Math. 3 (1), 33–107 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  4. V. N. Temlyakov, “Greedy approximation in Banach spaces,” in Banach Spaces and Their Applications in Analysis (de Gruyter, Berlin, 2007), pp. 193–208.

    MATH  Google Scholar 

  5. E. D. Livshits, “Convergence of greedy algorithms in Banach spaces,” Math. Notes 73 (3), 342–358 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  6. P. A. Borodin, “Greedy approximation by arbitrary sets,” Izv. Math. 84 (2), 246–261 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Diestel, Geometry of Banach Spaces (Springer, Berlin–New York, 1975).

    Book  MATH  Google Scholar 

  8. V. I. Bogachev and O. G. Smolyanov, Real and Functional Analysis (Regular and Chaotic Dynamics, Moscow–Izhevsk, 2020) [in Russian].

    Book  MATH  Google Scholar 

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Funding

This work was supported by the Russian Science Foundation under grant no. 22-21-00415.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    P. A. Borodin

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    P. A. Borodin

Authors
  1. P. A. Borodin
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Correspondence to P. A. Borodin.

Additional information

Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 483–488 https://doi.org/10.4213/mzm13667.

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Borodin, P.A. Weak Convergence of a Greedy Algorithm and the WN-Property. Math Notes 113, 475–479 (2023). https://doi.org/10.1134/S0001434623030197

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  • DOI: https://doi.org/10.1134/S0001434623030197

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