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Strong Partially Greedy Bases and Lebesgue-Type Inequalities

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Abstract

In this paper, we continue the study of Lebesgue-type inequalities for greedy algorithms. We introduce the notion of strong partially greedy Markushevich bases and study the Lebesgue-type parameters associated with them. We prove that this property is equivalent to that of being conservative and quasi-greedy, extending a similar result given in Dilworth et al. (Constr Approx 19:575–597, 2003) for Schauder bases. We also give a characterization of 1-strong partial greediness, following the study started in Albiac and Ansorena (Rev Matem Compl 30(1):13–24, 2017), Albiac and Wojtaszczyk (J Approx Theory 138:65–86, 2006).

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Notes

  1. We use the notation \(\Vert G\Vert =\sup _{x\not =0}\Vert G(x)\Vert /\Vert x\Vert \) and \(\Vert I-G\Vert =\sup _{x\not =0}\Vert x-G(x)\Vert /\Vert x\Vert \), even if \(G:\mathbb X\rightarrow {\mathbb {X}}\) is a nonlinear map.

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Authors and Affiliations

  1. IMAS - UBA - CONICET - Pab I, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428, Buenos Aires, Argentina

    M. Berasategui

  2. Departamento de Matemática Aplicada y Estadística, Facultad de Ciencias Económicas y Empresariales, Universidad San Pablo-CEU, CEU Universities, 28003, Madrid, Spain

    P. M. Berná

  3. Departamento de Matemática, Universidad de San Andrés, Vito Duma 284, 1644, Buenos Aires, Victoria, Argentina

    S. Lassalle

  4. IMAS - CONICET, Buenos Aires, Argentina

    S. Lassalle

Authors
  1. M. Berasategui
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  2. P. M. Berná
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  3. S. Lassalle
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Corresponding author

Correspondence to P. M. Berná.

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Communicated by Vladimir Temlyakov.

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The first and third authors were supported in part by CONICET PIP 11220130100483, ANPCyT PICT-2015-2299 and PICT-2018-04104. The second author was partially supported by the grants PID2019-105599GB-I00, MTM-2016-76566-P (Agencia Española de Investigación), 20906/PI/18 from Fundación Séneca (Región de Murcia, Spain) and has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 777822. The third author was also supported by PAI-UdeSA 2019.

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Berasategui, M., Berná, P.M. & Lassalle, S. Strong Partially Greedy Bases and Lebesgue-Type Inequalities. Constr Approx 54, 507–528 (2021). https://doi.org/10.1007/s00365-021-09531-8

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  • DOI: https://doi.org/10.1007/s00365-021-09531-8

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