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Embeddings and Lebesgue-Type Inequalities for the Greedy Algorithm in Banach Spaces

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Abstract

We obtain Lebesgue-type inequalities for the greedy algorithm for arbitrary complete seminormalized biorthogonal systems in Banach spaces. The bounds are given only in terms of the upper democracy functions of the basis and its dual. We also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces. Finally, the asymptotic optimality of these inequalities is illustrated in various examples of not necessarily quasi-greedy bases.

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Notes

  1. This is shown in [12] for the system \(\{{\mathbf {e}}_n - ({\mathbf {e}}_{2n} +{\mathbf {e}}_{2n+1})/2\}_{n=1}^\infty \), but the same arguments, with obvious modifications, work for the basis in (8.5).

  2. Slightly more elaborate computations actually lead to \(d(N)=N+1\).

  3. Slightly more elaborate computations, using the definition of \({\mathbf {y}}_n\) in (8.6), actually give \(d^*_{c_0}(N)=1\), and also \(D^*_{c_0}(N)=\log _2(N+1)\) if \(N+1=2^n\).

  4. We thank an anonymous referee for pointing out this fact.

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Acknowledgements

The research of the first, third and fourth authors is supported by Grants MTM-2013-40495-P and MTM-2016-76566-P (Spain). The research of the second author is supported by Grant MTM-2014-53009-P (Spain). The research of the first, third, and fourth authors is also partially supported by grant 19368/PI/14 (Fundación Séneca, Región de Murcia, Spain). The last author’s research is partially supported by the Simons Foundation travel award 210060.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Pablo M. Berná & Eugenio Hernández

  2. Departamento de Análisis Matemático, Universidad de Valencia, Campus de Burjassot, 46100, Valencia, Spain

    Oscar Blasco

  3. Departamento de Matemáticas, Universidad de Murcia, 30100, Murcia, Spain

    Gustavo Garrigós

  4. Department of Mathematics, University of Illinois, Urbana, IL, USA

    Timur Oikhberg

Authors
  1. Pablo M. Berná
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  2. Oscar Blasco
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  3. Gustavo Garrigós
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  4. Eugenio Hernández
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  5. Timur Oikhberg
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Corresponding author

Correspondence to Eugenio Hernández.

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Communicated by Vladimir N. Temlyakov.

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Berná, P.M., Blasco, O., Garrigós, G. et al. Embeddings and Lebesgue-Type Inequalities for the Greedy Algorithm in Banach Spaces. Constr Approx 48, 415–451 (2018). https://doi.org/10.1007/s00365-018-9415-9

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