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

Pablo M. Berná; Oscar Blasco; Gustavo Garrigós; Eugenio Hernández; Timur Oikhberg

doi:10.1007/s00365-018-9415-9

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

Pablo M. Berná, Oscar Blasco, Gustavo Garrigós, Eugenio Hernández, Timur Oikhberg

Mathematics

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

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.

Original language	English (US)
Pages (from-to)	415-451
Number of pages	37
Journal	Constructive Approximation
Volume	48
Issue number	3
DOIs	https://doi.org/10.1007/s00365-018-9415-9
State	Published - Dec 1 2018

Keywords

Biorthogonal system
Discrete Lorentz space
Greedy algorithm
Lebesgue-type inequality
Non-linear approximation
Quasi-greedy basis

ASJC Scopus subject areas

Analysis
General Mathematics
Computational Mathematics

Online availability

10.1007/s00365-018-9415-9

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title = "Embeddings and Lebesgue-Type Inequalities for the Greedy Algorithm in Banach Spaces",

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.",

keywords = "Biorthogonal system, Discrete Lorentz space, Greedy algorithm, Lebesgue-type inequality, Non-linear approximation, Quasi-greedy basis",

author = "Bern{\'a}, {Pablo M.} and Oscar Blasco and Gustavo Garrig{\'o}s and Eugenio Hern{\'a}ndez and Timur Oikhberg",

note = "Funding Information: 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 (Fundaci{\'o}n S{\'e}neca, Regi{\'o}n de Murcia, Spain). The last author{\textquoteright}s research is partially supported by the Simons Foundation travel award 210060. Publisher Copyright: {\textcopyright} 2018, Springer Science+Business Media, LLC, part of Springer Nature.",

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doi = "10.1007/s00365-018-9415-9",

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T1 - Embeddings and Lebesgue-Type Inequalities for the Greedy Algorithm in Banach Spaces

AU - Berná, Pablo M.

AU - Blasco, Oscar

AU - Garrigós, Gustavo

AU - Hernández, Eugenio

AU - Oikhberg, Timur

N1 - Funding Information: 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. Publisher Copyright: © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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.

AB - 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.

KW - Biorthogonal system

KW - Discrete Lorentz space

KW - Greedy algorithm

KW - Lebesgue-type inequality

KW - Non-linear approximation

KW - Quasi-greedy basis

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

Abstract

Keywords

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