Lebesgue-Type Inequalities for Quasi-greedy Bases

Gustavo Garrigós; Eugenio Hernández; Timur Oikhberg

doi:10.1007/s00365-013-9209-z

Lebesgue-Type Inequalities for Quasi-greedy Bases

Gustavo Garrigós, Eugenio Hernández, Timur Oikhberg

Mathematics

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

Abstract

We show that, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy N-term approximation ∥x-G _N x∥ and the best N-term approximation σ _N(x) is controlled by max{μ(N),k _N}, where μ(N) and k _N are well-known constants that quantify the democracy and conditionality of the basis. In particular, for democratic bases this bound is O(logN). We show with various examples that these bounds are actually attained.

Original language	English (US)
Pages (from-to)	447-470
Number of pages	24
Journal	Constructive Approximation
Volume	38
Issue number	3
DOIs	https://doi.org/10.1007/s00365-013-9209-z
State	Published - Dec 2013

Keywords

Bounded variation
Democracy functions
Lebesgue-type inequalities
Non-linear approximation
Quasi-greedy bases
Thresholding greedy algorithm

ASJC Scopus subject areas

Analysis
General Mathematics
Computational Mathematics

Online availability

10.1007/s00365-013-9209-z

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abstract = "We show that, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy N-term approximation ∥x-G N x∥ and the best N-term approximation σ N(x) is controlled by max{μ(N),k N}, where μ(N) and k N are well-known constants that quantify the democracy and conditionality of the basis. In particular, for democratic bases this bound is O(logN). We show with various examples that these bounds are actually attained.",

keywords = "Bounded variation, Democracy functions, Lebesgue-type inequalities, Non-linear approximation, Quasi-greedy bases, Thresholding greedy algorithm",

author = "Gustavo Garrig{\'o}s and Eugenio Hern{\'a}ndez and Timur Oikhberg",

note = "Funding Information: First author partially supported by Grants MTM2010-16518 and MTM2011-25377 (Spain). Second author supported by Grant MTM2010-16518 (Spain). Third author supported by Simons Foundation Travel Grant 210060, and by a COR grant from University of California System. ",

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AU - Garrigós, Gustavo

AU - Hernández, Eugenio

AU - Oikhberg, Timur

N1 - Funding Information: First author partially supported by Grants MTM2010-16518 and MTM2011-25377 (Spain). Second author supported by Grant MTM2010-16518 (Spain). Third author supported by Simons Foundation Travel Grant 210060, and by a COR grant from University of California System.

PY - 2013/12

Y1 - 2013/12

N2 - We show that, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy N-term approximation ∥x-G N x∥ and the best N-term approximation σ N(x) is controlled by max{μ(N),k N}, where μ(N) and k N are well-known constants that quantify the democracy and conditionality of the basis. In particular, for democratic bases this bound is O(logN). We show with various examples that these bounds are actually attained.

AB - We show that, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy N-term approximation ∥x-G N x∥ and the best N-term approximation σ N(x) is controlled by max{μ(N),k N}, where μ(N) and k N are well-known constants that quantify the democracy and conditionality of the basis. In particular, for democratic bases this bound is O(logN). We show with various examples that these bounds are actually attained.

KW - Bounded variation

KW - Democracy functions

KW - Lebesgue-type inequalities

KW - Non-linear approximation

KW - Quasi-greedy bases

KW - Thresholding greedy algorithm

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Lebesgue-Type Inequalities for Quasi-greedy Bases

Abstract

Keywords

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