Lebesgue-Type Inequalities for Quasi-greedy Bases

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...

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Bibliographic Details
Published in:Constructive approximation Vol. 38; no. 3; pp. 447 - 470
Main Authors: Garrigós, Gustavo, Hernández, Eugenio, Oikhberg, Timur
Format: Journal Article
Language:English
Published: Boston Springer US 01.12.2013
Subjects:
Analysis
Mathematics
Mathematics and Statistics
Numerical Analysis
41A46
46B15
Non-linear approximation
Bounded variation
41A65
Thresholding greedy algorithm
Quasi-greedy bases
Lebesgue-type inequalities
Democracy functions
41A17
ISSN:0176-4276, 1432-0940
Online Access:Get full text
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Summary: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 (log N ). We show with various examples that these bounds are actually attained.
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-013-9209-z