Characterization of 1-almost greedy bases

This article closes the cycle of characterizations of greedy-like bases in the “isometric” case initiated in Albiac and Wojtaszczyk (J. Approx. Theory 138(1):65–86, 2006 ) with the characterization of 1-greedy bases and continued in Albiac and Ansorena (J. Approx. Theory 201:7–12, 2016 ) with the ch...

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Bibliographic Details
Published in:Revista matemática complutense Vol. 30; no. 1; pp. 13 - 24
Main Authors: Albiac, F., Ansorena, J. L.
Format: Journal Article
Language:English
Published: Milan Springer Milan 01.01.2017
Springer Nature B.V
Subjects:
Algebra
Analysis
Applications of Mathematics
Banach spaces
Geometry
Mathematics
Mathematics and Statistics
Topology
Almost greedy basis
46B15
41A65
Quasi-greedy basis
Thresholding greedy algorithm
Unconditional basis
Property (A)
ISSN:1139-1138, 1988-2807
Online Access:Get full text
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Summary:This article closes the cycle of characterizations of greedy-like bases in the “isometric” case initiated in Albiac and Wojtaszczyk (J. Approx. Theory 138(1):65–86, 2006 ) with the characterization of 1-greedy bases and continued in Albiac and Ansorena (J. Approx. Theory 201:7–12, 2016 ) with the characterization of 1-quasi-greedy bases. Here we settle the problem of providing a characterization of 1-almost greedy bases in Banach spaces. We show that a basis in a Banach space is almost greedy with almost greedy constant equal to 1 if and only if it has Property (A). This fact permits now to state that a basis is 1-greedy if and only if it is 1-almost greedy and 1-quasi-greedy. As a by-product of our work we also provide a tight estimate of the almost greedy constant of a basis in the non-isometric case.
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ISSN:1139-1138
1988-2807
DOI:10.1007/s13163-016-0204-3