Characterization of Weight-Semi-greedy Bases

One classical result in greedy approximation theory is that almost-greedy and semi-greedy bases are equivalent in the context of Schauder bases in Banach spaces with finite cotype. This result was proved by Dilworth et al. (Studia Math 159:67–101, 2003) and, recently, in the study of Berná (J Math A...

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Bibliographic Details
Published in:The Journal of fourier analysis and applications Vol. 26; no. 1
Main Author: Berná, Pablo Manuel
Format: Journal Article
Language:English
Published: New York Springer US 01.02.2020
Springer
Springer Nature B.V
Subjects:
Abstract Harmonic Analysis
Approximations and Expansions
Banach spaces
Democracy
Equivalence
Fourier Analysis
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Partial Differential Equations
Signal,Image and Speech Processing
Weight
Weight-almost-greedy bases
46B15
Thresholding greedy algorithm
Semi-greedy bases
41A65
ISSN:1069-5869, 1531-5851
Online Access:Get full text
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Summary:One classical result in greedy approximation theory is that almost-greedy and semi-greedy bases are equivalent in the context of Schauder bases in Banach spaces with finite cotype. This result was proved by Dilworth et al. (Studia Math 159:67–101, 2003) and, recently, in the study of Berná (J Math Anal Appl 470:218–225, 2019), the author proved that the condition of finite cotype can be removed. One of the results in this paper is to show that the condition of Schauder can be relaxed using the ρ -admissibility, notion introduced in the study of Berná et al. (Rev Mat Complut; https://doi.org/10.1007/s13163-019-00328-9). On the other hand, in the study of Dilworth et al. (Tr Mat Inst Steklova 303: 120–141, 2018), the authors extend the notion of semi-greediness to the context of weights and proved the following: if w is a weight and B is a Schauder basis in a Banach space X with finite cotype, then w -semi-greediness and w -almost-greediness are equivalent notions. Here, we prove the same characterization but removing the condition of finite cotype. Also, we give some results improving the behavior of some constants in the relation between w -greedy-type bases and some w -democracy properties.
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ISSN:1069-5869
1531-5851
DOI:10.1007/s00041-020-09727-9