Twenty-five years of greedy bases

Although the basic idea behind the concept of a greedy basis had been around for some time, the formal development of a theory of greedy bases was initiated in 1999 with the publication of the article [S.V. Konyagin and V.N. Temlyakov, A remark on greedy approximation in Banach spaces, East J. Appro...

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Bibliographic Details
Published in:Journal of approximation theory Vol. 307; p. 106141
Main Authors: Albiac, Fernando, Ansorena, José L., Temlyakov, Vladimir
Format: Journal Article
Language:English
Published: Elsevier Inc 01.05.2025
Subjects:
Almost greedy
And quasi-greedy basis
Greedy
Nonlinear approximation
Thresholding greedy algorithm
41A46
46B15
41A25
41A65
Almost greedy
Thresholding greedy algorithm
Greedy
41A17
Nonlinear approximation
And quasi-greedy basis
ISSN:0021-9045
Online Access:Get full text
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Summary:Although the basic idea behind the concept of a greedy basis had been around for some time, the formal development of a theory of greedy bases was initiated in 1999 with the publication of the article [S.V. Konyagin and V.N. Temlyakov, A remark on greedy approximation in Banach spaces, East J. Approx. 5 (3) (1999), 365-379]. The theoretical simplicity of the thresholding greedy algorithm became a model for a procedure widely used in numerical applications and the subject of greedy bases evolved very rapidly from the point of view of approximation theory. The idea of studying greedy bases and related greedy algorithms attracted also the attention of researchers with a classical Banach space theory background. From the more abstract point of functional analysis, the theory of greedy bases and its derivates evolved very fast as many fundamental results were discovered and new ramifications branched out. Hundreds of papers on greedy-like bases and several monographs have been written since the appearance of the aforementioned foundational paper. After twenty-five years, the theory is very much alive and it continues to be a very active research topic both for functional analysts and for researchers interested in the applied nature of nonlinear approximation alike. This is why we believe it is a good moment to gather a selection of 25 open problems (one per year since 1999!) whose solution would contribute to advance the state of art of this beautiful topic.
ISSN:0021-9045
DOI:10.1016/j.jat.2024.106141