Twenty-five years of greedy bases
Fernando Albiac, Jose L. Ansorena, Vladimir Temlyakov

TL;DR
This paper reviews twenty-five years of research on greedy bases, highlighting key developments, and presents twenty-five open problems to guide future investigations in approximation theory and functional analysis.
Contribution
It provides a comprehensive overview of the evolution of greedy bases theory and introduces a curated list of open problems for future research.
Findings
Rapid development of greedy bases theory since 1999
Wide application of greedy algorithms in numerical methods
Ongoing active research with many open questions
Abstract
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 (1999), no. 3, 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…
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Taxonomy
TopicsChemistry and Chemical Engineering · History and advancements in chemistry
