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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| Published in: | Journal of approximation theory Vol. 307; p. 106141 |
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| ISSN: | 0021-9045 |
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| 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 (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. |
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| AbstractList | 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. |
| ArticleNumber | 106141 |
| Author | Ansorena, José L. Albiac, Fernando Temlyakov, Vladimir |
| Author_xml | – sequence: 1 givenname: Fernando orcidid: 0000-0001-7051-9279 surname: Albiac fullname: Albiac, Fernando email: fernando.albiac@unavarra.es organization: Department of Mathematics, Statistics, and Computer Sciences–InaMat2, Universidad Pública de Navarra, Campus de Arrosadía, Pamplona, 31006, Spain – sequence: 2 givenname: José L. surname: Ansorena fullname: Ansorena, José L. email: joseluis.ansorena@unirioja.es organization: Department of Mathematics and Computer Sciences, Universidad de La Rioja, Logroño, 26004, Spain – sequence: 3 givenname: Vladimir surname: Temlyakov fullname: Temlyakov, Vladimir email: temlyak@math.sc.edu organization: Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA |
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| Keywords | 41A46 46B15 41A25 41A65 Almost greedy Thresholding greedy algorithm Greedy 41A17 Nonlinear approximation And quasi-greedy basis |
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