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Sparse approximation using new greedy-like bases in superreflexive spaces

Volume 271 / 2023

Fernando Albiac, José L. Ansorena, Miguel Berasategui Studia Mathematica 271 (2023), 321-346 MSC: Primary 41A65; Secondary 41A46, 41A17, 46B15, 46B45. DOI: 10.4064/sm220506-3-2 Published online: 5 April 2023


This paper is devoted to theoretical aspects of optimality of sparse approximation. We undertake a quantitative study of new types of greedy-like bases that have recently arisen in the context of non-linear $m$-term approximation in Banach spaces as a generalization of the properties that characterize almost greedy bases, i.e., quasi-greediness and democracy. As a means to compare the efficiency of these new bases with already existing ones in regard to the implementation of the Thresholding Greedy Algorithm, we place emphasis on obtaining estimates for their sequence of unconditionality parameters. Using an enhanced version of the original Dilworth–Kalton–Kutzarova method (2003) for building almost greedy bases, we manage to construct bidemocratic bases whose unconditionality parameters satisfy significantly worse estimates than almost greedy bases even in Hilbert spaces.


  • Fernando AlbiacDepartment of Mathematics, Statistics,
    and Computer Science – InaMat2
    Universidad Pública de Navarra
    Campus de Arrosadía
    Pamplona, 31006 Spain
  • José L. AnsorenaDepartment of Mathematics
    and Computer Science
    Universidad de La Rioja
    Logroño, 26004 Spain
  • Miguel BerasateguiIMAS – UBA – CONICET – Pab I
    Facultad de Ciencias Exactas y Naturales
    Universidad de Buenos Aires
    1428 Buenos Aires, Argentina

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