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Greedy approximation for biorthogonal systems in quasi-Banach spaces

Volume 560 / 2021

Fernando Albiac, José L. Ansorena, Pablo M. Berná, Przemysław Wojtaszczyk Dissertationes Mathematicae 560 (2021), 1-88 MSC: Primary 53A55, 53B25, 53A15, 53A04, 53A05, 58K50; Secondary 53-08, 16W22, 14R20, 22E05, 35B06. DOI: 10.4064/dm817-11-2020 Published online: 29 April 2021


The general problem addressed in this work is the development of a systematic study of the thresholding greedy algorithm for general biorthogonal systems %(also known as Markushevich bases) in quasi-Banach spaces from a functional-analytic point of view. If $(\boldsymbol{x}_n,\boldsymbol{x}_n^*)_{n=1}^\infty$ is a biorthogonal system in $\boldsymbol{X}$ then for each $x\in \boldsymbol{X}$ we have a formal expansion $\sum_{n=1}^\infty \boldsymbol{x}_n^*(x)\boldsymbol{x}_n$. The thresholding greedy algorithm (with threshold $\varepsilon \gt 0$) applied to $x$ is formally defined as $\sum_{\{n \colon |\boldsymbol{x}_n^*(x)|\geq \varepsilon\}} \boldsymbol{x}_n^*(x) \boldsymbol{x}_n$. The properties of this operator give rise to the different classes of greedy-type bases. We revisit the concepts of greedy, quasi-greedy, and almost greedy bases in this comprehensive framework and provide the (non-trivial) extensions of the corresponding characterizations of those types of bases. As a by-product of our work, new properties arise, and the relations among them are carefully discussed.


  • Fernando AlbiacDepartment of Mathematics, Statistics, and Computer Sciences and InaMat$^2$
    Universidad Pública de Navarra
    Campus de Arrosadía
    31006 Pamplona, Spain
  • José L. AnsorenaDepartment of Mathematics and Computer Sciences
    Universidad de La Rioja
    26004 Logroño, Spain
  • Pablo M. BernáDepartamento de Matemática Aplicada y Estadística
    Facultad de Ciencias Económicas y Empresariales
    Universidad San Pablo-CEU, CEU Universities
    28003 Madrid, Spain
  • Przemysław WojtaszczykInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-656 Warszawa, Poland

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