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## Dissertationes Mathematicae

Artykuły w formacie PDF dostępne są dla subskrybentów, którzy zapłacili za dostęp online, po podpisaniu licencji Licencja użytkownika instytucjonalnego. Czasopisma do 2009 są ogólnodostępne (bezpłatnie).

## Greedy approximation for biorthogonal systems in quasi-Banach spaces

### Tom 560 / 2021

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 Opublikowany online: 29 April 2021

#### Streszczenie

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.

#### Autorzy

• Fernando AlbiacDepartment of Mathematics, Statistics, and Computer Sciences and InaMat$^2$
31006 Pamplona, Spain
e-mail
• José L. AnsorenaDepartment of Mathematics and Computer Sciences
26004 Logroño, Spain
e-mail
Facultad de Ciencias Económicas y Empresariales
e-mail
• Przemysław WojtaszczykInstitute of Mathematics