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On some characterizations of greedy-type bases

EXPOSITIONES MATHEMATICAE 2022年第4期40卷 1135-1158页

作者： Berna, Pablo M. Chu, Hung Viet CUNEF Univ Dept Metodos Cuantitat Madrid 28040 Spain Univ Illinois Dept Math Urbana IL 61820 USA

In 1999, S. V. Konyagin and V. N. Temlyakov introduced the so-called thresholding greedy algorithm. Since then, there have been many interesting and useful characterizations of greedy -type bases in Banach spaces. In this article, we study and extend several characterizations of greedy and almost greedy bases in the literature. Along the way, we give various examples to complement our main results. Furthermore, we propose a new version of the so-called Weak thresholding greedy algorithm (WTGA) and show that the convergence of this new algorithm is equivalent to the convergence of the WTGA.(c) 2022 Elsevier GmbH. All rights reserved.

关键词： thresholding greedy algorithm greedy bases Almost greedy bases

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On sequential greedy-type bases

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Annals of Functional Analysis 2025年第3期16卷 1-35页

作者： Berasategui, Miguel Berná, Pablo M. Chu, Hùng Việt IMAS-UBA-CONICET-Pab I Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Buenos Aires 1428 Argentina Departamento de Matemáticas CUNEF Universidad Madrid 28040 Spain Department of Mathematics Texas A & M University College Station 77843 TX United States

It is known that a basis is almost greedy if and only if the thresholding greedy algorithm gives essentially the smallest error term compared to errors from projections onto intervals or in other words, consecutive terms of N. In this paper, we fix a sequence (an)n=1∞ and compare the TGA against projections onto consecutive terms of the sequence and its shifts. We call the corresponding greedy-type condition the F(an)-almost greedy property. Our first result shows that the F(an)-almost greedy property is equivalent to the classical almost greedy property if and only if (an)n=1∞ is bounded. Then we establish an analog of the result for the strong partially greedy property. Finally, we show that under a certain projection rule and conditions on the sequence (an)n=1∞, we obtain a greedy-type condition that lies strictly between the almost greedy and strong partially greedy properties. © Tusi Mathematical Research Group (TMRG) 2025.

关键词： Bases Consecutive almost greediness thresholding greedy algorithm

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Elton's near unconditionality of bases as a threshold-free form of greediness

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JOURNAL OF FUNCTIONAL ANALYSIS 2023年第7期285卷

作者： Albiac, Fernando Ansorena, Jose L. Berasategui, Miguel Univ Publ Navarra Dept Math Stat & Comp Sci InaMat2 Campus Arrosadia Pamplona 31006 Spain Univ La Rioja Dept Math & Comp Sci Logrono 26004 Spain Univ Buenos Aires Fac Ciencias Exactas & Nat IMAS UBA CONICET Pab1 RA-1428 Buenos Aires Argentina

Elton's near unconditionality and quasi-greediness for largest coefficients are two properties of bases that made their appearance in functional analysis from very different areas of research. One of our aims in this note is to show that, oddly enough, they are connected to the extent that they are equivalent notions. We take advantage of this new description of the former property to further the study of the threshold function associated with near unconditionality. Finally, we make a contribution to the isometric theory of greedy bases by characterizing those bases that are 1-quasi-greedy for largest coefficients.& COPY;2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).

关键词： thresholding greedy algorithm Nearly unconditional bases

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On Weighted greedy-Type Bases

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BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY 2023年第4期54卷 49-49页

作者： Chu, Hung Viet Texas A&M Univ Dept Math College Stn TX 77843 USA

We study weights for the thresholding greedy algorithm, aiming to extend previous work on sequential weights sigma on N to weights omega on P(N). We revisit major results on weighted greedy-type bases in this new setting including characterizations of omega-(almost) greedy bases and the equivalence between omega-semi-greedy bases and omega-almost greedy bases. Some new results are encountered along the way. For example, we show that there exists an omega-greedy unconditional basis that is not sigma-almost greedy for any weight sequence sigma. Moreover, a basis is unconditional if and only if it is omega-greedy for some weight omega. Similarly, a basis is quasi-greedy if and only if it is omega-almost greedy for some weight omega.

关键词： thresholding greedy algorithm greedy Almost greedy Semi-greedy Partially greedy Weight

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Characterization of Weight-Semi-greedy Bases

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2020年第1期26卷 1-21页

作者： Berna, Pablo Manuel Univ Autonoma Madrid Dept Matemat E-28049 Madrid Spain

One classical result in greedy approximation theory is that almost-greedy and semi-greedy bases are equivalent in the context of Schauder bases in Banach spaces with finite cotype. This result was proved by Dilworth et al. (Studia Math 159:67-101, 2003) and, recently, in the study of Berna (J Math Anal Appl 470:218-225, 2019), the author proved that the condition of finite cotype can be removed. One of the results in this paper is to show that the condition of Schauder can be relaxed using the rho-admissibility, notion introduced in the study of Berna et al. (Rev Mat Complut;https://***/10.1007/s13163-019-00328-9). On the other hand, in the study of Dilworth et al. (Tr Mat Inst Steklova 303: 120-141, 2018), the authors extend the notion of semi-greediness to the context of weights and proved the following: if w is a weight and B is a Schauder basis in a Banach space Xwith finite cotype, then w-semi-greediness and w-almost-greediness are equivalent notions. Here, we prove the same characterization but removing the condition of finite cotype. Also, we give some results improving the behavior of some constants in the relation between w-greedy-type bases and some w-democracy properties.

关键词： thresholding greedy algorithm Weight-almost-greedy bases Semi-greedy bases

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Lebesgue-type inequalities in greedy approximation

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JOURNAL OF FUNCTIONAL ANALYSIS 2021年第5期280卷 108885-108885页

作者： Dilworth, Stephen Garrigos, Gustavo Hernandez, Eugenio Kutzarova, Denka Temlyakov, Vladimir Univ South Carolina Dept Math Columbia SC 29208 USA Univ Murcia Dept Matemat Murcia 30100 Spain Univ Autonoma Madrid Dept Matemat Madrid 28049 Spain Univ Illinois Dept Math Urbana IL 61807 USA Bulgarian Acad Sci Inst Math & Informat Sofia Bulgaria Steklov Inst Math Moscow Russia Lomonosov Moscow State Univ Moscow Russia Moscow Ctr Fundamental & Appl Math Moscow Russia

We present new results regarding Lebesgue-type inequalities for the Weak Chebyshev greedy algorithm (WCGA) in uniformly smooth Banach spaces. We improve earlier bounds in [19] for dictionaries satisfying a new property introduced here. We apply these results to derive optimal bounds in two natural examples of sequence spaces. In particular, optimality is obtained in the case of the multivariate Haar system in L-p with 1 < p <= 2, under the Littlewood-Paley norm. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Non-linear approximation Weak Chebyshev greedy algorithm thresholding greedy algorithm Uniformly smooth Banach space

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A remark on approximation with polynomials and greedy bases

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2019年第2期478卷 466-475页

作者： Berna, Pablo M. Perez, Antonio Univ Autonoma Madrid Dept Matemat E-28049 Madrid Spain UCM UC3M UAM Inst Ciencias MatematCSIC C Nicolas Cabrera 13-15Campus Cantoblanco Madrid 28049 Spain

We investigate properties of the m-th error of approximation by polynomials with constant coefficients 9,,(x) and with modulus-constant coefficients D-m(x) introduced by Bernd and Blasco (D-m* (x) to study greedy bases in Banach spaces. We characterize when lim inf (x) and lim inf n D-m(x) are equivalent to parallel to x parallel to in terms of the democracy and superdemocracy functions, and provide sufficient conditions ensuring that D-m*9,(x) = extending previous very particular results. (C) 2019 Elsevier Inc. All rights reserved.

关键词： thresholding greedy algorithm greedy bases Almost greedy bases

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Unconditional and quasi-greedy bases in Lp with applications to Jacobi polynomials Fourier series

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REVISTA MATEMATICA IBEROAMERICANA 2019年第2期35卷 561-574页

作者： Albiac, Fernando Ansorena, Jose L. Ciaurri, Oscar Varona, Juan L. Univ Publ Navarra InaMat Inst Advance Mat Campus Arrosadia Pamplona 31006 Spain Univ Publ Navarra Dept Estadist Informat & Matemat Campus Arrosadia Pamplona 31006 Spain Univ La Rioja Dept Math & Comp Sci Logrono 26006 Spain

We show that the decreasing rearrangement of the Fourier series with respect to the Jacobi polynomials for functions in L-p does not converge unless p = 2. As a by-product of our work on quasi-greedy bases in L-p(mu), we show that no normalized unconditional basis in L-p, p not equal 2, can be semi-normalized in L-q for q not equal p, thus extending a classical theorem of Kadets and Pelczynski from 1962.

关键词： thresholding greedy algorithm unconditional basis quasi-greedy basis L-p-spaces Jacobi polynomials

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Equivalence between almost-greedy and semi-greedy bases

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2019年第1期470卷 218-225页

作者： Berna, P. M. Univ Autonoma Madrid Dept Matemat E-28049 Madrid Spain

In [3] it was proved that almost-greedy and semi-greedy bases are equivalent in the context of Banach spaces with finite cotype. In this paper we show this equivalence for general Banach spaces.(C) 2018 Elsevier Inc. ... 详细信息

关键词： thresholding greedy algorithm Almost-greedy bases Semi-greedy bases

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Conditional Quasi-greedy Bases in Non-superreflexive Banach Spaces

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CONSTRUCTIVE APPROXIMATION 2019年第1期49卷 103-122页

作者： Albiac, Fernando Ansorena, Jose L. Wojtaszczyk, Przemyslaw Univ Publ Navarra Math Dept Campus Arrosadia Pamplona 31006 Spain Univ La Rioja Dept Math & Comp Sci Logrono 26004 Spain Univ Warsaw Interdisciplinary Ctr Math & Computat Modelling Ul Prosta 69 PL-02838 Warsaw Poland Polish Acad Sci Inst Math Ul Sniadeckich 8 PL-00656 Warsaw Poland

For a conditional quasi-greedy basis B in a Banach space, the associated conditionality constants km[B] verify the estimate km[B]=O(logm). Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-greedy bases in some special class of spaces. It is known that every quasi-greedy basis in a superreflexive Banach space verifies km[B]=O((logm)1-E) for some 0greedy bases in superreflexive spaces. We prove that if a Banach space X is not superreflexive, then there is a quasi-greedy basis B in a Banach space Y finitely representable in X with km[B]approximate to logm. As a consequence, we obtain that for every 2greedy basis B with km[B]approximate to logm. We also tackle the corresponding problem for Schauder bases and show that if a space is nonsuperreflexive, then it possesses a basic sequence B with km[B]approximate to m.

关键词： thresholding greedy algorithm Conditional basis Conditionality constants Quasi-greedy basis Type Cotype Reflexivity Superreflexivity Super property Finite representability Banach spaces

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