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The thresholding greedy algorithm versus approximations with sizes bounded by certain functions f

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2024年第2期539卷

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

Let X be a Banach space and ( e n ) infinity n =1 be a basis. For a function f in a large collection F (closed under composition), we define and characterize f- greedy and f- almost greedy bases. We study relations among these bases as f varies and show that while a basis is not almost greedy, it can be f- greedy for some f is an element of F. Furthermore, we prove that for all non-identity function f is an element of F, we have the surprising equivalence f- greedy double left arrow double right arrow f- almost greedy. We give various examples of Banach spaces to illustrate our results. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： thresholding greedy algorithm Almost greedy Bases

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Lebesgue inequalities for Chebyshev thresholding greedy algorithms

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REVISTA MATEMATICA COMPLUTENSE 2020年第3期33卷 695-722页

作者： Berna, P. M. Blasco, O. Garrigos, G. Hernandez, E. Oikhberg, T. Univ Autonoma Madrid Dept Matemat Madrid 28049 Spain Univ Valencia Dept Anal Matemat Campus Burjassot Valencia 46100 Spain Univ Murcia Dept Matemat Murcia 30100 Spain Univ Illinois Dept Math Urbana IL 61807 USA

We establish estimates for the Lebesgue parameters of the Chebyshev weak thresholding greedy algorithm in the case of general bases in Banach spaces. These generalize and slightly improve earlier results in Dilworth et al. (Rev Mat Complut 28(2):393-409, 2015), and are complemented with examples showing the optimality of the bounds. Our results also clarify certain bounds recently announced in Shao and Ye (J Inequal Appl 2018(1):102, 2018), and answer some questions left open in that paper.

关键词： thresholding Chebyshev greedy algorithm thresholding greedy algorithm Quasi-greedy basis Semi-greedy bases

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Twenty-five years of greedy bases

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JOURNAL OF APPROXIMATION THEORY 2025年 307卷

作者： Albiac, Fernando Ansorena, Jose L. Temlyakov, Vladimir 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 South Carolina Dept Math Columbia SC 29208 USA Russian Acad Sci Steklov Math Inst Moscow Russia Lomonosov Moscow State Univ Moscow Russia Moscow Ctr Fundamental & Appl Math Moscow Russia

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. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Nonlinear approximation thresholding greedy algorithm greedy Almost greedy And quasi-greedy basis

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On approximation spaces and greedy-type bases

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ANNALS OF FUNCTIONAL ANALYSIS 2025年第1期16卷 1-39页

作者： Berna, Pablo M. Chu, Hung Viet Hernandez, Eugenio CUNEF Univ Dept Matemat Madrid 28040 Spain Texas A&M Univ Dept Math College Stn TX 77843 USA Univ Autonoma Madrid Dept Matemat Madrid 28049 Spain

The purpose of this paper is to introduce omega\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-Chebyshev-greedy and omega\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-partially greedy approximation classes and study their relation with omega\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-approximation spaces, where the latter are a generalization of the classical approximation spaces. The relation gives us sufficient conditions of when certain continuous embeddings imply different greedy-type properties. Along the way, we generalize a result by P. Wojtaszczyk as well as characterize semi-greedy Schauder bases in quasi-Banach spaces, generalizing a previous result by the first author.

关键词： Approximation spaces greedy bases thresholding greedy algorithm

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On consecutive greedy and other greedy-like type of bases

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RESEARCH IN THE MATHEMATICAL SCIENCES 2024年第4期11卷 1-17页

作者： Berasategui, Miguel Berna, Pablo M. Chu, Hung Viet Univ Buenos Aires Fac Ciencias Exactas & Nat IMAS UBA CONICET Pab1 RA-1428 Buenos Aires Argentina CUNEF Univ Dept Metodos Cuantitat Madrid 28040 Spain Texas A&M Univ Dept Math College Stn TX 77843 USA

We continue our study of the thresholding greedy algorithm when we restrict the vectors involved in our approximations so that they either are supported on intervals of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {N}$$\end{document} or have constant coefficients. We introduce and characterize what we call consecutive greedy bases and provide new characterizations of almost greedy and squeeze symmetric Schauder bases. Moreover, we investigate some cases involving greedy-like properties with constant 1 and study the related notion of Property (A, tau\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}).

关键词： thresholding greedy algorithm Consecutive greedy bases Squeeze symmetric bases

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Counterexamples in isometric theory of symmetric and greedy bases

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JOURNAL OF APPROXIMATION THEORY 2024年 297卷

作者： Albiac, Fernando Ansorena, Jose L. Blasco, Oscar Chu, Hung Viet Oikhberg, Timur 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 Valencia Dept Math Anal Burjassot 46100 Spain Texas A&M Univ Dept Math College Stn TX 77843 USA Univ Illinois Dept Math Urbana IL 61820 USA

We continue the study initiated in Albiac and Wojtaszczyk (2006) of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to 1. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be 1-suppression unconditional, thus settling Problem 4.4 from Albiac and Ansorena (2017). In particular, our construction demonstrates that bases with Property (A) need not be 1-greedy even with the additional assumption that they are unconditional and symmetric. We also exhibit a finite-dimensional counterpart of this example, and show that, at least in the finite-dimensional setting, Property (A) does not pass to the dual. As a by-product of our arguments, we prove that a symmetric basis is unconditional if and only if it is total, thus generalizing the well-known result that symmetric Schauder bases are unconditional. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://***/licenses/by/4.0/).

关键词： thresholding greedy algorithm greedy basis Property (A) Suppression unconditional basis Symmetric basis

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Schreier families and 9-(almost) greedy bases

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CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES 2024年第4期76卷 1379-1399页

作者： Beanland, Kevin Chu, Hung Viet Washington & Lee Univ Dept Math Lexington VA 24450 USA Univ Illinois Dept Math Urbana IL USA

Let $\mathcal {F}$ be a hereditary collection of finite subsets of $\mathbb {N}$ . In this paper, we introduce and characterize $\mathcal {F}$ -(almost) greedy bases. Given such a family $\mathcal {F}$ , a basis $(e_n)_n$ for a Banach space X is called $\mathcal {F}$ -greedy if there is a constant $C\geqslant 1$ such that for each $x\in X$ , $m \in \mathbb {N}$ , and $G_m(x)$ , we have $$ \begin{align*} \|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}. \end{align*} $$ Here, $G_m(x)$ is a greedy sum of x of order m, and $\mathbb {K}$ is the scalar field. From the definition, any $\mathcal {F}$ -greedy basis is quasi-greedy, and so the notion of being $\mathcal {F}$ -greedy lies between being greedy and being quasi-greedy. We characterize $\mathcal {F}$ -greedy bases as being $\mathcal {F}$ -unconditional, $\mathcal {F}$ -disjoint democratic, and quasi-greedy, thus generalizing the well-known characterization of greedy bases by Konyagin and Temlyakov. We also prove a similar characterization for $\mathcal {F}$ -almost greedy ***, we provide several examples of bases that are nontrivially $\mathcal {F}$ -greedy. For a countable ordinal $\alpha $ , we consider the case $\mathcal {F}=\mathcal {S}_{\alpha }$ , where $\mathcal {S}_{\alpha }$ is the Schreier family of order $\alpha $ . We show that for each $\alpha $ , there is a basis that is $\mathcal {S}_{\alpha }$ -greedy but is not $\mathcal {S}_{\alpha +1}$ -greedy. In other words, we prove that none of the following implications can be reversed: for two countable ordinals $\alpha < \beta $ , $$ \begin{align*} \mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_{\alpha}\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_{\beta}\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}. \end{align*} $$

关键词： thresholding greedy algorithm Schreier unconditional Schreier families

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Linear versus nonlinear forms of partial unconditionality of bases

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JOURNAL OF FUNCTIONAL ANALYSIS 2024年第9期287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 IMAS UBA CONICET Pab 1 Fac Ciencias Exactas & Nat RA-1428 Buenos Aires Argentina

The main results in this paper contribute to bringing to the fore novel underlying connections between the contemporary concepts and methods springing from greedy approximation theory with the well-established techniques of classical Banach spaces. We do that by showing that bounded-oscillation unconditional bases, introduced by Dilworth et al. in 2009 in the setting of their search for extraction principles of subsequences verifying partial forms of unconditionality, are the same as truncation quasi-greedy bases, a new breed of bases that appear naturally in the study of the performance of the thresholding greedy algorithm in Banach spaces. We use this identification to provide examples of bases that exhibit that bounded-oscillation unconditionality is a stronger condition than Elton's near unconditionality. We also take advantage of our arguments to provide examples that allow us to tell apart certain types of bases that verify either debilitated unconditionality conditions or weaker forms of quasi-greediness in the context of abstract approximation theory. (c) 2024 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 Partial unconditionality Bounded-oscillation unconditionality

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Existence of Almost greedy Bases in Mixed-Norm Sequence and Matrix Spaces, Including Besov Spaces

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CONSTRUCTIVE APPROXIMATION 2024年第2期60卷 253-283页

作者： Albiac, Fernando Ansorena, Jose L. Bello, Glenier Wojtaszczyk, Przemyslaw Univ Publ Navarra Dept Math Stat & Comp Sci Pamplona 31006 Spain Univ Publ Navarra INAMAT2 Pamplona 31006 Spain Univ La Rioja Dept Math & Comp Sci Logrono 26004 Spain Univ Autonoma Madrid Dept Matemat Madrid 28049 Spain ICMAT CSIC UAM UC3M UCM Madrid 28049 Spain Polish Acad Sci Inst Math Ul Sniadeckich 8 PL-00656 Warsaw Poland

We prove that the sequence spaces l(p) circle plus l(q) and the spaces of infinite matrices l(p)(l(q)), l(q)(l(p)) and (circle plus(infinity)(n=1) l(p)(n))(lq), which are isomorphic to certain Besov spaces, have an almost greedy basis whenever 0 < p < 1 < q < infinity. More precisely, we custom-buildal most greedy bases in such a way that the Lebesgue parameters grow in a prescribed manner. Our arguments critically depend on the extension of the Dilworth-Kalton-Kutzarova method from Dilworth et al. (Stud Math 159(1):67-101, 2003), which was originally designed for constructing almost greedy bases in Banach spaces, to makeit valid for direct sums of mixed-normed spaces with nonlocally convex components. Additionally, we prove that the fundamental functions of all almost greedy bases of these spaces grow as (m(1/q))(m=1)(infinity).

关键词： thresholding greedy algorithm Conditional basis Quasi-greedy basis Almost greedy basis Subsymmetric basis l(p)-Space

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Bidemocratic Bases and Their Connections with Other greedy-Type Bases

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CONSTRUCTIVE APPROXIMATION 2023年第1期57卷 125-160页

作者： Albiac, Fernando Ansorena, Jose L. Berasategui, Miguel Berna, Pablo M. Lassalle, Silvia 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 IMAS UBA CONICET Pab 1 Fac Ciencias Exactas & Nat RA-1428 Buenos Aires DF Argentina CUNEF Univ Dept Metodos Cuantitativos Madrid 28040 Spain Univ San Andres Dept Matemat Vito Duma 284 RA-1644 Buenos Aires DF Argentina IMAS CONICET Buenos Aires DF Argentina

In nonlinear greedy approximation theory, bidemocratic bases have traditionally played the role of dualizing democratic, greedy, quasi-greedy, or almost greedy bases. In this article we shift the viewpoint and study them for their own sake, just as we would with any other kind of greedy-type bases. In particular we show that bidemocratic bases need not be quasi-greedy, despite the fact that they retain a strong unconditionality flavor which brings them very close to being quasi-greedy. Our constructive approach gives that for each 1 < p < infinity the space L-p has a bidemocratic basis which is not quasi-greedy. We also present a novel method for constructing conditional quasi-greedy bases which are bidemocratic, and provide a characterization of bidemocratic bases in terms of the new concepts of truncation quasi-greediness and partially demo-cratic bases.

关键词： Nonlinear approximation thresholding greedy algorithm Quasi-greedy basis Democracy

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