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 am...
详细信息
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.
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 e...
详细信息
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.
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 a...
详细信息
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 greedyalgorithms 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.
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{u...
详细信息
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.
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{amsma...
详细信息
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}).
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 ...
详细信息
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/).
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...
详细信息
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 $ , $$ \begin{align*} \mbox{quasi-greedy}\ \Longleftarrow\ \mathcal{S}_{\alpha}\mbox{-greedy}\ \Longleftarrow\ \mathcal{S}_{\beta}\mbox{-greedy}\ \Longleftarrow\ \mbox{greedy}. \end{align*} $$
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 techniq...
详细信息
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/).
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 al...
详细信息
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).
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 te...
详细信息
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.\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} In this paper, we fix a sequence (an)n=1 infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_n)_{n=1}<^>\infty $$\end{document} 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}_{(a_n)}$$\end{document}-almost greedy property. Our first result shows that the F(an)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}_{(a_n)}$$\end{document}-almost greedy property is equivalent to the classical almost greedy property if and only if (an)n=1 infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_n)_{n=1}<^>\infty $$\end{document} is bounded. Then we establish an analog of the result for the strong
暂无评论