Let $X$ be a Banach space and $(e_n)_{n=1}^\infty$ be a basis. For a function $f$ in a large collection $\mathcal{F}$ (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 $f\in \mathcal{F}$. Furthermore, we prove that for all non-identity function $f\in \mathcal{F}$, we have the surprising equivalence $$f\mbox{-greedy}\ \Longleftrightarrow \ f\mbox{-almost greedy}.$$ We give various examples of Banach spaces to illustrate our results.
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 [9], and are complemented with examples showing the optimality of the bounds. Our results also correct certain bounds recently announced in [18], and answer some questions left open in that paper.
The goal of this paper is to study the performance of the Thresholding Greedy Algorithm (TGA) when we increase the size of greedy sums by a constant factor $λ\geqslant 1$. We introduce the so-called $λ$-almost greedy and $λ$-partially greedy bases. The case when $λ= 1$ gives us the classical definitions of almost greedy and (strong) partially greedy bases. We show that a basis is almost greedy if and only if it is $λ$-almost greedy for all (some) $λ\geqslant 1$. However, for each $λ> 1$, there exists an unconditional basis that is $λ$-partially greedy but is not $1$-partially greedy. Furthermo…
Albiac and Wojtaszczyk introduced property (A) to characterize $1$-greedy bases. Later, Dilworth et al. generalized the concept to $C$-property (A), where the case $C = 1$ gives property (A). They (among other results) characterized greedy bases by unconditionality and $C$-property (A). In this paper, we extend the definition of the so-called A-property constant to (A,$τ$)-property constants and use the extension to obtain new estimates for various Lebesgue parameters. Furthermore, we study the relation among (A,$τ$)-property constants and other well-known constants when $τ$ varies.
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.
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 $\mathbb N$ 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,$τ$).
In this paper we study a new class of bases, weaker than quasi-greedy bases, which retain their unconditionality properties and can provide the same optimality for the thresholding greedy algorithm. We measure how far these bases are from being unconditional and use this concept to give a new characterization of nearly unconditional bases.
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. We revisit the concepts of greedy, quasi-greedy, and almost greedy bases in this comprehensive framework and provide the (nontrivial) extensions of the corresponding characterizations of those types of bases. As a by-product of our work, new properties arise, and the relations amongst them are carefully discussed.
In this paper, we study weights for the Thresholding Greedy Algorithm (TGA). While previous work focused on sequential weights $ς= (s_n)_{n\in\mathbb{N}}$ on each positive integer, we study a more general weight $ω= (w_A)_{A\subset\mathbb{N}}$ on each set $A\subset \mathbb{N}$. We define and characterize $ω$-(almost) greedy bases. Furthermore, we leverage existing results to show that there exists an $ω$-greedy unconditional basis that is not $ς$-almost greedy for any weight sequence $ς$. Last but not least, we show the equivalence between $ω$-semi-greedy bases and $ω$-almost greedy bases when…
We define a family of weak thresholding greedy algorithms for the multivariate Haar basis for $L_1[0,1]^d$ ($d \ge 1$). We prove convergence and uniform boundedness of the weak greedy approximants for all $f \in L_1[0,1]^d$.
We introduce and study the notion of weak semi-greedy systems -which is inspired in the concepts of semi-greedy and Branch semi-greedy systems and weak thresholding sets-, and prove that in the context Markushevich bases in infinite dimensional Banach spaces, the notions of \textit{ semi-greedy, branch semi-greedy, weak semi-greedy, and almost greedy} Markushevich bases are all equivalent. This completes and extends some results from \cite{Berna2019}, \cite{Dilworth2003b}, and \cite{Dilworth2012}. We also exhibit an example of a semi-greedy system that is neither almost greedy nor a Markushevi…
The purpose of this paper is to quantify the size of the Lebesgue constants $(L_m)_{m=1}^{\infty}$ associated with the thresholding greedy algorithm in terms of a new generation of parameters that modulate accurately some features of a general basis. This fine-tuning of constants allows us to provide an answer to the question raised by Temlyakov in 2011 to find a natural sequence of greedy-type parameters for arbitrary bases in Banach (or quasi-Banach) spaces which combined linearly with the sequence of unconditionality parameters $(k_m)_{m=1}^{\infty}$ determines the growth of $(L_m)_{m=1}^{\…
This paper is devoted to theoretical aspects on optimality of sparse approximation. We undertake a quantitative study of new types of greedy-like bases that have recently arisen in the context of nonlinear $m$-term approximation in Banach spaces as a generalization of the properties that characterize almost greedy bases, i.e., quasi-greediness and democracy. As a means to compare the efficiency of these new bases with already existing ones in regards to the implementation of the Thresholding Greedy Algorithm, we place emphasis on obtaining estimates for their sequence of unconditionality param…
The theory of greedy-like bases started in 1999 when S. V. Konyagin and V. N. Temlyakov introduced in \cite{KT} the famous Thresholding Greedy Algorithm. Since this year, different greedy-like bases appeared in the literature, as for instance: quasi-greedy, almost-greedy and greedy bases. The purpose of this paper is to introduce some new characterizations of 1-greedy bases. Concretely, given a basis $\mathcal B=(\mathbf x_n)_{n\in\mathbb N}$ in a Banach space $\mathbb X$, we know that $\mathcal B$ is $C$-greedy with $C>0$ if $\Vert f-\mathcal G_m(f)\Vert\leq Cσ_m(f)$ for every $f\in\mathbb X$…
