This article closes the cycle of characterizations of greedy-like bases in the isometric case initiated in [F. Albiac and P. Wojtaszczyk, Characterization of $1$-greedy bases, J. Approx. Theory 138 (2006)] with the characterization of $1$-greedy bases and continued in [F. Albiac and J. L. Ansorena, Characterization of $1$-quasi-greedy bases, arXiv:1504.04368v1 [math.FA] (2015)] with the characterization of $1$-quasi-greedy bases. Here we settle the problem of providing a characterization of $1$-almost greedy bases in Banach spaces. We show that a (semi-normalized) basis in a Banach space is almost greedy with almost greedy constant equal to $1$ if and only if it has Property (A). This fact permits now to state that a basis is $1$-greedy if and only if it is $1$-almost greedy and $1$-quasi-greedy. As a by-product of our work we also provide a tight characterization of almost greedy bases.
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.
We shall present new characterizations of partially greedy and almost greedy bases. A new class of basis (which we call reverse partially greedy basis) arises naturally from these characterizations of partially greedy bases. We also give characterizations for $1$-partially greedy and $1$-reverse partially greedy bases.
We introduce the notion of a \textit{weight-almost greedy} basis and show that a basis for a real Banach space is $w$-almost greedy if and only if it is both quasi-greedy and $w$-democratic. We also introduce the notion of \textit{weight-semi-greedy} basis and show that a $w$-almost greedy basis is $w$-semi-greedy and that the converse holds if the Banach space has finite cotype.
In 2003, S. J. Dilworth et al. ([8]) introduced the notion of almost-greedy (resp. partially-greedy) bases. These bases were characterized in terms of quasi-greediness and democracy (resp. conservativeness). In this paper we will show a new functional characterization of these type of bases in general Banach spaces following the spirit of the characterization of greediness proved in [5].
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 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…
We prove that the fundamental function of any almost greedy basis of $L_p$, $1<p<\infty$, grows as either $(m^{1/p})_{m=1}^\infty$ or $(m^{1/2})_{m=1}^\infty$.
We construct a uniformly bounded orthonormal almost greedy basis for $L_p([0,1])$, $1<p<\infty$. The example shows that it is not possible to extend Orlicz's theorem, stating that there are no uniformly bounded orthonormal unconditional bases for $L_p([0,1])$, $p\not=2$, to the class of almost greedy bases.
The aim of this paper is to show that almost greedy bases induce tighter embeddings in superreflexive Banach spaces than in general Banach spaces. More specifically, we show that an almost greedy basis in a superreflexive Banach space $\mathbb{X}$ induces embeddings that allow squeezing $\mathbb{X}$ between two superreflexive Lorentz sequence spaces that are close to each other in the sense that they have the same fundamental function.
We prove that the sequence spaces $\ell_p\oplus\ell_q$ and the spaces of infinite matrices $\ell_p(\ell_q)$, $\ell_q(\ell_p)$ and $(\bigoplus_{n=1}^\infty \ell_p^n)_{\ell_q}$, which are isomorphic to certain Besov spaces, have an almost greedy basis whenever $0<p<1<q<\infty$. More precisely, we custom-build almost 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 [S. J. Dilworth, N. J. Kalton, and D. Kutzarova, On the existence of almost greedy bases in Banach spaces…
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 $$\|x - G_m(x)\|\ \leqslant\ C \inf\left\{\left\|x-\sum_{n\in A}a_ne_n\right\|\,:\, |A|\leqslant m, A\in \mathcal{F}, (a_n)\subset \mathbb{K}\right\}.$$ Here $G_m(x)$ is a greedy sum of $x$ of order $m$, and $\mathbb{K}$ i…
Partially greedy bases in Banach spaces were introduced by Dilworth et al. as a strictly weaker notion than the (almost) greedy bases. In this paper, we study two natural ways to strengthen the definition of partial greediness. The first way produces what we call the consecutive almost greedy property, which turns out to be equivalent to the almost greedy property. Meanwhile, the second way reproduces the PG property for Schauder bases but a strictly stronger property for general bases.
It is known that for a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space $\mathbb{X}$, the associated sequence $(k_{m}[\mathcal{B}])_{m=1}^{\infty}$ of its conditionality constants verifies the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$ and that if the reverse inequality $\log m =\mathcal{O}(k_m[\mathcal{B}])$ holds then $\mathbb{X}$ is non-superreflexive. Indeed, it is known that a quasi-greedy basis in a superreflexive quasi-Banach space fulfils the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)^{1-ε}$ for some $ε>0$. However, in the existing literature one finds very f…
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…
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 investigate various aspects of the "weighted" greedy algorithm with respect to a Schauder basis. For a weight w, we describe w-greedy, w-almost-greedy and w-partially-greedy bases, and show some properties of w-semi-greedy bases. To achieve these goals, we introduce and study the w-Property (A).
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,$τ$).
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 S. J. Dilworth, N. J. Kalton and D. Kutzarova in [7] and, recently, the first author in [2] proved that the condition of finite cotype can be removed in this result. In [11], the authors extend the notion of semi-greediness to the context of weights and proved the following: if w is a weight and $\mathcal B$ is a Schauder basis in a Banach space $\mathbb X$ with finite cotype, then w-semi-g…
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.
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 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 $1<p<\infty$ the space $\ell_p$ has a bidemocratic basis which …
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 $f\in\mathbb X$…
Tsirelson's space $\mathcal{T}$ made its appearance in Banach space theory in 1974 soon to become one of the most significant counterexamples in the theory. Its structure broke the ideal pattern that analysts had conceived for a generic Banach space, thus giving rise to the era of pathological examples. Since then, many authors have contributed to the study of different aspects of this special space with an eye on better understanding its idiosyncrasies. In this paper we are concerned with the greedy-type basis structure of $\mathcal{T}$, a subject that had not been previously explored in the …
The fact that finite direct sums of two or more mutually different spaces from the family $\{\ell_{p} : 1\le p<\infty\}\cup c_{0}$ fail to have greedy bases is stated in [Dilworth et al., Greedy bases for Besov spaces, Constr. Approx. 34 (2011), no. 2, 281-296]. However, the concise proof that the authors give of this fundamental result in greedy approximation relies on a fallacious argument, namely the alleged uniqueness of unconditional basis up to permutation of the spaces involved. The main goal of this note is to settle the problem by providing a correct proof. For that we first show that…
It is known that for a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space $\mathbb{X}$, the associated sequence $(k_{m}[\mathcal{B}])_{m=1}^{\infty}$ of its conditionality constants verifies the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$ and that if the reverse inequality $\log m =\mathcal{O}(k_m[\mathcal{B}])$ holds then $\mathbb{X}$ is non-superreflexive. However, in the existing literature one finds very few instances of non-superreflexive spaces possessing quasi-greedy basis with conditionality constants as large as possible. Our goal in this article is to fill this gap.…
We study sparse approximation by greedy algorithms. We prove the Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA), a generalization of the Weak Orthogonal Matching Pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. The results are proved for redundant dictionaries satisfying certain conditions. Then we apply these general results to the case of bases. In particular, we prove that the WCGA provides almost optimal sparse approximation for the trigonometric system in $L_p$, $2\le p<\infty$…
We continue with the study of greedy-type bases in quasi-Banach spaces started in [3]. In this paper, we study partially-greedy bases focusing our attention in two main results: -Characterization of partially-greedy bases in quasi-Banach spaces in terms of quasi-greediness and different conservative-like properties. -Given a $C$-partially-greedy basis in a quasi-Banach space, there exists a "renorming" such that the basis is $1$-partially-greedy.
We show that for quasi-greedy bases in real Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N-term error of approximation times a constant which depends on the democracy functions and the quasi-greedy constant of the basis.
We show that a (semi-normalized) basis in a Banach space is quasi-greedy with quasi-greedy constant equal to 1 if and only if it is unconditional with suppression-unconditional constant equal to 1.
We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N- term error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C logN. We show with two examples that this bound is attained for quasi-greedy democratic bases.
Our aim is to investigate the properties of existence and uniqueness of greedy bases in Banach spaces. We show the non-existence of greedy basis in some Nakano spaces and Orlicz sequence spaces and produce the first-known examples of non-trivial spaces (i.e., different from $c_0$, $\ell_1$, and $\ell_2$) with a unique greedy basis.
We shall present a new characterization of greedy bases and 1-greedy bases in terms of certain functionals defined using distances to one dimensional subspaces generated by the basis. We also introduce a new property that unifies the notions of unconditionality and democracy and allows us to recover a better dependence on the constants.
The purpose of this paper is to introduce $ω$-Chebyshev-greedy and $ω$-partially greedy approximation classes and to study their relation with $ω$-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.
In this paper, motivated by the notion of $w$-Property $(A)$ defined in [2], we introduce the notions of $w$-left Property $(A)$ and $w$-right Property $(A)$. We also introduce the notions of $w$-partially greedy basis (using a characterization of partially greedy basis from [4]) and $w$-reverse partially greedy basis. The main aim of this paper is to study $(i)$ some characterizations of $w$-partially greedy and $w$-reverse partially greedy basis $(ii)$ conditions on the weight sequences when $w$-left Property $(A)$ and (or) $w$-right Property $(A)$ implies $w$-Property $(A)$.
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.
In this paper, we establish new advances in the theory started by T. Oikhberg in [15] where the author joins greedy approximation theory with the use of sequences with gaps. Concretely, we address and partially answer three open questions related to quasi-greedy bases for sequences with gaps posed in [15, Section 6].
In this paper we continue the study of Lebesgue-type inequalities for greedy algorithms. We introduce the notion of strong partially greedy Markushevich bases and study the Lebesgue-type parameters associated with them. We prove that this property is equivalent to that of being conservative and quasi-greedy, extending a similar result given in [9] for Schauder bases. We also give a characterization of 1-strong partial greediness, following the study started in [1,3].
We give a complete characterization of $2π$-periodic weights $w$ for which the usual trigonometric system forms a quasi-greedy basis for $L^p(\bT;w)$, i.e., bases for which simple thresholding approximants converge in norm. The characterization implies that this can happen only for $p=2$ and whenever the system forms a quasi-greedy basis, the basis must actually be a Riesz basis.
We show that non of the spaces $(\bigoplus_{n=1}^\infty\ell_p)_{\ell_q}$, $1\le p\not= q<\infty$, have a greedy basis. This solves a problem raised by Dilworth, Freeman, Odell and Schlumprect. Similarly, the spaces $(\bigoplus_{n=1}^\infty\ell_p)_{c_0}$, $1\le p<\infty$, and $(\bigoplus_{n=1}^\infty c_o)_{\ell_q}$, $1\le q<\infty$, do not have greedy bases. It follows from that and known results that a class of Besov spaces on $\R^n$ lack greedy bases as well.
We present various estimates for the Lebesgue constants of the thresholding greedy algorithm, in the case of general bases in Banach spaces. We show the optimality of these estimates in some situations. Our results recover and slightly improve various estimates appearing earlier in the literature.
We prove thatthe Banach space $(\oplus_{n=1}^\infty \ell_p^n)_{\ell_q}$, which is isomorphic to certain Besov spaces, has a greedy basis whenever $1\leq p \leq\infty$ and $1<q<\infty$. Furthermore, the Banach spaces $(\oplus_{n=1}^\infty \ell_p^n)_{\ell_1}$, with $1<p\le \infty$, and $(\oplus_{n=1}^\infty \ell_p^n)_{c_0}$, with $1\le p<\infty$ do not have a greedy bases. We prove as well that the space $(\oplus_{n=1}^\infty \ell_p^n)_{\ell_q}$ has a 1-greedy basis if and only if $1\leq p=q\le \infty$.
We study the problem of improving the greedy constant or the democracy constant of a basis of a Banach space by renorming. We prove that every Banach space with a greedy basis can be renormed, for a given $\vare>0$, so that the basis becomes $(1+\vare)$-democratic, and hence $(2+\vare)$-greedy, with respect to the new norm. If in addition the basis is bidemocratic, then there is a renorming so that in the new norm the basis is $(1+\vare)$-greedy. We also prove that in the latter result the additional assumption of the basis being bidemocratic can be removed for a large class of bases. Applicat…
We use new methods, specific of non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a $p$-Banach space for $0<p<1$ are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy space $H_p(\mathbb{D})$ for $0<p<1$ are democratic while, in contrast, no quasi-greedy basis of $H_p(\mathbb{D}^d)$ for $d\ge 2$ is, solving thus a problem that was raised in [F. Albiac, J. L. Ansorena, and P. Wojtaszczyk, \textit{Quasi-greedy bases in $\ell_p$ ($0<p<1$) are democratic}, J. Funct. Anal. \textbf{280} (2021), no. 7, 1…
We prove that, given $2<p<\infty$, the Fourier coefficients of functions in $L_2(\mathbb{T}, \lvert t \rvert^{1-2/p}\, dt)$ belong to $\ell_p$, and that, given $1<p<2$, the Fourier series of sequences in $\ell_p$ belong $L_2(\mathbb{T}, \lvert t \rvert^{2/p-1}\, dt)$. Then, we apply these results to the study of conditional Schauder bases and conditional almost greedy bases in Banach spaces. Specifically, we prove that, for every $1<p<\infty$ and every $0\le α<1$, there is a Schauder basis of $\ell_p$ whose conditionality constants grow as $(m^α)_{m=1}^\infty$, and there is an almost greedy ba…
