Related papers: Weight-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.
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…
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…
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…
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…
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.
For two countable ordinals $\alpha$ and $\beta$, a basis of a Banach space $X$ is said to be $(\alpha, \beta)$-quasi-greedy if it is 1) quasi-greedy, 2) $\mathcal{S}_\alpha$-unconditional but not $\mathcal{S}_{\alpha+1}$-unconditional, and…
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…
We construct two counterexamples that resolve long-standing open problems on greedy approximation theory with respect to bases, posed in [F. Albiac et al., Dissertationes Math. 560 (2021)] and restated in [F. Albiac, J. L. Ansorena, V.…
We introduce and study the notion of weak weight-semi-greedy Markushevich bases - which extends the concepts of weight semi-greedy and weak semi-greedy Markushevich bases. In particular, we study conditions under which such bases are weight…
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,…
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…
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…
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…
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…
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…
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…
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…
The list of known Banach spaces whose linear geometry determines the (nonlinear) democracy functions of their quasi-greedy bases to the extent that they end up being democratic, reduces to $c_0$, $\ell_2$, and all separable…
The purpose of this paper is to introduce $\omega$-Chebyshev-greedy and $\omega$-partially greedy approximation classes and to study their relation with $\omega$-approximation spaces, where the latter are a generalization of the classical…