arXiv subjects

Miguel Berasategui

Publications and source records attributed to Miguel Berasategui.

25 records · Page 2Linked to original sources

arXiv2022

Sparse approximation using new greedy-like bases in superreflexive spaces

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 parameters. Using an enhanced version of the original method from [S. J. Dilworth, N. J. Kalton, and D. Kutzarova, On the existence of almost greedy bases in Banach spaces, Studia Math. 159 (2003), no. 1, 67-101] for building almost greedy bases, we manage to construct bidemocratic bases whose unconditionality parameters satisfy significantly worse estimates than almost greedy bases even in Hilbert spaces.

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arXiv2022

Extensions of democracy-like properties for sequences with gaps

In \cite{O2015}, T. Oikhberg introduced and studied variants of the greedy and weak greedy algorithms for sequences with gaps. In this paper, we extend some of the notions that appear naturally in connection with these algorithms to the context of sequences with gaps. In particular, we will consider sequences of natural numbers for which the inequality $n_{k+1}\le \C n_{k}$ or $n_{k+1}\le \C+n_k$ holds for a positive constant $\C$ and all $k$, and find conditions under which the extended notions are equivalent their regular counterparts.

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arXiv2021

Weak greedy algorithms and the equivalence between semi-greedy and almost greedy Markushevich bases

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 Markushevich basis, showing that the Markushevich condition cannot be dropped from the equivalence result. In some cases, we obtain improved upper bounds for the corresponding constants of the systems.

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arXiv2021

Weaker forms of unconditionality of bases in greedy approximation

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.

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arXiv2021

Bidemocratic bases and their connections with other greedy-type bases

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 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 democratic bases.

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arXiv2020

Unconditional Frames of Translates in $L_p(\mathbb{R}^d)$

We show that, for $1<p \le 2$, the space $L_p(\mathbb{R}^d)$ does not admit unconditional Schauder frames $\left\lbrace f_i,f_i'\right\rbrace_{i\in\mathbb{N}}$ where $\left\lbrace f_i\right\rbrace$ is a sequence of translates of finitely many functions and $\left\lbrace f_i'\right\rbrace$ is seminormalized. In fact, the only subspaces of $L_p(\mathbb{R}^d)$ admitting such Banach frames are those isomorphic to $\ell_p$. On the other hand, if $2<p<+\infty$ and $\left\lbrace λ_{i}\right\rbrace_{i\in\mathbb{N}}\subseteq \mathbb{R}^d$ is an unbounded sequence, there is a subsequence $\left\lbrace λ_{m_i}\right\rbrace_{i\in\mathbb{N}}$, a function $f\in L_p(\mathbb{R}^d)$, and a seminormalized sequence of bounded functionals $\left\lbrace f_i'\right\rbrace_{i\in\mathbb{N}}$ such that $\left\lbrace T_{λ_{m_i}}f,f_i'\right\rbrace_{i\in\mathbb{N}}$ is an unconditional Schauder frame for $L_p(\mathbb{R}^d)$.

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arXiv2020

Strong partially greedy bases and Lebesgue-type inequalities

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].

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