arXiv subjects
Publications and source records attributed to Miguel Berasategui.
We characterize the approximation spaces of a broad class of bases - which includes almost greedy bases - in terms of weighted Lorentz spaces. For those bases, we also find necessary and sufficient conditions under which the approximation spaces and greedy classes are the same.
We study the boundedness of averaging projections associated with symmetric Schauder bases in quasi-Banach spaces. Although this property is standard in the Banach setting, it is far from clear in the absence of local convexity and, indeed, fails for a broad class of quasi-Banach spaces with a symmetric basis, including $\ell_p$ for $0<p<1$. Our main result shows that, nevertheless, the canonical basis of an entire class of weighted Lorentz sequence spaces, including the spaces $\ell_{p,q}$ for $0<q<1<p<\infty$, has uniformly bounded averaging projections. Thus, bounded averaging projections do not characterize local convexity among quasi-Banach spaces with symmetric bases. As applications, we obtain new consequences for the structure of special bases. In particular, as a byproduct of our approach, we derive new examples of conditional and almost greedy bases in nonlocally convex spaces.
We investigate the problem of improving the greedy-type constant of a basis by means of an equivalent renorming of the ambient Banach space. Our main result shows that if a Banach space admits an unconditional and bidemocratic basis whose fundamental function satisfies certain regularity properties, then the space can be renormed so that the basis becomes isometrically greedy. The renorming simultaneously ensures lattice $1$-unconditionality, isometric bidemocracy, and allows prescribing the fundamental function up to a suitable regularization. As a principal application, we resolve a long-standing problem posed by Albiac--Wojtaszczyk in 2006 by proving that for each $1<p<\infty$ the $L_p$-normalized Haar system can be made $1$-greedy under an equivalent norm of $L_p$. Further applications include isometric greedy renormings for bases of Besov spaces, mixed-norm direct sums, and for a wide class of subsymmetric and conditional bases, including spreading models and the canonical basis of Schlumprecht space. These results show that isometric greedy renormings arise in far greater generality than previously known.
This paper develops a general theory for first-order descent methods whose search directions are restricted to a prescribed dictionary in a reflexive Banach space. Instead of assuming that the linear span of the dictionary is dense, as in the classical Proper Generalized Decomposition framework of Falcó and Nouy or in the universality approach of Berná and Falcó, we introduce a geometric condition based on norming sets that guarantees density through a duality argument. This makes it possible to treat dictionaries arising from tensor formats, neural network units, and other nonlinear or parameterized approximation families within a unified setting. On the algorithmic side, we analyze a simple greedy update rule in which each iterate is obtained by minimizing the energy functional along one direction from the dictionary. Under mild differentiability, Lipschitz continuity, and ellipticity assumptions on the objective, we derive explicit quantitative descent bounds and sharp convergence rates. These include algebraic rates that improve those of classical steepest-descent schemes in Banach spaces, as well as arbitrarily high polynomial rates and exponential convergence in a critical regime. The results apply broadly to convex variational problems, high-dimensional approximation, and structured optimization methods that rely on restricted or compressed search directions.
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. Temlyakov, J. Approx. Theory 307 (2025)]. Our first result exhibits a quasi-Banach space $\mathbb{X}$ with an almost greedy basis which, when transported to the Banach envelope of $\mathbb{X}$, ceases to be quasi-greedy. This shows that the passage to the Banach envelope, although it preserves linear and lattice structure, may radically disrupt the performance of the thresholding greedy algorithm, to the extent that in some respects it could perform better in a quasi-Banach space than in its Banach envelope. Our second result constructs an almost greedy Markushevich basis in a nonlocally convex quasi-Banach space $\mathbb{Y}$ which fails to be a Schauder basis under any reordering. Together, these examples highlight that local convexity and the Banach envelope construction play an unexpectedly active role in shaping greedy approximation phenomena, revealing structural differences between Banach and quasi-Banach spaces that go beyond the classical theory of bases.
We prove that if $\mathcal{X}$ is a quasi-greedy Markushevich basis of a Banach space $\mathbb{X}$, its dual basis $\mathcal{X}^*$ spans a norming subspace of $\mathbb{X}^*$. We also prove this result for weaker forms of quasi-greediness, and study the cases of other greedy-like properties from the literature.
We continue the study of Lebesgue-type parameters for various greedy algorithms in quasi-Banach spaces. First, we introduce a parameter that can be used with the quasi-greedy parameter to obtain the exact growth of the Lebesgue parameter for strong partially greedy bases. Second, we establish a new upper bound for the Lebesgue parameter for semi-greedy bases using the quasi-greedy and the squeeze symmetry parameters. Finally, we answer several open questions regarding the optimal power in various bounds proved in [F. Albiac, J. L. Ansorena, and P. M. Berná, New parameters and Lebesgue-type estimates in greedy approximation, Forum Math. Sigma 10 (2022), 1-39].
For the past 25 years, one of the most studied algorithms in the field of Nonlinear Approximation Theory has been the Thresholding Greedy Algorithm. In this paper, we propose new summability methods for this algorithm, generating two new types of greedy-like bases - namely Cesàro quasi-greedy and de la Vallée-Poussin-quasi-greedy bases. We analyze the connection between these types of bases and the well-known quasi-greedy bases, and leave some open problems for future research. In addition, as a consequence of our techniques for handling these summability methods, we answer a question posed by P. Wojtaszczyk in [16], by giving a categorial proof of equivalence between the uniform boundedness of the greedy sums and the convergence of the thresholding greedy algorithm.
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 $\mathbb{N}$. In this paper, we fix a sequence $(a_n)_{n=1}^\infty$ and compare the TGA against projections onto consecutive terms of the sequence and its shifts. We call the corresponding greedy-type condition the $\mathcal{F}_{(a_n)}$-almost greedy property. Our first result shows that the $\mathcal{F}_{(a_n)}$-almost greedy property is equivalent to the classical almost greedy property if and only if $(a_n)_{n=1}^\infty$ is bounded. Then we establish an analog of the result for the strong partially greedy property. Finally, we show that under a certain projection rule and conditions on the sequence $(a_n)_{n=1}^\infty$, we obtain a greedy-type condition that lies strictly between the almost greedy and strong partially greedy properties.
Property~(A) is a week symmetry condition that plays a fundamental role in the characterization of greedy-type bases in the isometric case, i.e., when the constants involved in the study of the efficiency of the thresholding greedy algorithm in Banach spaces are sharp. In this note we build examples of Banach spaces with Schauder bases that have Property~(A) but fail to be unconditional, thus settling a long standing problem in the area. As a by-product of our work we hone our construction to produce counterexamples that solve other open questions in the isometric theory of greedy bases.
Greedy bases are those bases where the Thresholding Greedy Algorithm (introduced by S. V. Konyagin and V. N. Temlyakov) produces the best possible approximation up to a constant. In 2017, Berná and Blasco gave a characterization of these bases using polynomials with constant coefficients. In this paper, we continue this study improving some optimization problems and extending some results to the context of quasi-Banach spaces.
In [25], T. Oikhberg introduced and studied variants of the greedy and weak greedy algorithms for sequences with gaps, with a focus on the $\mathbf n$-$t$-quasi-greedy property that is based on them. Building upon this foundation, our current work aims to further investigate these algorithms and bases while introducing new ideas for two primary purposes. Firstly, we aim to prove that for $\mathbf n$ with bounded quotient gaps, $\mathbf n$-$t$-quasi-greedy bases are quasi-greedy bases. This generalization extends the result previously established in [7] to the context of Markushevich bases and, also, completes the answer to a question from [25]. The second objective is to extend certain approximation properties of the greedy algorithm to the context of sequences with gaps and study if there is a relationship between this new extension and the usual convergence.
The main results in this paper contribute to bring 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 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.
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 almost greedy. We also define the notion of weak weight almost greedy bases, and prove that this formally weaker concept is equivalent to that concept of weight almost greedy bases. Finally, we study some parameters involving the weak thresholding and Chebyshevian greedy algorithms.
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,$τ$).
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.
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].
Elton's near unconditionality and quasi-greediness for largest coefficients are two properties of bases that made their appearance in functional analysis from very different areas of research. One of our aims in this note is to show that, oddly enough, they are connected to the extent that they are equivalent notions. We take advantage of this new description of the former property to further the study of the threshold function associated with near unconditionality. Finally, we made a contribution to the isometric theory of greedy bases by characterizing those bases that are $1$-quasi-greedy for largest coefficients.