Embargo
Twenty-five years of greedy bases
(Elsevier, 2024-12-21) Albiac Alesanco, Fernando José; Ansorena, José L.; Temlyakov, Vladimir; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2
Although the basic idea behind the concept of a greedy basis had been around for some time, the formal development of a theory of greedy bases was initiated in 1999 with the publication of the article [S.V. Konyagin and V.N. Temlyakov, A remark on greedy approximation in Banach spaces, East J. Approx. 5 (3) (1999), 365-379]. The theoretical simplicity of the thresholding greedy algorithm became a model for a procedure widely used in numerical applications and the subject of greedy bases evolved very rapidly from the point of view of approximation theory. The idea of studying greedy bases and related greedy algorithms attracted also the attention of researchers with a classical Banach space theory background. From the more abstract point of functional analysis, the theory of greedy bases and its derivates evolved very fast as many fundamental results were discovered and new ramifications branched out. Hundreds of papers on greedy-like bases and several monographs have been written since the appearance of the aforementioned foundational paper. After twenty-five years, the theory is very much alive and it continues to be a very active research topic both for functional analysts and for researchers interested in the applied nature of nonlinear approximation alike. This is why we believe it is a good moment to gather a selection of 25 open problems (one per year since 1999!) whose solution would contribute to advance the state of art of this beautiful topic.
Open Access
The uniqueness of unconditional basis of the 2-convexified Tsirelson space, revisited
(Springer, 2024-10-13) Albiac Alesanco, Fernando José; Ansorena, José L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
One of the hallmarks in the study of the classification of Banach spaces with a unique (normalized) unconditional basis was the unexpected result by Bourgain, Casazza, Lindenstrauss, and Tzafriri from their 1985 Memoir that the 2-convexified Tsirelson space T(2) had that property (up to equivalence and permutation). Indeed, on one hand, finding a “pathological” space (i.e., not built out as a direct sum of the only three classical sequence spaces with a unique unconditional basis) shattered the hopeful optimism of attaining a satisfactory description of all Banach spaces which enjoy that important structural feature. On the other hand it encouraged furthering a research topic that had received relatively little attention until then. After forty years, the advances on the subject have shed light onto the underlying patterns shared by those spaces with a unique unconditional bases belonging to the same class, which has led to reproving the original theorems with fewer technicalities. Our motivation in this note is to revisit the aforementioned result on the uniqueness of unconditional basis of T(2) from the current state-of-art of the subject and to fill in some details that we missed from the original proof.
Open Access
Greedy approximation for biorthogonal systems in quasi-Banach spaces
(Instytut Matematyczny, 2021) Albiac Alesanco, Fernando José; Ansorena, José L.; Berná, Pablo M.; Wojtaszczyk, Przemyslaw; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
The general problem addressed in this work is the development of a systematic study of the thresholding greedy algorithm for general biorthogonal systems in quasi-Banach spaces from a functional-analytic point of view. If (Formula Presented) is a biorthogonal system in X then for each x ∈ X we have a formal expansion (Formula Presented). The thresholding greedy algorithm (with threshold ε > 0) applied to x is formally defined as (Formula Presented). The properties of this operator give rise to the different classes of greedy-type bases. We revisit the concepts of greedy, quasi-greedy, and almost greedy bases in this comprehensive framework and provide the (non-trivial) extensions of the corresponding characterizations of those types of bases. As a by-product of our work, new properties arise, and the relations among them are carefully discussed.
Open Access
Embeddability of ℓp and bases in Lipschitz free p-spaces for 0 < p ≤ 1
(Elsevier, 2020) Albiac Alesanco, Fernando José; Ansorena, José L.; Cúth, Marek; Doucha, Michal; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
Our goal in this paper is to continue the study initiated by the authors in of the geometry of the Lipschitz free p-spaces over quasimetric spaces for 0 < p ≤ 1, denoted Fp(M). Here we develop new techniques to show that, by analogy with the case p = 1, the space p embeds isomorphically in Fp(M) for 0 < p < 1. Going further we see that despite the fact that, unlike the case p = 1, this embedding need not be complemented in general, complementability of p in a Lipschitz free p-space can still be attained by imposing certain natural restrictions to M. As a by-product of our discussion on bases in Fp([0, 1]), we obtain examples of p-Banach spaces for p < 1 that are not based on a trivial modification of Banach spaces, which possess a basis but fail to have an unconditional basis.
Open Access
Structure of the Lipschitz free p-spaces Fp(Zd) and Fp(Rd) for 0 < p ≤ 1
(Springer, 2021) Albiac Alesanco, Fernando José; Ansorena, José L.; Cúth, Marek; Doucha, Michal; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
Our aim in this article is to contribute to the theory of Lipschitz free p-spaces for 0 < p ≤ 1 over the Euclidean spaces Rd and Zd. To that end, on one hand we show that Fp(Rd) admits a Schauder basis for every p ∈ 2 (0, 1], thus generalizing the corresponding result for the case p = 1 by H_ajek and Perneck_a [20, Theorem 3.1] and answering in the positive a question that was raised in [3]. Explicit formulas for the bases of both Fp(Rd) and its isomorphic space Fp([0, 1]d) are given. On the other hand we show that the well-known fact that F(Z) is isomorphic to l1 does not extend to the case when p < 1, that is, Fp(Z) is not isomorphic to lp when 0 < p < 1.
Open Access
Counterexamples in isometric theory of symmetric and greedy bases
(Elsevier, 2024) Albiac Alesanco, Fernando José; Ansorena, José L.; Blasco, Óscar; Chu, Hùng Việt; Oikhberg, Timur; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
We continue the study initiated in Albiac and Wojtaszczyk (2006) of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to 1. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be 1-suppression unconditional, thus settling Problem 4.4 from Albiac and Ansorena (2017). In particular, our construction demonstrates that bases with Property (A) need not be 1-greedy even with the additional assumption that they are unconditional and symmetric. We also exhibit a finite-dimensional counterpart of this example, and show that, at least in the finite-dimensional setting, Property (A) does not pass to the dual. As a by-product of our arguments, we prove that a symmetric basis is unconditional if and only if it is total, thus generalizing the well-known result that symmetric Schauder bases are unconditional.
Open Access
Democracy of quasi-greedy bases in p-Banach spaces with applications to the efficiency of the thresholding greedy algorithm in the hardy spaces Hp(Dd)
(Cambridge University Press, 2023) Albiac Alesanco, Fernando José; Ansorena, José L.; Bello, Glenier; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2
We use new methods, specific for non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a -Banach space for 0 < p < p are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy Hp(D) space for 0 < p < 1 are democratic while, in contrast, no quasi-greedy basis of Hp(Dd) for d > 2 is, solving thus a problem that was raised in [7]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided.
Open Access
On certain subspaces of p for 0 < p ≤ 1 and their applications to conditional quasi-greedy bases in p-Banach spaces
(Springer, 2021) Albiac Alesanco, Fernando José; Ansorena, José L.; Wojtaszczyk, Przemyslaw; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
We construct for each 0
Open Access
Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces
(Kluwer Academic Publishers, 2022) Albiac Alesanco, Fernando José; Ansorena, José L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
This paper is devoted to providing a unifying approach to the study of the uniqueness of unconditional bases, up to equivalence and permutation, of infinite direct sums of quasi-Banach spaces. Our new approach to this type of problem permits to show that a wide class of vector-valued sequence spaces have a unique unconditional basis up to a permutation. In particular, solving a problem from Albiac and Leránoz (J Math Anal Appl 374(2):394-401, 2011. https://doi.org/10.1016/j.jmaa.2010.09.048) we show that if X is quasi-Banach space with a strongly absolute unconditional basis then the infinite direct sum -1(X) has a unique unconditional basis up to a permutation, even without knowing whether X has a unique unconditional basis or not. Applications to the uniqueness of unconditional structure of infinite direct sums of non-locally convex Orlicz and Lorentz sequence spaces, among other classical spaces, are also obtained as a by-product of our work.
Open Access
The structure of greedy-type bases in Tsirelson's space and its convexifications
(Scuola Normale Superiore, 2024-10-09) Albiac Alesanco, Fernando José; Ansorena, José L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2
Tsirelson's space 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 T , a subject that had not been previously explored in the literature. More specifically, we show that Tsirelson's space and its convexifications T (p) for 0 < p < 1 have uncountably many non-equivalent greedy bases. We also investigate the conditional basis structure of spaces T (p) in the range of 0 < p < 1 and prove that they have uncountably many non-equivalent conditional almost greedy bases.