Open Access
Conditional quasi-greedy bases in non-superreflexive Banach spaces
(Springer, 2019) Albiac Alesanco, Fernando José; Ansorena, José L.; Wojtaszczyk, Przemyslaw; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
For a conditional quasi-greedy basis B in a Banach space, the associated conditionality constants km[B] verify the estimate km[B]=O(logm). Answering a question raised by Temlyakov, Yang, and Ye, several authors have studied whether this bound can be improved when we consider quasi-greedy bases in some special class of spaces. It is known that every quasi-greedy basis in a superreflexive Banach space verifies km[B]=O((logm)1-E) for some 0
Open Access
Projections and unconditional bases in direct sums of ℓp SPACES, 0
(Wiley, 2021) Albiac Alesanco, Fernando José; Ansorena, José L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
We show that every unconditional basis in a finite direct sum ⊕p∈Aℓp , with A ⊂ (0,∞], splits into unconditional bases of each summand. This settles a 40 years old question raised in 'A. Ortyński, Unconditional bases in ℓp ⊕ ℓq, 0< p < q <1, Math. Nachr. 103 (1981), 109–116'. As an application we obtain that for any A ⊂ (0,1] finite, the spaces Z = ⊕p∈A ℓp,Z ⊕ ℓ2, and Z ⊕ c0 have a unique unconditional basis up to permutation.
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
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
Submultiplicative norms in CC(K) spaces with applications to uniform algebras
(Springer, 2024) Albiac Alesanco, Fernando José; Blasco, Óscar; Briem, E.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
Certain algebra norms on the algebra of functions on a two-point set are defined and used to construct a new family of parametric algebra norms on the space CC(K) of continuous complex-valued functions on a compact Hausdorff space. As a by-product of our work we transfer our construction to uniform algebras to obtain a new collection of norms with special properties.
Open Access
Lipschitz free p-spaces for 0 < p < 1
(Springer, 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
This paper initiates the study of the structure of a new class of p-Banach spaces, 0
Open Access
Linear versus nonlinear forms of partial unconditionality of bases
(Elsevier, 2024-07-22) Albiac Alesanco, Fernando José; Ansorena, José L.; Berasategui, Miguel; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
The main results in this paper contribute to bringing 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-oscillation 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.
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
Embargo
Twenty-five years of greedy bases
(Elsevier, 2025-05-01) 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 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.