# Person: Albiac Alesanco, Fernando José

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Albiac Alesanco

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Fernando José

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Matemáticas

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0000-0001-7051-9279

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Publication Open Access Building highly conditional almost greedy and quasi-greedy bases in Banach spaces(Elsevier, 2019) Albiac Alesanco, Fernando José; Ansorena, José L.; Dilworth, S. J.; Kutzarova, Denka; Estadística, Informática y Matemáticas; Estatistika, Informatika eta MatematikaShow more It is known that for a conditional quasi-greedy basis B in a Banach space X, the associated sequence (k(m)[B](m=1)(infinity) of its conditionality constants verifies the estimate k(m)[B] = O(log m) and that if the reverse inequality log m =O(k(m)[B]) holds then X is non-superreflexive. Indeed, it is known that a quasi-greedy basis in a superreflexive quasi-Banach space fulfils the estimate k(m)[B] =O(log m)(1-epsilon) for some epsilon > 0. However, in the existing literature one finds very few instances of spaces possessing quasi-greedy basis with conditionality constants "as large as possible." Our goal in this article is to fill this gap. To that end we enhance and exploit a technique developed by Dilworth et al. in [15] and craft a wealth of new examples of both non-superreflexive classical Banach spaces having quasi-greedy bases B with k(m)[B] = O(log m) and superreflexiye classical Banach spaces having for every epsilon > 0 quasi-greedy bases B with k(m)[B] = O(log m)(1-epsilon). Moreover, in most cases those bases will be almost greedy.Show more Publication Open Access New parameters and Lebesgue-type estimates in greedy approximation(Cambridge University Press, 2022) Albiac Alesanco, Fernando José; Ansorena, José L.; Berná, Pablo M.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate PublikoaShow more The purpose of this paper is to quantify the size of the Lebesgue constants (𝑳𝑚)∞ 𝑚=1 associated with the thresholding greedy algorithm in terms of a new generation of parameters that modulate accurately some features of a general basis. This fine tuning of constants allows us to provide an answer to the question raised by Temlyakov in 2011 to find a natural sequence of greedy-type parameters for arbitrary bases in Banach (or quasi-Banach) spaces which combined linearly with the sequence of unconditionality parameters (𝒌𝑚)∞ 𝑚=1 determines the growth of (𝑳𝑚)∞ 𝑚=1. Multiple theoretical applications and computational examples complement our study.Show more Publication Open Access Asymptotic greediness of the Haar system in the spaces Lp[0 , 1] , 1< p< ∞(Springer, 2019) Albiac Alesanco, Fernando José; Ansorena, José L.; Berná, Pablo M.; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y MatemáticasShow more Our aim in this paper is to attain a sharp asymptotic estimate for the greedy constant Cg[H(p), Lp] of the (normalized) Haar system H(p) in Lp[0 , 1] for 1 < p < ∞. We will show that the super-democracy constant of H(p) in Lp[0 , 1] grows as p∗= max { p, p/ (p- 1) } as p∗ goes to ∞. Thus, since the unconditionality constant of H(p) in Lp[0 , 1] is p∗- 1 , the well-known general estimates for the greedy constant of a greedy basis obtained from the intrinsic features of greediness (namely, democracy and unconditionality) yield that p∗≲Cg[H(p),Lp]≲(p∗)2. Going further, we develop techniques that allow us to close the gap between those two bounds, establishing that, in fact, Cg[H(p), Lp] ≈ p∗. Our work answers a question that was raised by Hytonen (2015).Show more Publication Open Access Elton's near unconditionality of bases as a threshold-free form of greediness(Elsevier, 2023) Albiac Alesanco, Fernando José; Ansorena, José L.; Berasategui, Miguel; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2Show more 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 make a contribution to the isometric theory of greedy bases by characterizing those bases that are 1-quasi-greedy for largest coefficients.Show more Publication 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 MatemáticasShow more 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.Show more Publication 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 PublikoaShow more 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.Show more Publication 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 PublikoaShow more 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.Show more Publication Open Access Bidemocratic bases and their connections with other greedy-type bases(Springer, 2023) Albiac Alesanco, Fernando José; Ansorena, José L.; Berasategui, Miguel; Berná, Pablo M.; Lassalle, Silvia; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate PublikoaShow more 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 < infinity the space L-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 demo-cratic bases.Show more Publication 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 MatemáticasShow more We construct for each 0Show more Publication Open Access Unconditional and quasi-greedy bases in L-p with applications to Jacobi polynomials Fourier series(European Mathematical Society, 2019) Albiac Alesanco, Fernando José; Ansorena, José L.; Ciaurri, Óscar; Varona, Juan L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta MatematikaShow more We show that the decreasing rearrangement of the Fourier series with respect to the Jacobi polynomials for functions in L-p does not converge unless p = 2. As a by-product of our work on quasi-greedy bases in L-p(µ), we show that no normalized unconditional basis in L-p, p not equal 2, can be semi-normalized in L-q for q not equal p, thus extending a classical theorem of Kadets and Pelczynski from 1968.Show more

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