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
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 - INAMAT2
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.
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 Publikoa
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.
Open Access
Sparse approximation using new greedy-like bases in superreflexive spaces
(Instytut Matematyczny, 2023) Albiac Alesanco, Fernando José; Ansorena, José L.; Berasategui, Miguel; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2
This paper is devoted to theoretical aspects of optimality of sparse approximation. We undertake a quantitative study of new types of greedy-like bases that have recently arisen in the context of non-linear 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 regard 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 Dilworth-Kalton-Kutzarova method (2003) 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.
Open Access
Weak forms of unconditionality of bases in greedy approximation
(Instytut Matematyczny, 2022) 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
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.