BUSCADOR RECOLECTA

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., F. Albiac acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L.Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Analisis Vectorial, Multilineal y Aproximacion. M. Berasategui and S. Lassalle were supported by ANPCyT PICT-2018-04104 and CONICET PIP 1609. P. M. Berna by Grants PID2019-105599GB-I00 (Agencia Estatal de Investigacion, Spain) and 20906/PI/18 from Fundacion Seneca (Region de Murcia, Spain). S. Lassalle was also supported in part by CONICET PIP0483 and PAI UdeSA 2020-2021. Open Access funding provided by Universidad Pública de Navarra.

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., F. Albiac acknowledges the support of the Spanish Ministry for Science and Innovation underGrant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aproximacion. P. M. Berna acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-105599GB-I00 and the Grant 20906/PI/18 from Fundacion Seneca (Region de Murcia, Spain). Open Access funding provided by Universidad Publica de Navarra

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., F. Albiac acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities Grant PGC2018-095366-B-I00 for Análisis Vectorial, Multilineal y Aplicaciones. P. M. Berná acknowledges the support of the Spanish Ministry for Economy and Competitivity Grants MTM-2016-76566-P and PID2019-105599GB-100 (Agencia Estatal de Investigación). P. M. Berná was also supported by Grant 20906/PI/18 from Fundación Séneca (Región de Murcia, Spain). P. Wojtaszczyk was partially supported by National Science Centre, Poland, grant UMO-2016/21/B/ST1/00241. This work was supported by EPSRC grant number EP/R014604/1.

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., F. Albiac acknowledges the support of the Spanish Ministry for Science and Innovation under Grant PID2019-107701GB-I00 for Operators, lattices, and structure of Banach spaces. F. Albiac and J. L. Ansorena acknowledge the support of the Spanish Ministry for Science, Innovation, and Universities under Grant PGC2018-095366-B-I00 for Analisis Vectorial, Multilineal y Aproximacion. M. Berasategui and S. Lassalle were supported by ANPCyT PICT-2018-04104 and CONICET PIP 1609. P. Berna by Grant PID2019-105599GB-I00 (Agencia Estatal de Investigacion, Spain) and 20906/PI/18 from Fundacion Seneca (Region de Murcia, Spain). S. Lassalle was also supported in part by CONICET PIP 0483 and PAI UdeSA 2020-2021.