Weight-Almost Greedy Bases (original) (raw)
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On the existence of almost greedy bases in Banach spaces
Studia Mathematica, 2003
We consider several greedy conditions for bases in Banach spaces that arise naturally in the study of the Thresholding Greedy Algorithm (TGA). In particular, we continue the study of almost greedy bases begun in [3]. We show that almost greedy bases are essentially optimal for n-term approximation when the TGA is modified to include a Chebyshev approximation. We prove that if a Banach space X has a basis and contains a complemented subspace with a symmetric basis and finite cotype then X has an almost greedy basis. We show that c 0 is the only L ∞ space to have a quasi-greedy basis. The Banach spaces which contain almost greedy basic sequences are characterized. Contents 8. Quasi-greedy bases in L ∞ spaces 31 References 37
Extensions and New Characterizations of Some Greedy-Type Bases
Bulletin of the Malaysian Mathematical Sciences Society, 2023
Partially greedy bases in Banach spaces were introduced by Dilworth et al. as a strictly weaker notion than the (almost) greedy bases. In this paper, we study two natural ways to strengthen the definition of partial greediness. The first way produces what we call the consecutive almost greedy property, which turns out to be equivalent to the almost greedy property. Meanwhile, the second way reproduces the PG property for Schauder bases but a strictly stronger property for general bases.
2021
We continue the study of the implication from semi-greedy to almost greedy Markushevich bases, and improve known results. We also introduce and study the notion of weak weight-semi-greedy bases which extends the concepts of weight semi-greedy and weak semi-greedy bases. In particular, we study conditions under which such bases are weight almost greedy. Finally, we define weak weight almost greedy bases, and prove that this natural extension is equivalent to the concept of weight almost greedy bases.
Characterization of greedy bases in Banach spaces
Journal of Approximation Theory, 2017
We shall present a new characterization of greedy bases and 1-greedy bases in terms of certain functionals defined using distances to one dimensional subspaces generated by the basis. We also introduce a new property that unifies the notions of unconditionality and democracy and allows us to recover a better dependence on the constants.
Existence and uniqueness of greedy bases in Banach spaces
arXiv: Functional Analysis, 2015
Our aim is to investigate the properties of existence and uniqueness of greedy bases in Banach spaces. We show the non-existence of greedy basis in some Nakano spaces and Orlicz sequence spaces and produce the first-known examples of non-trivial spaces (i.e., different from c_0c_0c0, ell1\ell_1ell1, and ell2\ell_2ell_2) with a unique greedy basis.
2020
We introduce and study the notion of weak semi-greedy systems -which is inspired in the concepts of semi-greedy and Branch semi-greedy systems and weak thresholding sets-, and prove that in the context Markushevich bases in infinite dimensional Banach spaces, the notions of \textit{ semi-greedy, branch semi-greedy, weak semi-greedy, and almost greedy} Markushevich bases are all equivalent. This completes and extends some results from \cite{Berna2019}, \cite{Dilworth2003b}, and \cite{Dilworth2012}. We also exhibit an example of a semi-greedy system that is neither almost greedy nor a Markushevich basis, showing that the Markushevich condition cannot be dropped from the equivalence result. In some cases, we obtain improved upper bounds for the corresponding constants of the systems.
Banach spaces with a unique greedy basis
Journal of Approximation Theory, 2016
The purpose of this article is to undertake an in-depth study of the properties of existence and uniqueness of greedy bases in Banach spaces. We show that greedy bases fail to exist for a range of neo-classical spaces within the family of Nakano and Orlicz sequence spaces and find the first-known cases of non-trivial spaces (i.e., different from c 0 , ℓ 1 , and ℓ 2) with a unique greedy basis. The variety and nature of those examples evince that a complete classification of Banach spaces with a unique greedy basis cannot be expected.
Bidemocratic bases and their connections with other greedy-type bases
2021
Abstract. 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 < ∞ the space lp 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 democratic bases.
Renorming spaces with greedy bases
Journal of Approximation Theory, 2014
We study the problem of improving the greedy constant or the democracy constant of a basis of a Banach space by renorming. We prove that every Banach space with a greedy basis can be renormed, for a given ε > 0, so that the basis becomes (1 + ε)-democratic, and hence (2 + ε)-greedy, with respect to the new norm. If in addition the basis is bidemocratic, then there is a renorming so that in the new norm the basis is (1 + ε)-greedy. We also prove that in the latter result the additional assumption of the basis being bidemocratic can be removed for a large class of bases. Applications include the Haar systems in Lp[0, 1], 1 < p < ∞, and in dyadic Hardy space H 1 , as well as the unit vector basis of Tsirelson space.
Weak greedy algorithms and the equivalence between semi-greedy and almost greedy Markushevich bases
2020
We introduce and study the notion of weak semi-greedy systems -which is inspired in the concepts of semi-greedy and branch semi-greedy systems and weak thresholding sets-, and prove that in infinite dimensional Banach spaces, the notions of semi-greedy, branch semi-greedy, weak semigreedy, and almost greedy Markushevich bases are all equivalent. This completes and extends some results from [5], [9], and [13]. We also exhibit an example of a semi-greedy system that is neither almost greedy nor a Markushevich basis, showing that the Markushevich condition cannot be dropped from the equivalence result. In some cases, we obtain improved upper bounds for the corresponding constants of the systems.