Weak weight-semi-greedy bases (original) (raw)
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Weak weight-semi-greedy Markushevich bases
arXiv (Cornell University), 2021
The main purpose of this paper is to study weight-semi-greedy Markushevich bases, and in particular, find conditions under which such bases are weight-almost greedy. In this context, we prove that, for a large clase of weights, the two notions are equivalent. We also show that all weight semigreedy bases are truncation quasi-greedy and weight-superdemocratic. In all of the above cases, we also bring to the context of weights the weak greedy and Chebyshev greedy algorithms-which are frequently studied in the literature on greedy approximation. In the course of our work a new property arises naturally and its relation with squeeze symmetric and bidemocratic bases is given. In addition, we study some parameters involving the weak thresholding and Chebyshevian greedy algorithms. Finally, we give examples of conditional bases with some of the weighted greedy-type conditions we study.
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.
Weak forms of unconditionality of bases in greedy approximation
Studia Mathematica
In this paper 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.
Proceedings of the Steklov Institute of Mathematics
We introduce the notion of a weight-almost greedy basis and show that a basis for a real Banach space is w-almost greedy if and only if it is both quasi-greedy and w-democratic. We also introduce the notion of weight-semi-greedy basis and show that a w-almost greedy basis is w-semi-greedy and that the converse holds if the Banach space has finite cotype.
The weighted property (A) and the greedy algorithm
Journal of Approximation Theory
We investigate various aspects of the "weighted" greedy algorithm with respect to a Schauder basis. For a weight w, we describe w-greedy, w-almost-greedy, and w-partiallygreedy bases, and examine some properties of w-semi-greedy bases. To achieve these goals, we introduce and study the w-Property (A).
On consecutive greedy and other greedy-like type of bases
arXiv (Cornell University), 2023
We continue our study of the Thresholding Greedy Algorithm when we restrict the vectors involved in our approximations so that they either are supported on intervals of N or have constant coefficients. We introduce and characterize what we call consecutive greedy bases and provide new characterizations of almost greedy and squeeze symmetric Schauder bases. Moreover, we investigate some cases involving greedy-like properties with constant 1 and study the related notion of Property (A, τ).
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.
Strong Partially Greedy Bases and Lebesgue-Type Inequalities
Constructive Approximation
In this paper we continue the study of Lebesgue-type inequalities for greedy algorithms. We introduce the notion of strong partially greedy Markushevich bases and study the Lebesgue-type parameters associated with them. We prove that this property is equivalent to that of being conservative and quasi-greedy, extending a similar result given in [9] for Schauder bases. We also give a characterization of 1-strong partial greediness, following the study started in [1, 3].
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.
Some inequalities for the tensor product of greedy bases and weight-greedy bases
In this paper we study properties of bases that are important in nonlinear m-term approximation with regard to these bases. It is known that the univariate Haar basis is a greedy basis for L p ([0, 1)), 1 < p < ∞. This means that a greedy type algorithm realizes nearly best m-term approximation for any individual function. It is also known that the multivariate Haar basis that is a tensor product of the univariate Haar bases is not a greedy basis. This means that in this case a greedy algorithm provides a m-term approximation that may be equal to the best m-term approximation multiplied by a growing (with m) to infinity factor. There are known results that describe the behavior of this extra factor for the Haar basis. In this paper we extend these results to the case of a basis that is a tensor product of the univariate greedy bases for L p ([0, 1)), 1 < p < ∞. Also, we discuss weight-greedy bases and prove a criterion for weight-greedy bases similar to the one for greedy bases.