On consecutive greedy and other greedy-like type of bases (original) (raw)
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The Thresholding Greedy Algorithm, Greedy Bases, and Duality
Constructive Approximation, 2003
Some new conditions that arise naturally in the study of the Thresholding Greedy Algorithm are introduced for bases of Banach spaces. We relate these conditions to best n-term approximation and we study their duality theory. In particular, we obtain a complete duality theory for greedy bases. Recently, Konyagin and Temlyakov [6] introduced the Thresholding Greedy Algorithm (TGA) (G n ) ∞ n=1 , where G n (x) is obtained by taking the largest n coefficients (precise definitions are given in Section 2). The TGA provides a theoretical model for the thresholding procedure that is used in image compression and other applications.
Lebesgue inequalities for the greedy algorithm in general bases
Revista Matemática Complutense, 2017
We present various estimates for the Lebesgue constants of the thresholding greedy algorithm, in the case of general bases in Banach spaces. We show the optimality of these estimates in some situations. Our results recover and slightly improve various estimates appearing earlier in the literature.
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.
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).
Weaker forms of unconditionality of bases in greedy approximation
arXiv (Cornell University), 2021
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.
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.
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.
Greedy approximations with regard to bases
Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006, 2007
This paper is a survey of recent results on greedy approximations with regard to bases. The theory of greedy approximations is a part of nonlinear approximations. The standard problem in this regard is the problem of m-term approximation where one fixes a basis and seeks to approximate a target function by a linear combination of m terms of the basis. When the basis is a wavelet basis or a basis of other waveforms, then this type of approximation is the starting point for compression algorithms. We are interested in the quantitative aspects of this type of approximation. Introducing the concept of best m-term approximation we obtain a lower bound for the accuracy of any method providing m-term approximation. It is known that a problem of simultaneous optimization over many parameters (like in best m-term approximation) is a very difficult problem. We would like to have an algorithm for constructing m-term approximants that adds at each step only one new element from the basis and keeps elements of the basis obtained at the previous steps. The primary object of our discussion is the Thresholding Greedy Algorithm (TGA) with regard to a given basis. The TGA, applied to a function f , picks at the mth step an element with the mth biggest coefficient (in absolute value) of the expansion of f in the series with respect to the basis. We show that this algorithm is very good for a wavelet basis and is not that good for the trigonometric system. We discuss in detail the behavior of the TGA with regard to the trigonometric system. We also discuss one example of an algorithm from a family of very general greedy algorithms that works in the case of a redundant system instead of a basis. It turns out that this general greedy algorithm is very good for the trigonometric system.
Lebesgue-Type Inequalities for Quasi-greedy Bases
Constructive Approximation, 2013
We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best Nterm error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C log N. We show with two examples that this bound is attained for quasi-greedy democratic bases. We define the quasi-greedy constant K of the basis B to be the least K such that (1.2) holds for all permutations π satisfying (1.1).
Extensions of greedy-like bases for sequences with gaps
arXiv (Cornell University), 2020
In [25], T. Oikhberg introduced and studied variants of the greedy and weak greedy algorithms for sequences with gaps, with a focus on the n-t-quasigreedy property that is based on them. Building upon this foundation, our current work aims to further investigate these algorithms and bases while introducing new ideas for two primary purposes. Firstly, we aim to prove that for n with bounded quotient gaps, n-t-quasi-greedy bases are quasi-greedy bases. This generalization extends the result previously established in [7] to the context of Markushevich bases and, also, completes the answer to a question from [25]. The second objective is to extend certain approximation properties of the greedy algorithm to the context of sequences with gaps and study if there is a relationship between this new extension and the usual convergence.