Equivalence between almost-greedy and semi-greedy bases

Abstract

In [3] it was proved that almost-greedy and semi-greedy bases are equivalent in the context of Banach spaces with finite cotype. In this paper we show this equivalence for general Banach spaces.2000 Mathematics Subject Classification. 46B15, 41A65.

Characterization of Weight-Semi-greedy Bases

“…As in the Remark 1.6, the last bound is an improvement respect to the bound C sg = (C 3 q C d ) using Proposition 6.1. Hence, for w = (1, 1, ...), we recover the result proved in [2] as we say in the following corollary. The structure of the paper is the following: in Section 2, we write and study some preliminary results that we will use in the proof of the main results.…”

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One classical result in greedy approximation theory is that almost-greedy and semigreedy bases are equivalent in the context of Schauder bases in Banach spaces with finite cotype. This result was proved by S. J. Dilworth, N. J. Kalton and D. Kutzarova in [7] and, recently, the first author in [2] proved that the condition of finite cotype can be removed in this result. In [11], the authors extend the notion of semi-greediness to the context of weights and proved the following: if w is a weight and B is a Schauder basis in a Banach space X with finite cotype, then w-semi-greediness and w-almost-greediness are equivalent notions.

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“…The first authors that studied the relation between semi-greediness and almost-greediness were S. J. Dilworth, N. J. Kalton and D. Kutzarova in [7]. Recently, the first author proved in [2] that the condition of finite cotype can be removed. Focusing our attention in the weighted case, in [11], the authors extend the definition of semigreediness.…”

Lebesgue inequalities for Chebyshev thresholding greedy algorithms

“…Then the sequence {e n } ∞ n=1 is L 1 (µ)-null in X. Indeed, this follows from case (2), and the fact that C(K) is dense in L 1 (µ).…”

“…When L ch m = O(1) the system B is called semi-greedy; see [7]. We remark that the first author recently established that a Schauder basis B is semi-greedy if and only if is quasigreedy and democratic; see [2].…”