The weighted property (A) and the greedy algorithm

Abstract

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).2000 Mathematics Subject Classification. 46B15, 41A65.

Weight-partially greedy bases and weight-property (A)

“…Thus from Theorem 2.1 it follows that if a basis (e n ) is quasi-greedy and has w-left Property (A) then (e n ) is w-partially greedy. We now prove this result using the arguments similar to [2] to get better estimates in terms of the constant. Proof.…”

“…It was proved in [2] that a basis is w-partially greedy if and only if it is quasi-greedy and wconservative. Thus from Theorem 2.1 it follows that the w-partially greedy basis considered in this paper is equivalent to the one considered in [2].…”

“…In [6] the authors studied weight-greedy bases. Recently, in [5] and [2], the authors studied weight-almost greedy bases and weight-partially greedy bases (using the definition from [3]). In [6], [5] and [2] the authors proved a criterion for weight-greedy, weight-almost greedy and weight-partially greedy bases similar to the one for greedy, almost greedy and partially greedy bases respectively.…”

“…Recently, in [5] and [2], the authors studied weight-almost greedy bases and weight-partially greedy bases (using the definition from [3]). In [6], [5] and [2] the authors proved a criterion for weight-greedy, weight-almost greedy and weight-partially greedy bases similar to the one for greedy, almost greedy and partially greedy bases respectively. In [2] the notion of w-Property (A) was introduced and a characterization of (i) w-greedy bases in terms of unconditionality and w-Property (A) (ii) w-almost greedy bases in terms of quasi-greediness and w-Property (A) was proved.…”

“…Remark 2.2. In [2] a basis (e n ) of a Banach space X was defined to be w-partially greedy if for all m, r such that w({1, · · · , m}) ≤ w(Λ r (x)), there exists a constant C such that…”