Trigonometric Quasi-Greedy Bases for $L^p(T;w)$

Abstract

Let B = {en}n∈N be a bounded Schauder basis for a Banach space X, i.e., a basis for which 0 ρ(j). Then the greedy m-term approximant to x is given by Gm(x) = ∑m j=1 e ∗ ρ(j)(x)eρ(j). The question is whether the greedy algorithm is convergent. This is clearly the case for an unconditional basis where the expansion x = ∑∞ k=1 e ∗ k(x)ek converges regardless of the ordering. However, Temlyakov and Konyagin [7] showed that the greedy algorithm may also converge for certain conditional bases. This lead them to define so-called quasi-greedy bases, see [7]. The definition of a quasi-greedy basis in [7] is slightly technical, but it was shown by Wojtaszczyk [11] to be equivalent to to the following statement which we use as definition.