Greedy expansions in Banach spaces

Abstract

We study convergence and rate of convergence of expansions of elements in a Banach space X into series with regard to a given dictionary $\mathcal{D}$ . For convenience we assume that $\mathcal{D}$ is symmetric: $g\in\mathcal{D}$ implies $-g\in\mathcal{D}$ . The primary goal of this paper is to study representations of an element f∈X by a series

$$f\sim\sum_{j=1}^{\infty}c_{j}(f)g_{j}(f),\quad g_{j}(f)\in\mathcal{D},\ c_{j}(f)>0,\ j=1,2,\dots.$$

In building such a representation we should construct two sequences: {g _j (f)} ^∞_j=1 and {c _j (f)} ^∞_j=1 . In this paper the construction of {g _j (f)} ^∞_j=1 will be based on ideas used in greedy-type nonlinear approximation. This explains the use of the term greedy expansion. We use a norming functional $F_{f_{m-1}}$ of a residual f _m−1 obtained after m−1 steps of an expansion procedure to select the mth element $g_{m}(f)\in \mathcal{D}$ from the dictionary. This approach has been used in previous papers on greedy approximation. The greedy expansions in Hilbert spaces are well studied. The corresponding convergence theorems and estimates for the rate of convergence are known. Much less is known about greedy expansions in Banach spaces. The first substantial result on greedy expansions in Banach spaces has been obtained recently by Ganichev and Kalton. They proved a convergence result for the L _p , 1<p<∞, spaces. In this paper we find a simple way of selecting coefficients c _m (f) that provides convergence of the corresponding greedy expansions in any uniformly smooth Banach space. Moreover, we obtain estimates for the rate of convergence of such greedy expansions for $f\in A_{1}(\mathcal{D})$ – the closure (in X) of the convex hull of $\mathcal{D}$ .