Simultaneous greedy approximation in Banach spaces

Abstract

We study nonlinear m-term approximation with regard to a redundant dictionary $\mathcal{D}$ in a Banach space. It is known that in the case of Hilbert space H the pure greedy algorithm (or, more generally, the weak greedy algorithm) provides for each $f\in H$ and any dictionary $\mathcal{D}$ an expansion into a series

$$f=\sum _{j=1}^{\infty }{c}_j(f){\varphi }_j(f),{\varphi }_j(f)\in \mathcal{D},j=1,2,\dots$$

with the Parseval property: $\parallel f{\parallel }^2={\sum }_j|c_j(f){|}^2$. The orthogonal greedy algorithm (or, more generally, the weak orthogonal greedy algorithm) has been introduced in order to enhance the rate of convergence of greedy algorithms. Recently, we have studied analogues of the PGA and WGA for a given finite number of functions $f^1,\dots ,f^N$ with a requirement that the dictionary elements ${\varphi }_j$ of these expansions are the same for all $f^i$, $i=1,\dots ,N$. We have studied convergence and rate of convergence of such expansions which we call simultaneous expansions. The goal of this paper is twofold. First, we work in a Hilbert space and enhance the convergence of the simultaneous greedy algorithms by introducing an analogue of the orthogonalization process, and we give estimates on the rate of convergence. Then, we study simultaneous greedy approximation in a more general setting, namely, in uniformly smooth Banach spaces.