Relaxation in Greedy Approximation

We study greedy algorithms in a Banach space from the point of view of convergence and rate of convergence. There are two well-studied approximation methods: the Weak Chebyshev Greedy Algorithm (WCGA) and the Weak Relaxed Greedy Algorithm (WRGA). The WRGA is simpler than the WCGA in the sense of computational complexity. However, the WRGA has limited applicability. It converges only for elements of the closure of the convex hull of a dictionary. In this paper we study algorithms that combine good features of both algorithms, the WRGA and the WCGA. In the construction of such algorithms we use different forms of relaxation. First results on such algorithms have been obtained in a Hilbert space by A. Barron, A. Cohen, W. Dahmen, and R. DeVore. Their paper was a motivation for the research reported here.