Greedy algorithms in Banach spaces

We study efficiency of approximation and convergence of two greedy type algorithms in uniformly smooth Banach spaces. The Weak Chebyshev Greedy Algorithm (WCGA) is defined for an arbitrary dictionary D and provides nonlinear m-term approximation with regard to D. This algorithm is defined inductively with the mth step consisting of two basic substeps: (1) selection of an mth element ϕmc from D, and (2) constructing an m-term approximant Gmc. We include the name of Chebyshev in the name of this algorithm because at the substep (2) the approximant Gmc is chosen as the best approximant from Span(ϕ1c,...,ϕmc). The term Weak Greedy Algorithm indicates that at each substep (1) we choose ϕmc as an element of D that satisfies some condition which is “tm-times weaker” than the condition for ϕmc to be optimal (tm=1). We got error estimates for Banach spaces with modulus of smoothness ρ(u)≤γuq, 1<q≤2. We proved that for any f from the closure of the convex hull of D the error of m-term approximation by WCGA is of order (1+t1p+⋅⋅⋅+tmp)−1/p, 1/p+1/q=1. Similar results are obtained for Weak Relaxed Greedy Algorithm (WRGA) and its modification. In this case an approximant Grm is a convex linear combination of 0,ϕ1r,...,ϕrm. We also proved some convergence results for WCGA and WRGA.