Greedy algorithms in Banach spaces

Abstract

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 ϕ ^c_m from D, and (2) constructing an m-term approximant G ^c_m . We include the name of Chebyshev in the name of this algorithm because at the substep (2) the approximant G ^c_m is chosen as the best approximant from Span(ϕ ^c₁ ,...,ϕ ^c_m ). The term Weak Greedy Algorithm indicates that at each substep (1) we choose ϕ ^c_m as an element of D that satisfies some condition which is “t _m -times weaker” than the condition for ϕ ^c_m to be optimal (t _m =1). We got error estimates for Banach spaces with modulus of smoothness ρ(u)≤γu ^q, 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+t ₁ ^p+⋅⋅⋅+t _m ^p)^−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 G ^r _m is a convex linear combination of 0,ϕ₁ ^r,...,ϕ^r _m . We also proved some convergence results for WCGA and WRGA.