Greedy expansions in Hilbert spaces

Abstract

We study the rate of convergence of expansions of elements in a Hilbert space H into series with regard to a given dictionary D. The primary goal of this paper is to study representations of an element f ∈ H by a series f ∼ ∑j=1∞cj(f)gj(f), \(g_j \left( f \right) \in \mathcal{D}\). Such a representation involves two sequences: {gj(f)}j=1∞ and {cj(f)j=1∞. In this paper the construction of {gj(f)}j=1∞ is based on ideas used in greedy-type nonlinear approximation, hence the use of the term greedy expansion. An interesting open problem questions, “What is the best possible rate of convergence of greedy expansions for f ∈ A1(D)?” Previously it was believed that the rate of convergence was slower than \(m^{ - \tfrac{1} {4}}\). The qualitative result of this paper is that the best possible rate of convergence of greedy expansions for \(f \in A_1 \left( \mathcal{D} \right)\) is faster than \(m^{ - \tfrac{1} {4}}\). In fact, we prove it is faster than \(m^{ - \tfrac{2} {7}}\).