Greedy expansions in Hilbert spaces

We study the rate of convergence of expansions of elements in a Hilbert space $H$ into series with regard to a given dictionary $\mathcal D$. The primary goal of this paper is to study representations of an element $f\in H$ by a series $f\sim\sum_{j=1}^\infty c_j(f)g_j(f)$, $g_j(f)\in\mathcal D$. Such a representation involves two sequences: $\{g_j(f)\}_{j=1}^\infty$ and $\{c_j(f)\}_{j=1}^\infty$. In this paper the construction of $\{g_j(f)\}_{j=1}^\infty$ 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 $f\in A_1(\mathcal D)$?" Previously it was believed that the rate of convergence was slower than $m^{-\frac14}$. The qualitative result of this paper is that the best possible rate of convergence of greedy expansions for $f\in A_1(\mathcal D)$ is faster than $m^{-\frac14}$. In fact, we prove it is faster than $m^{-\frac27}$.