Some Remarks on Nonlinear Approximation with Schauder Bases
Résumé
We study the approximation classes A_s^\alpha and G_s^\alpha associated with nonlinear m-term approximation and greedy approximation by elements from a quasi-normed Schauder basis in a separable Banach space. We show that there is always a two-sided embedding K_s^\tau_p \hookrightarrow A_s^\alpha \hookrightarrow K_s^\tau_q, where K_s^\tau denotes the associated smoothness space. We provide estimates of \tau_p and \tau_q in terms of quantitative properties of the basis. The lower and upper estimates are sharp for so-called quasi-greedy bases, but may not coincide with each other to completely characterize A_s^\alpha. For a quasi-greedy and democratic basis, a complete characterization G_s^\alpha = K_s^1/\alpha(w) is obtained where w is a weight depending on the properties of the basis. For greedy bases, G_s^\alpha = A_s^\alpha but the converse is not true. The results in this paper can be considered a generalization of the characterization for an orthonormal basis B in a Hilbert space H, where is it well known that A_s^\alpha(B) = K_s^\tau(B), with \alpha = 1/\tau-1/2 and s \in (0,\infty].