Democracy functions and optimal embeddings for approximation spaces

Abstract

We prove optimal embeddings for nonlinear approximation spaces $\mathcal{A}^{\alpha}_q$ , in terms of weighted Lorentz sequence spaces, with the weights depending on the democracy functions of the basis. As applications we recover known embeddings for N-term wavelet approximation in L p , Orlicz, and Lorentz norms. We also study the greedy classes ${\mathcal{G}_{q}^{\alpha}}$ introduced by Gribonval and Nielsen, obtaining new counterexamples which show that ${\mathcal{G}_{q}^{\alpha}}\not=\mathcal{A}^{\alpha}_q$ for most non-democratic unconditional bases.