Radial Time-Frequency Analysis and Embeddings of Radial Modulation Spaces

In this paper we construct frames of Gabor type for the space Lrad(R ) of radial L-functions, and more generally, for subspaces of modulation spaces consisting of radial distributions. Hereby, each frame element itself is a radial function. This construction is based on a generalization of the so called Feichtinger-Gröchenig theory – sometimes also called coorbit space theory – which was developed in an earlier article. We show that this new type of Gabor frames behaves better in linear and nonlinear approximation in a certain sense than usual Gabor frames when approximating a radial function. Moreover, we derive new embedding theorems for coorbit spaces restricted to invariant vectors (functions) and apply them to modulation spaces of radial distributions. As a special case this result implies that the Feichtinger algebra (S0)rad(R ) = M rad(R ) restricted to radial functions is embedded into the Sobolev space H (d−1)/2 rad (R ). Moreover, for d ≥ 2 the embedding (S0)rad(R) →֒ Lrad(Rd) is compact. 2000 AMS subject classification: 42C40, 46E35, 41A46