## Generalized coorbit theory, Banach frames, and the relation to $${\alpha}$$-modulation spaces.(English)Zbl 1215.42035

Summary: This paper is concerned with generalizations and specific applications of the coorbit space theory based on group representations modulo quotients, which has been developed quite recently. We show that the general theory applied to the affine Weyl-Heisenberg group gives rise to families of smoothness spaces that can be identified with $${\alpha}$$-modulation spaces.

### MSC:

 42C15 General harmonic expansions, frames 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 46E15 Banach spaces of continuous, differentiable or analytic functions 57S25 Groups acting on specific manifolds 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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