Coorbit Spaces with Voice in a Fréchet Space

Abstract

We set up a new general coorbit space theory for reproducing representations of a locally compact second countable group G that are not necessarily irreducible nor integrable. Our basic assumption is that the kernel associated with the voice transform belongs to a Frechet space $$\mathcal T$$ of functions on G, which generalizes the classical choice $$\mathcal T=L_w^1(G)$$ . Our basic example is $$ \mathcal T=\bigcap _{p\in (1,+\infty )} L^p(G)$$ , or a weighted versions of it. By means of this choice it is possible to treat, for instance, Paley-Wiener spaces and coorbit spaces related to Shannon wavelets and Schrodingerlets.