Coorbit spaces with voice in a Fr\'echet space
  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 Fr\'echet 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 Schr\"odingerlets. 
Stephan Dahlke, Filippo De Mari, Ernesto De Vito, Demetrio Labate,   Gabrielle Steidl, Gerd Teschke, Stefano Vigogna
math.FA