  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.
