Coorbit spaces with voice in a Fr\'echet space

TL;DR

This paper develops a new coorbit space theory for group representations using a Fréchet space of functions, broadening the scope beyond classical integrable cases and enabling analysis of spaces like Paley-Wiener and Shannon wavelet-related coorbit spaces.

Contribution

It introduces a general framework for coorbit spaces with kernels in a Fréchet space, extending classical theory to non-irreducible and non-integrable representations.

Findings

Allows treatment of Paley-Wiener spaces

Enables analysis of Shannon wavelets and Schr"odingerlets

Generalizes classical coorbit space theory

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 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.