On Coorbit Fr\'echet Spaces

TL;DR

This paper introduces a new approach to coorbit space theory by constructing coorbit spaces as Fréchet spaces, which simplifies discretization and atomic decompositions when traditional methods face difficulties due to non-integrable representations.

Contribution

It proposes a novel framework for coorbit spaces using Fréchet spaces, overcoming discretization challenges in non-integrable group representations.

Findings

Fréchet space construction simplifies coorbit space discretization.

Atomic decompositions are more natural in the Fréchet space setting.

The approach extends coorbit theory to cases with non-integrable representations.

Abstract

This paper is concerned with a new approach to coorbit space theory. Usually, coorbit spaces are defined by collecting all distributions for which the voice transform associated with a square-integrable group representation possesses a certain decay, usually measured in a Banach space norm such as weighted $L_p$-norms. Unfortunately, in cases where the representation does not satisfy certain integrability conditions, one is faced with a bottleneck, namely that the discretization of the coorbit spaces is surprisingly difficult. It turns out that in these cases the construction of coorbit spaces as Fr\'echet spaces is much more convenient since then atomic decompositions can be established in a very natural way.