On the Atomic Decomposition of Coorbit Spaces with Non-Integrable Kernel

TL;DR

This paper investigates the extension of coorbit space theory to cases with non-integrable kernels, clarifying conditions for atomic decompositions and Banach frames in these generalized settings.

Contribution

It provides new insights into constructing atomic decompositions and Banach frames for coorbit spaces with non-integrable kernels, expanding the applicability of the theory.

Findings

Atomic decompositions are possible for non-integrable kernels under certain conditions.

Banach frames can be constructed for these generalized coorbit spaces.

The paper clarifies the theoretical foundation for non-integrable kernel cases.

Abstract

This paper ist concerned with recent progress in the context of coorbit space theory. Based on a square integrable group representation, the coorbit theory provides new families of associated smoothness spaces, where the smoothness of a function is measured by the decay of the associated voice transform. Moreover, by discretizing the representation, atomic decomposi- tions and Banach frames can be constructed. Usually, the whole machinery works well if the associated reproducing kernel is integrable with respect to a weighted Haar measure on the group. In recent studies, it has turned out that to some extent coorbit spaces can still be established if this condition is violated. In this paper, we clarify in which sense atomic decompositions and Banach frames for these generalized coorbit spaces can be obtained.