Coorbit Spaces for Dual Pairs

TL;DR

This paper develops an abstract framework for constructing Banach spaces of distributions via group representations, generalizing coorbit space theory and applying it to Bergman and Besov spaces with sampling and wavelet characterizations.

Contribution

It introduces a broad, unified approach to coorbit spaces that encompasses new examples like Bergman and Besov spaces, extending the original theory.

Findings

Bergman spaces on the unit disc are included in the new framework.

Atomic decompositions for Bergman spaces are achieved through sampling.

A wavelet characterization of Besov spaces on the forward light cone is provided.

Abstract

In this paper we present an abstract framework for construction of Banach spaces of distributions from group representations. This generalizes the theory of coorbit spaces initiated by H.G. Feichtinger and K. Gr\"ochenig in the 1980's. Spaces that can be described by this new technique include the whole Banach-scale of Bergman spaces on the unit disc. For these Bergman spaces we show that atomic decompositions can be constructed through sampling. We further present a wavelet characterization of Besov spaces on the forward light cone.