Gabor frame characterisations of generalised modulation spaces

TL;DR

This paper extends Gabor frame characterisations to a broad class of generalized modulation spaces, providing atomic decompositions and new window constructions, thus generalising classical results beyond traditional assumptions.

Contribution

It introduces a novel approach using twisted convolution to characterise these spaces, overcoming limitations of existing methods.

Findings

Spaces admit atomic Gabor decompositions

Characterised by Gabor coefficient summability

Constructed large class of admissible windows

Abstract

We obtain Gabor frame characterisations of modulation spaces defined via a class of translation-modulation invariant Banach spaces of distributions that was recently introduced in $[10]$. We show that these spaces admit an atomic decomposition through Gabor expansions and that they are characterised by summability properties of their Gabor coefficients. Furthermore, we construct a large space of admissible windows. This generalises several fundamental results for the classical modulation spaces $ M^{p,q}_{w}$. Due to the absence of solidity assumptions on the Banach spaces defining these modulation spaces, the methods used for the spaces $M^{p,q}_{w}$ (or, more generally, in coorbit space theory) fail in our setting and we develop here a new approach based on the twisted convolution.