Gabor frame characterisations of generalised modulation spaces

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 spacesM w . Due to the absence of solidity assumptions on the Banach spaces defining these modulation spaces, the methods used for the spaces M w (or, more generally, in coorbit space theory) fail in our setting and we develop here a new approach based on the twisted convolution.