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We obtain Gabor frame characterisations of modulation spaces defined via aclass of translation-modulation invariant Banach spaces of distributions thatwas recently introduced in $[10]$. We show that these spaces admit an atomicdecomposition through Gabor expansions and that they are characterised bysummability properties of their Gabor coefficients. Furthermore, we construct alarge space of admissible windows. This generalises several fundamental resultsfor the classical modulation spaces $ M^{p,q}_{w}$. Due to the absence ofsolidity assumptions on the Banach spaces defining these modulation spaces, themethods used for the spaces $M^{p,q}_{w}$ (or, more generally, in coorbit spacetheory) fail in our setting and we develop here a new approach based on thetwisted convolution.

Gabor frame characterisations of generalised modulation spaces