Gabor frame characterizations of generalized modulation spaces

Abstract

We obtain Gabor frame characterizations of modulation spaces defined via a broad class of translation-modulation invariant Banach spaces of distributions. We show that these spaces admit an atomic decomposition through Gabor expansions and that they are characterized by summability properties of their Gabor coefficients. Furthermore, we construct a large space of admissible windows. This generalizes and unifies several fundamental results for the classical modulation spaces [Formula: see text] and the amalgam spaces [Formula: see text]. Due to the absence of solidity assumptions on the Banach spaces defining these generalized modulation spaces, the methods used for the spaces [Formula: see text] (or, more generally, in coorbit space theory) fail in our setting and we develop here a new approach based on the twisted convolution.