Sequence Space Representations for Translation-Modulation Invariant Function and Distribution Spaces

We provide sequence space representations for the test function space D E and the distribution space D E ′ associated to a Banach space E belonging to a broad class of translation-modulation invariant Banach spaces of distributions. The spaces D E and D E ′ generalize the classical Schwartz spaces D L p and D L p ′ , respectively. Our proof is based on Gabor frame characterizations of D E and D E ′ , which are also established here and are of independent interest. We recover in a unified way some known sequence space representations as well as obtain several new ones.