Sequence space representations for translation-modulation invariant function and distribution spaces

Abstract: We provide sequence space representations for the test function space $\mathcal{D}{E}$ and the distribution space $\mathcal{D}^{\prime}{E}$ associated to a Banach space $E$ belonging to a broad class of translation-modulation invariant Banach spaces of distributions. The spaces $\mathcal{D}{E}$ and $\mathcal{D}^{\prime}{E}$ generalize the classical Schwartz spaces $\mathcal{D}{L^p}$ and $\mathcal{D}^{\prime}{L^p}$, respectively. Our proof is based on Gabor frame characterizations of $\mathcal{D}{E}$ and $\mathcal{D}^{\prime}{E}$, 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.