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 ${\mathcal{D}}_E^{\prime }$ 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 ${\mathcal{D}}_E^{\prime }$ generalize the classical Schwartz spaces ${\mathcal{D}}_{L^p}$ and ${\mathcal{D}}_{L^p}^{\prime }$, respectively. Our proof is based on Gabor frame characterizations of ${\mathcal{D}}_E$ and ${\mathcal{D}}_E^{\prime }$, 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.

17 pages

References in corpus (3)

• Translation-modulation invariant Banach spaces of ultradistributions
• Modulation spaces associated to tensor products of amalgam spaces
• Sequence space representations for spaces of smooth functions and distributions via Wilson bases