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

TL;DR

This paper introduces sequence space representations for generalized translation-modulation invariant function and distribution spaces, extending classical Schwartz space concepts using Gabor frame characterizations.

Contribution

It provides a unified framework for representing these spaces via sequences, generalizing classical spaces and establishing new Gabor frame characterizations.

Findings

Unified sequence space representations for generalized spaces

New Gabor frame characterizations of distribution spaces

Recovery of classical and new sequence space representations

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.