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arXiv · 2111.00334

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

Andreas Debrouwere·Lenny Neyt
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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.

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Andreas Debrouwere, Lenny Neyt. 2021-10-30. Sequence space representations for translation-modulation invariant function and distribution spaces. https://doi.org/10.1007/s00041-022-09981-z

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