Translation-modulation invariant Banach spaces of ultradistributions

TL;DR

This paper introduces a new class of translation-modulation invariant Banach spaces of ultradistributions with stability under Fourier transform, tensor products, and algebraic structures, expanding the framework of ultradistribution analysis.

Contribution

It defines and studies a novel class of ultradistribution Banach spaces invariant under translation and modulation, with applications to modulation spaces and algebraic structures.

Findings

Spaces are stable under Fourier transform and tensor products.

They possess Banach convolution and multiplication module structures.

New modulation spaces of ultradistributions are introduced.

Abstract

We introduce and study a new class of translation-modulation invariant Banach spaces of ultradistributions. These spaces show stability under Fourier transform and tensor products; furthermore, they have a natural Banach convolution module structure over a certain associated Beurling algebra, as well as a Banach multiplication module structure over an associated Wiener-Beurling algebra. We also investigate a new class of modulation spaces, the Banach spaces of ultradistributions $\mathcal{M}^F$ on $\mathbb{R}^{d}$, associated to translation-modulation invariant Banach spaces of ultradistributions $F$ on $\mathbb{R}^{2d}$.