Convolutors of translation-modulation invariant Banach spaces of ultradistributions

TL;DR

This paper investigates a class of ultradistributions invariant under translation and modulation, establishing a structural theorem and exploring extensions of convolution within these spaces.

Contribution

It provides the first structural theorem for ultradistributions whose convolutions lie in a translation-modulation invariant Banach space.

Findings

Established a structural theorem for these ultradistribution spaces.

Extended the concept of convolution in the context of ultradistributions.

Analyzed properties of ultradistributions under translation and modulation invariance.

Abstract

We study the space of tempered ultradistributions whose convolutions with test functions are all contained in a given translation-modulation invariant Banach space of ultradistributions. Our main result will be the first structural theorem for the aforementioned space. As an application we consider several extensions of convolution.