Equivalence of several definitions of convolution of Roumieu ultradistributions

TL;DR

This paper proves the equivalence of multiple definitions of convolution for Roumieu ultradistributions, using tensor product techniques within the framework of locally convex spaces.

Contribution

It establishes the equivalence of various convolution definitions for Roumieu ultradistributions, advancing the theoretical understanding of ultradistribution convolution.

Findings

Multiple convolution definitions are shown to be equivalent.

Tensor product methods are effectively applied to ultradistribution theory.

The results deepen the mathematical foundation of ultradistribution convolutions.

Abstract

The equivalence of several definitions of convolution of two Roumieu ultradistributions is proved. For that purpose, the $\varepsilon$ tensor product of $\dot{\tilde{\mathcal{B}}}^{\{M_p\}}$ and a locally convex space $E$ is considered.