Convolution of Roumieu ultradistributions in sequential approach

TL;DR

This paper establishes equivalent conditions for the convolvability of Roumieu ultradistributions using sequential criteria, extending known results from distribution and Beurling ultradistribution spaces.

Contribution

It introduces new sequential conditions for convolvability of Roumieu ultradistributions and proves their equivalence to existing convolution notions, also analyzing properties of convolution and ultradifferential operators.

Findings

Equivalent convolvability conditions for Roumieu ultradistributions.

Extension of convolution properties to ultradistribution spaces.

Validation of ultradifferential operator properties in this context.

Abstract

We consider several general sequential conditions for convolvability of two Roumieu ultradistributions and prove that they are equivalent to the convolvability of these ultradistributions in the sense of Pilipovic and Prangoski. The discussed conditions, based on two classes of approximate units and corresponding sequential conditions of integrability of Roumieu ultradistributions, are analogous to the known convolvability conditions in the space of distributions and in the space of ultradistributions of Beurling type. Moreover, the useful property of the convolution and ultradiferential operator is proved.