Convolution with the kernel $e^{s\langle x\rangle^q}, q\geq 1, s>0$ within ultradistribution spaces

TL;DR

This paper investigates the conditions under which convolution with a specific exponential kernel is well-defined within Roumieu ultradistribution spaces, expanding understanding of ultradistribution operations.

Contribution

It establishes the existence criteria for convolution with exponential kernels in Roumieu ultradistribution spaces, a novel contribution to ultradistribution theory.

Findings

Convolution with the kernel $e^{sraket{x}^q}$ is well-defined under certain conditions.

Provides new criteria for convolution existence in ultradistribution spaces.

Enhances the mathematical framework for ultradistribution operations.

Abstract

We consider the existence of convolution of Roumieu type ultradistribution with the kernel $e^{s(1+|x|^2)^{q/2}}, q\geq 1$, $s\in\RR\backslash\{0\}$.