Generators for rings of compactly supported distributions

TL;DR

This paper characterizes when a set of distributions supported in a convex cone can generate the entire ring of such distributions under convolution, providing an analytic criterion and answering an open question.

Contribution

It establishes a necessary and sufficient condition for generators of the ring of compactly supported distributions in a convex cone, extending H"ormander's results and solving an open problem.

Findings

Provides an analytic criterion for generators of the distribution ring.

Answers an open question by Yutaka Yamamoto.

Extends the theory of rings of analytic functions to distributions.

Abstract

Let $C$ denote a closed convex cone $C$ in $\mathbb{R}^d$ with apex at 0. We denote by $\mathcal{E}'(C)$ the set of distributions having compact support which is contained in $C$. Then $\mathcal{E}'(C)$ is a ring with the usual addition and with convolution. We give a necessary and sufficient analytic condition on $\hat{f}_1,..., \hat{f}_n$ for $f_1,...,f_n \in \mathcal{E}'(C)$ to generate the ring $\mathcal{E}'(C)$. (Here $\hat{\cdot}$ denotes Fourier-Laplace transformation.) This result is an application of a general result on rings of analytic functions of several variables by H\"ormander. En route we answer an open question posed by Yutaka Yamamoto.