Generators for rings of compactly supported distributions

Let $C$ denote a closed convex cone $C$ in ${\mathbb{R}}^d$ with apex at 0. We denote by ${\mathcal{E}}^{\mathrm{\prime }}(C)$ the set of distributions having compact support which is contained in $C$. Then ${\mathcal{E}}^{\mathrm{\prime }}(C)$ is a ring with the usual addition and with convolution. We give a necessary and sufficient analytic condition on ${\widehat{f}}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\widehat{f}}_n$ for $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_n\in {\mathcal{E}}^{\mathrm{\prime }}(C)$ to generate the ring ${\mathcal{E}}^{\mathrm{\prime }}(C)$. (Here $\widehat{\cdot }$ 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.