Generators for Rings of Compactly Supported Distributions

Abstract

Let \({{\tt C}}\) denote a closed convex cone in \({\mathbb R^d}\) with apex at 0. We denote by \({\mathcal E'({\tt C})}\) the set of distributions on \({\mathbb R^d}\) having compact support contained in \({{\tt C}}\). Then \({\mathcal E'({\tt C})}\) is a ring with the usual addition and with convolution. We give a necessary and sufficient analytic condition on \({\widehat{f}_1,\dots, \widehat{f}_n}\) for \({f_1,\dots ,f_n \in \mathcal E'({\tt C})}\) to generate the ring \({\mathcal E'({\tt C})}\). (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 Lars Hörmander. En route we answer an open question posed by Yutaka Yamamoto.