On an infinite convolution product of measures (Q5938540)

This paper concerns infinite convolution products of complex probability Radon measure with bounded total variation on \(n\)-dimensional Euclidean plane \(\mathbb{R}^n\), and it is proved that they converge to a hyperfunction under a weak assumption on supports.NEWLINENEWLINENEWLINEMain results involves a complex Radon measure \(u\) on \(\mathbb{R}^n\) with compact support and \(\|u\|\) denotes its total variation, while \(\text{supp }u\) is its support and \(u* v\) denotes the convolution product of \(u\) and \(v\). \(K\) is the compact subset of \(\mathbb{R}^n\) and Sato hyperfunction [cf. \textit{M. Morimoto}, ``An introduction to Sato's hyperfunctions'', transl. from Math. Monogr. 129 (1993; Zbl 0811.46034)] on \(\mathbb{R}^n\) with compact support contained in \(K\) is considered. Proof of the theorem is aided by Fourier-Borel transform of the hyperfunction and the Paley-Wiener-Ehrenpreis theorem for hyperfunctions [cf. \textit{M. Morimoto}, loc. cit.].