Uchida, Motoo On an infinite convolution product of measures. (English) Zbl 0984.46027 Proc. Japan Acad., Ser. A 77, No. 1, 20-21 (2001). 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.Main 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. 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. M. Morimoto, loc. cit.]. Reviewer: P.K.Banerji (Jodhpur) MSC: 46F15 Hyperfunctions, analytic functionals 46F20 Distributions and ultradistributions as boundary values of analytic functions 46F10 Operations with distributions and generalized functions 40A99 Convergence and divergence of infinite limiting processes 44A35 Convolution as an integral transform Keywords:convolution products; Radon measure; hyperfunction; Fourier-Borel transform; Paley-Wiener-Ehrenpreis theorem Citations:Zbl 0811.46034 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Deslauriers, G., and Dubuc, S.: Interpolation dyadique. Fractals, dimensions non entières et ap-plications (ed. Cherbit, G.). Masson, Paris, pp. 44-45 (1987). · Zbl 0644.42004 [2] Lawton, W.: Infinite convolution products and refinable diatributions on Lie groups. Trans. Amer. Math. Soc. 352 , 2913-2936 (2000). · Zbl 0940.43002 · doi:10.1090/S0002-9947-00-02409-0 [3] Lawton, W.: Infinite convolution products of mea-sures and Taylor expantions (Talk at Dept. Math., Osaka Univ., 26 June 2000). [4] Morimoto, M.: Theory of the Sato Hyper-functions. Kyōritsu-Shuppan. Tokyo (1976) (in Japanese); Transl. Math. Monogr., vol. 129, Amer Math. Soc. (1993) (English transl.). [5] Schapire, P.: Théorie des Hyperfonctions. Lecture Notes Math., vol. 126, Springer, Berlin-Heidelberg-New York (1970). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.