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On an infinite convolution product of measures. (English) Zbl 0984.46027

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.].

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

Citations:

Zbl 0811.46034
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References:

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[2] Lawton, W.: Infinite convolution products and refinable diatributions on Lie groups. Trans. Amer. Math. Soc. 352 , 2913-2936 (2000). · Zbl 0940.43002
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[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.).
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