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Bochner’s theorem for finite-dimensional conelike semigroups. (English) Zbl 0790.43008

A subset \(S\) of some real vector space \(V\) is called conelike, if for every \(s \in S\) there exists a number \(a(s) \in \mathbb{R}_ +\) such that \(\alpha s \in S\) for all \(\alpha \geq a(s)\). If \(S\) is also a semigroup and \(V\) is of finite dimension, we can prove the following extension of Bochner’s classical theorem: Each bounded continuous positive definite function on \(S\) is a mixture of continuous characters. These continuous characters are identified in terms of a canonically associated subspace of \(V\) and the dual cone of the hermitian part of \(S\). We then give a probabilistic application.

MSC:

43A35 Positive definite functions on groups, semigroups, etc.
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
60E07 Infinitely divisible distributions; stable distributions
60E10 Characteristic functions; other transforms
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References:

[1] Berg, C.: Positive definite and related functions on semigroups. In: Hofmann, K.H., Lawson, J.D., Pym, J.S. (eds.) The analytical and topological theory of semigroups, pp. 253-278. Berlin New York: de Gruyter 1990
[2] Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic analysis on semigroups. (Grad. texts Math., vol. 100) Berlin Heidelberg New York: Springer 1984 · Zbl 0619.43001
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