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On the Simplex of Completely Monotonic Functions on a Commutative Semigroup

  • N. J. Fine (a1) and P. H. Maserick (a1)

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Bernstein's classical integral representation theorem for completely monotonie functions can be proved most elegantly, on a commutative semigroup with identity, by the integral version of the Kreĭn-Milman theorem [2]. The key to this approach is the identification (as exponentials) of the extremal points of the normalized completely monotonie functions. Alternate proofs of this identification are given in § 1. The first (Corollary 1.3) is based on the Kreĭn-Milman theorem and the second (see remarks following Corollary 1.5) is derived from elementary analytic techniques. Other interesting facts about completely monotonie functions are mentioned in passing. For example, we observe that the normalized completely monotonie functions form a simplex (Corollary 1.4). In Corollary 1.6 we note that the product of completely monotonie functions corresponds to the convolution of their representing measures. Thus the normalized completely monotonie functions form an affine semigroup [3].

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References

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1. Alfsen, E. M., On the geometry of Choquet simplexes, Math. Scand. 15 (1964), 97110.
2. Bauer, H., Konvexitdt in Topologischen Vektorraumen, Lecture notes University of Hamburg, Hamburg, West Germany.
3. Cohen, H. and Collins, H. S., Affine semigroups, Trans. Amer. Math. Soc. 93 (1959), 97113.
4. Davis, P. J., Interpolation and approximation (Blaisdell, New York-Toronto-London, 1963).
5. Fan, K., Les fonctions définies-positives et les fonctions complètement monotones. Leurs applications au calcul des probabilités et à la théorie des espaces distanciés, Mémor. Sci. Math., no. 114 (Gauthier-Villars, Paris, 1950).
6. Phelps, R. R., Lectures on Choquet1 s theorem (Van Nostrand, Princeton, New Jersey, 1966).
7. Ross, K., A note on extending semicharacters on semigroups, Proc. Amer. Math. Soc. 10 (1959), 579583.
8. Widder, D. V., The Laplace transform, Princeton Mathematical Series, Vol. 6 (Princeton Univ. Press, Princeton, N.J., 1941).
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On the Simplex of Completely Monotonic Functions on a Commutative Semigroup

  • N. J. Fine (a1) and P. H. Maserick (a1)

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