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On the Simplex of Completely Monotonic Functions on a Commutative Semigroup

Published online by Cambridge University Press:  20 November 2018

N. J. Fine  and
P. H. Maserick
N. J. Fine
Affiliation:
The Pennsylvania State University, University Park, Pennsylvania
P. H. Maserick
Affiliation:
The Pennsylvania State University, University Park, Pennsylvania
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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].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 317 - 326
Copyright
Copyright © Canadian Mathematical Society 1970

References

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