𝐵𝑉-functions, positive-definite functions and moment problems

Maserick, P. H.

doi:10.1090/S0002-9947-1975-0380272-2

$\textrm {BV}$-functions, positive-definite functions and moment problems
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Trans. Amer. Math. Soc. 214 (1975), 137-152 Request permission

Abstract:

Let S be a commutative semigroup with identity 1 and involution. A complex valued function f on S is defined to be positive definite if ${\Pi _j}{\Delta _j}f(1) \geqslant 0$ where the ${\Delta _j}$’s belong to a certain class of linear sums of shift operators. For discrete groups the positive definite functions defined herein are shown to be the classically defined positive definite functions. An integral representation theorem is proved and necessary and sufficient conditions for a function to be the difference of two positive-definite functions, i.e. a BV-function, are given. Moreover the BV-function defined herein agrees with those previously defined for semilattices, with respect to the identity involution. Connections between the positive-definite functions and completely monotonic functions are discussed along with applications to moment problems.

References

Konvexität in topologischen Vektorräumen

A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
G. G. Lorentz, Bernstein polynomials, Mathematical Expositions, No. 8, University of Toronto Press, Toronto, 1953. MR 0057370
N. J. Fine and P. H. Maserick, On the simplex of completely monotonic functions on a comutative semigroup, Canadian J. Math. 22 (1970), 317–326. MR 257688, DOI 10.4153/CJM-1970-039-4
J. Kist and P. H. Maserick, BV-functions on semilattices, Pacific J. Math. 37 (1971), 711–723. MR 306417, DOI 10.2140/pjm.1971.37.711
R. J. Lindahl and P. H. Maserick, Positive-definite functions on involution semigroups, Duke Math. J. 38 (1971), 771–782. MR 291826, DOI 10.1215/S0012-7094-71-03893-2
P. H. Maserick, Moment and BV-functions on commutative semigroups, Trans. Amer. Math. Soc. 181 (1973), 61–75. MR 396835, DOI 10.1090/S0002-9947-1973-0396835-2
M. A. Naĭmark, Normirovannye kol′tsa, Izdat. “Nauka”, Moscow, 1968 (Russian). Second edition, revised. MR 0355602
Stephen E. Newman, Measure algebras and functions of bounded variation on idempotent semigroups, Trans. Amer. Math. Soc. 163 (1972), 189–205. MR 308686, DOI 10.1090/S0002-9947-1972-0308686-4
Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470

The Laplace transform

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