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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by P. H. Maserick

Trans. Amer. Math. Soc. 214 (1975), 137-152

DOI: https://doi.org/10.1090/S0002-9947-1975-0380272-2

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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.

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The Laplace transform

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