-functions, positive-definite functions and moment problems

Author:
P. H. Maserick

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

MSC:
Primary 43A35; Secondary 44A10, 44A50

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

MathSciNet review:
0380272

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Abstract | References | Similar Articles | Additional Information

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 where the '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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Additional Information

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

Keywords:
BV-functions,
positive-definite function,
completely monotonic function,
semigroup,
moment problem,
semicharacter,
integral representation,
finite difference,
vector lattice,
Banach algebra,
convolution of measures

Article copyright:
© Copyright 1975
American Mathematical Society