$\textrm {BV}$functions, positivedefinite functions and moment problems
Author:
P. H. Maserick
Journal:
Trans. Amer. Math. Soc. 214 (1975), 137152
MSC:
Primary 43A35; Secondary 44A10, 44A50
DOI:
https://doi.org/10.1090/S00029947197503802722
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 ${\Pi _j}{\Delta _j}f(1) \geqslant 0$ where the ${\Delta _j}$âs 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 positivedefinite functions, i.e. a BVfunction, are given. Moreover the BVfunction defined herein agrees with those previously defined for semilattices, with respect to the identity involution. Connections between the positivedefinite functions and completely monotonic functions are discussed along with applications to moment problems.

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Additional Information
Keywords:
BVfunctions,
positivedefinite 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