Exponentially bounded positive definite functions

Abstract

Equivalent conditions for scalar (or operator valued) positive definite functions, on a commutative semigroup $S$ with identity $e$, to admit a disintegration with respect to a regular positive (operator valued) measure supported by an arbitrary compact subset of semicharacters are given. The theory links to the theory of $\tau$-positive functions presented previously by the second author and comparisons between the two are given. Old and new theorems to classical and modern moment problems are obtained as a consequence.