Positive Definite Functions and Moment Functions

Abstract

Throughout this section, S = (S, +, *) will be an abelian semigroup with involution and zero element 0. In Chapter 4 we introduced the convex cones P^b(S) ⊆ P^e(S) ⊆ P(S) ⊆ ℂ^S and every function ϕ ∈ P^e(S) has a representation

$$\varphi \left( s \right) = \int_{s * } {\rho (s)d\mu } \left( \rho \right)$$

with a uniquely determined measure μ ∈ M ^c₊ (S*). Our aim is to examine for which ϕ∈P(S) there exists a representation as above with μ∈M₊(S*) of not necessarily compact support.