Positive Definite Functions and Moment Functions

Throughout this section, S = (S, +, *) will be an abelian semigroup with involution and zero element 0. In Chapter 4 we introduced the convex cones Pb(S) ⊆ Pe(S) ⊆ P(S) ⊆ ℂS and every function ϕ ∈ Pe(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.