Extension of positive definite functions.

Let Ω ⊂ R n be an open, connected subset of R n , and let F : Ω − Ω → C , where Ω − Ω = { x − y : x , y ∈ Ω } , be a continuous positive definite function. We give necessary and sufficient conditions for F to have an extension to a continuous positive definite function defined on the entire Euclidean space R n . The conditions are formulated in terms of existence of a unitary representations of R n whose generators extend a certain system of unbounded Hermitian operators defined on a Hilbert space associated to F . Different positive definite extensions correspond to different unitary representations. [ABSTRACT FROM AUTHOR]