  Let $\Omega\subset\mathbb{R}^n$ be an open, connected subset of
$\mathbb{R}^n$, and let $F\colon\Omega-\Omega\to\mathbb{C}$, where
$\Omega-\Omega=\{x-y\colon x,y\in\Omega\}$, 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 $\mathbb{R}^n$. The conditions are formulated in terms of
strong commutativity of a system of certain unbounded selfadjoint operators
defined on a Hilbert space associated to $F$. When a positive definite function
$F$ is extendable, we show that it is characterized by existence of associated
unitary representations of $\mathbb{R}^n$. Different positive definite
extensions correspond to different unitary representations. We prove that each
such unitary representation has simple spectrum. We give necessary and
sufficient conditions for a continuous positive definite function to have
exactly one extension. Our proof regarding extensions of positive definite
functions carries over mutatis mutandis to the case of conditionally negative
definite functions, which has applications to Gaussian stochastic processes,
whose increments in mean-square are stationary (e.g., fractional Brownian
motion).
