Extremal and approximation problems for positive definite functions

Let $\Omega$ be an open 0-symmetric subset of $\mathbb R^d$ which contains 0 and
f a continuous positive definite function vanishing off O, that is,
supp f is contained in the closure of $\Omega$. The problem is to approximate
f by a continuous positive definite function F supported in $\Omega$. We prove
this when 1. d=1. 2 $\Omega$ is strictly star-shaped 3. f is a radial function.
We also consider the following problem: Given a measure $\mu$
supported in $\Omega$, does  there exist an extremal function for the problem
$\sup \int f d\mu$, where the sup is taken over the cone of continuous
 positive definite functions f supported in $\Omega$ with f(0)=1?