  In the general setting of a locally compact Abelian group $G$, the Delsarte
extremal problem asks for the supremum of integrals over the collection of
continuous positive definite functions $f: G \to \mathbb{R}$ satisfying $f(0) =
1$ and having $\mathrm{supp} f_{+} \subset \Omega$ for some measurable subset
$\Omega$ of finite measure. In this paper, we consider the question of the
existence of an extremal function for the Delsarte extremal problem. In
particular, we show that there exists an extremal function for the Delsarte
problem when $\Omega$ is closed, extending previously known existence results
to a larger class of functions.
