On the existence of an extremal function in the delsarte extremal problem

This paper is concerned with a Delsarte type extremal problem. Denote by $\mathcal{P}(G)$ the set of positive definite continuous functions on a locally compact abelian group $G$. We consider the function class, which was originally introduced by Gorbachev, \begin{multline*} \mathcal{G}(W, Q)_G = \left\{ f \in \mathcal{P}(G) \cap L^1(G) ~ : \right. ~ \left. f(0) = 1, ~ \text{supp}f_+ \subseteq W,~ \text{supp}\hat{f} \subseteq Q \right\} \end{multline*} where $W\subseteq G$ is closed and of finite Haar measure and $Q\subseteq \hat{G}$ is compact. We also consider the related Delsarte type problem of finding the extremal quantity \begin{equation*} \mathcal{D}(W,Q)_G = \sup \left\{ \int_{G} f(g) d\lambda_G(g) ~ : ~ f \in \mathcal{G}(W,Q)_G\right\}. \end{equation*} The main objective of the current paper is to prove the existence of an extremal function for the Delsarte type extremal problem $\mathcal{D}(W,Q)_G$. The existence of the extremal function has recently been established by Berdysheva and R\'ev\'esz in the most immediate case where $G=\mathbb{R}^d$. So the novelty here is that we consider the problem in the general setting of locally compact abelian groups. In this way our result provides a far reaching generalization of the former work of Berdysheva and R\'ev\'esz.