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This paper is concerned with a Delsarte-type extremal problem. Denote by $${\mathcal {P}}(G)$$ 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, $$\begin{aligned}&{\mathcal {G}}(W, Q)_G = \left\{ f \in {\mathcal {P}}(G) \cap L^1(G)~:\right. \\&\qquad \qquad \qquad \qquad \qquad \left. f(0) = 1, ~ {\text {supp}}{f_+} \subseteq W,~ {\text {supp}}{\widehat{f}} \subseteq Q \right\} \end{aligned}$$ G ( W , Q ) G = f ∈ P ( G ) ∩ L 1 ( G ) : f ( 0 ) = 1 , supp f + ⊆ W , supp f ^ ⊆ Q where $$W\subseteq G$$ W ⊆ G is closed and of finite Haar measure and $$Q\subseteq {\widehat{G}}$$ Q ⊆ G ^ is compact. We also consider the related Delsarte-type problem of finding the extremal quantity $$\begin{aligned} {\mathcal {D}}(W,Q)_G = \sup \left\{ \int _{G} f(g) \mathrm{d}\lambda _G(g) ~ : ~ f \in {\mathcal {G}}(W,Q)_G\right\} . \end{aligned}$$ D ( W , Q ) G = sup ∫ G f ( g ) d λ G ( g ) : f ∈ G ( W , Q ) G . The main objective of the current paper is to prove the existence of an extremal function for the Delsarte-type extremal problem $${\mathcal {D}}(W,Q)_G$$ D ( W , Q ) G . The existence of the extremal function has recently been established by Berdysheva and Révész in the most immediate case where $$G={\mathbb {R}}^d$$ G = 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 Révész.