On the Existence of an Extremal Function in the Delsarte Extremal Problem

This paper is concerned with a Delsarte-type extremal problem. Denote by 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, 𝒢(W, Q)_G = { f ∈𝒫(G) ∩ L^1(G) :. . f(0) = 1, suppf_+⊆ W, suppf⊆ Q } where W⊆ G is closed and of finite Haar measure and Q⊆G is compact. We also consider the related Delsarte-type problem of finding the extremal quantity 𝒟(W,Q)_G = sup{∫ _G f(g) dλ _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 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=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.