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 , supp f + ⊆ W , supp f ^ ⊆ Q <graphic href="9_2020_1626_Article_Equ15.gif"></graphic> 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) d λ G (g) : f ∈ G (W , Q) G. <graphic href="9_2020_1626_Article_Equ16.gif"></graphic> 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. [ABSTRACT FROM AUTHOR]