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

TL;DR

This paper proves the existence of an extremal function in a Delsarte extremal problem within the general setting of locally compact abelian groups, extending previous results limited to Euclidean spaces.

Contribution

It generalizes the existence of extremal functions in Delsarte problems from Euclidean spaces to all locally compact abelian groups.

Findings

Established the existence of extremal functions in general locally compact abelian groups.

Extended previous Euclidean space results to a broader group setting.

Provides a theoretical foundation for further extremal problems in harmonic analysis.

Abstract

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.