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

TL;DR

This paper investigates the existence of extremal functions in the Delsarte extremal problem within locally compact Abelian groups, establishing existence results for closed support sets and extending previous findings.

Contribution

The paper proves the existence of extremal functions for the Delsarte problem when the support set is closed, broadening the class of sets for which existence is known.

Findings

Existence of extremal functions is confirmed for closed support sets.

Extends previous results to a larger class of support sets.

Provides a theoretical foundation for further analysis of the Delsarte problem.

Abstract

In the general setting of a locally compact Abelian group $G$, the Delsarte extremal problem asks for the supremum of integrals over the collection of continuous positive definite functions $f: G \to \mathbb{R}$ satisfying $f(0) = 1$ and having $\mathrm{supp} f_{+} \subset \Omega$ for some measurable subset $\Omega$ of finite measure. In this paper, we consider the question of the existence of an extremal function for the Delsarte extremal problem. In particular, we show that there exists an extremal function for the Delsarte problem when $\Omega$ is closed, extending previously known existence results to a larger class of functions.