Duality for Delsarte's extremal problem on compact Gelfand pairs

TL;DR

This paper investigates Delsarte-type extremal problems on compact Gelfand pairs, establishing duality results and connecting to classical problems in number theory, sphere packing, and statistics.

Contribution

It formulates dual problems for positive definite functions on compact Gelfand pairs and proves strong duality, extending classical extremal problems to a broader setting.

Findings

Established duality for Delsarte-type problems on compact Gelfand pairs.

Connected extremal problems to classical applications in number theory and sphere packing.

Proved strong duality results for these infinite-dimensional linear programming problems.

Abstract

We study Delsarte-type problems for positive definite functions on compact Gelfand pairs as infinite-dimensional linear programming problems. This setup includes, as a particular case, the case of compact Abelian groups. Depending on the restriction on the signs of the functions, we obtain two important particular cases, the Tur\'an and Delsarte problems. These problems have been studied in relation to number theory, sphere packing, and statistics. In this paper, we describe their duals and prove a strong duality statement.