On extremal problems of Delsarte type for positive definite functions on LCA groups

TL;DR

This paper develops a unifying framework for extremal problems related to positive definite functions on locally compact Abelian groups, including Delsarte and Turán problems, and proves existence results under broad conditions.

Contribution

It introduces a new variation of extremal problems and establishes conditions for the existence of extremal functions in this generalized setting.

Findings

Unified approach to Delsarte and Turán extremal problems

Proved existence of extremal functions under general topological conditions

Extended previous results to a broader class of problems

Abstract

A unifying framework for some extremal problems on locally compact Abelian groups is considered, special cases of which include the Delsarte and Tur\'an extremal problems. A slight variation of the extremal problem is introduced and the different formulations are studied for equivalence. Extending previous work, a general result on existence of extremal functions for the new variant is proved under a certain general topological condition.