The point value maximization problem for positive definite functions supported in a given subset of a locally compact group

TL;DR

This paper extends the classical extremal problem of maximizing point values of positive definite functions, originally studied for trigonometric polynomials, to all locally compact groups, including non-commutative ones.

Contribution

It generalizes existing results from Abelian groups to all locally compact groups, including non-commutative cases.

Findings

Extended extremal problem to non-commutative locally compact groups.

Unified framework for positive definite functions supported in subsets.

Generalized classical results to broader group settings.

Abstract

The century old extremal problem, solved by Carath\'eodory and Fej\'er, concerns a nonnegative trigonometric polynomial normalized by a0 = 1, and the quantity to be maximized is the coefficient a1. In the complex exponential form, the coefficient sequence (ck) will be supported in [-n; n] and normalized by c0 =1. Reformulating, nonnegativity of T translates to positive definiteness of the sequence (ck), and the extremal problem becomes a maximization problem for the value at 1 of a normalized positive definite function c : Z --> C, supported in [-n; n]. Boas and Katz, Arestov, Berdysheva and Berens, Kolountzakis and R\'ev\'esz and recently Krenedits and R\'ev\'esz investigated the problem in increasing generality, reaching analogous results for all locally compact Abelian groups. We prove an extension to all the known results in not necessarily commutative locally compact groups.