The Caratheodory-Fej\'er type extremal problem on locally compact Abelian groups

TL;DR

This paper extends the classical Carathéodory-Fejér extremal problem to locally compact Abelian groups, generalizing previous results from Euclidean spaces and tori to a broader algebraic setting.

Contribution

It generalizes the Carathéodory-Fejér extremal problem to locally compact Abelian groups, broadening the scope of previous Euclidean and torus results.

Findings

Extended extremal problem to locally compact Abelian groups.

Connected continuous and discrete extremal problems.

Unified framework for classical and new cases.

Abstract

We consider the extremal problem of maximizing a point value jf(z)j at a given point z 2 G by some positive definite and continuous function f on an Abelian group G, where for a given symmetric open set 3 z, f vanishes outside and is normalized by f(0) = 1. Denote the extremal value as CG(; z). This extremal problem was investigated in R and Rd and for a 0-symmetric convex body in a paper of Boas and Kac in 1943. Arestov and Berdysheva extended the investigation to Td, where T := R=Z. Kolountzakis and R?ev?esz gave a more general setting, considering arbitrary open sets, in all the classical groups above. Also they observed, that such extremal problems occurred in certain special cases and in a different, but equivalent formulation already a century ago in the work of Carath?eodory and Fej?er. Moreover, following observations of Boas and Kac, Kolountzakis and R?ev?esz showed how the general problem can be reduced to equivalent discrete problems of "Carath?eodory-Fej?er type" on Z or Zm := Z=mZ. We extend the results of Kolountzakis and R?ev?esz to locally compact Abelian groups.