Caratheodory-Fejer type extremal problems on locally compact Abelian groups

We consider the extremal problem of maximizing a point value vertical bar f (z) vertical bar at a given point Z is an element of G by some positive definite and continuous function f on a locally compact Abelian group (LCA group) G, where for a given symmetric open set Omega there exists z, f vanishes outside Omega and is normalized by f(0) = 1. This extremal problem was investigated in R and R-d and for Omega a 0-symmetric convex body in a paper of Boas and Kac in 1945. Arestov, Berdysheva and Berens extended the investigation to T-d, where T := R/Z. Kolountzalds and Revesz 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 Caratheodory and Fejer. Moreover, following observations of Boas and Kac, Kolountzakis and Revesz showed how the general problem can be reduced to equivalent discrete problems of "Caratheodory-Fejer type" on Z or Z(m) := Z/mZ. We extend their results to arbitrary LCA groups. (C) 2015 Elsevier Inc. All rights reserved.