  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.
