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We consider the extremal problem of maximizing a point value jf(z)j at agiven point z 2 G by some positive definite and continuous function f on anAbelian group G, where for a given symmetric open set 3 z, f vanishes outsideand is normalized by f(0) = 1. Denote the extremal value as CG(; z). Thisextremal problem was investigated in R and Rd and for a 0-symmetric convex bodyin a paper of Boas and Kac in 1943. Arestov and Berdysheva extended theinvestigation to Td, where T := R=Z. Kolountzakis and R?ev?esz gave a moregeneral setting, considering arbitrary open sets, in all the classical groupsabove. Also they observed, that such extremal problems occurred in certainspecial cases and in a different, but equivalent formulation already a centuryago in the work of Carath?eodory and Fej?er. Moreover, following observationsof Boas and Kac, Kolountzakis and R?ev?esz showed how the general problem canbe reduced to equivalent discrete problems of "Carath?eodory-Fej?er type" on Zor Zm := Z=mZ. We extend the results of Kolountzakis and R?ev?esz to locallycompact Abelian groups.

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