© The London Mathematical Society 2006
TURÁN'S EXTREMAL PROBLEM FOR POSITIVE DEFINITE FUNCTIONS ON GROUPS
School of Mathematics, Georgia Institute of Technology 686 Cherry Street NW, Atlanta, GA 30332, USA and Department of Mathematics, University of Crete Knossos Ave., 714 09 Iraklio, Greece kolount{at}member.ams.org
Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences, 1364 Budapest, Hungary revesz{at}renyi.hu
Received 26 October 2004.
We study the following question: given an open set 
, symmetric about 0, and a continuous, integrable, positive definite function f, supported in and with f(0) = 1, how large can 
f be? This problem has been studied so far mostly for convex domains in Euclidean space. In this paper we study the question in arbitrary locally compact abelian groups and for more general domains. Our emphasis is on finite groups as well as Euclidean spaces and 
d. We exhibit upper bounds for f assuming geometric properties of of two types: (a) packing properties of and (b) spectral properties of . Several examples and applications of the main theorems are shown. In particular, we recover and extend several known results concerning convex domains in Euclidean space. Also, we investigate the question of estimating f over possibly dispersed sets solely in dependence of the given measure m :=|| of . In this respect we show that in 
and the integral is maximal for intervals.