Delsarte's Extremal Problem and Packing on Locally Compact Abelian Groups

Abstract

Let G be a locally compact Abelian group, and let Ω + , Ω − be two open sets in G. We investigate the constantand its negative part f − is supported in Ω − . In the case when Ω + = Ω − =: Ω, the problem is exactly the so-called Turán problem for the set Ω. When Ω − = G, i.e., there is a restriction only on the set of positivity of f , we obtain the Delsarte problem. The Delsarte problem in R d is the sharpest Fourier analytic tool to study packing density by translates of a given "master copy" set, which was studied first in connection with packing densities of Euclidean balls.We give an upper estimate of the constant C(Ω + , Ω − ) in the situation when the set Ω + satisfies a certain packing type condition. This estimate is given in terms of the asymptotic uniform upper density of sets in locally compact Abelian groups.