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 constantis the class of positive definite functions f on G such that f (0) = 1, the positive part f + of f is supported in Ω + , and 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.

Locations

• ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE - View
• Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) - View - PDF
• arXiv (Cornell University) - View - PDF