Delsarte’s extremal problem and packing on locally compact Abelian groups

Berdysheva, Elena E. and Révész, Szilárd (2023) Delsarte’s extremal problem and packing on locally compact Abelian groups. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE, 24 (2). pp. 1007-1052. ISSN 0391-173X

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Abstract

Let G be a locally compact Abelian group, and let Ω+, Ω− be two open sets in G. We investigate the constant C(Ω+, Ω−) = sup R G f : f ∈ F(Ω+, Ω−) , where F(Ω+, Ω−) is 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.

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