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.

Integral comparisons of nonnegative positive definite functions on LCA groups

“…Lemma 1 Let f ∈ D ∩ L 1 (G) be arbitrary. 1 Then there exists a so-called Boas-Kac squareroot g ∈ L 2 (G) satisfying g g = f . Furthermore, if f ∈ D c , then we also have suppg G.…”

“…So the natural setup is to consider doubly positive functions in this sort of extremal problems. Therefore, in the present paper we discuss extremal problems in the general setting of LCA (locally compact abelian) groups, taking arbitrary regular Borel measures μ, ν of finite total variation and doubly positive functions f vanishing at infinity in place of f in (1). The question of finding conditions for (1) to hold was posed by Halász in oral communication to us.…”

“…Therefore, in the present paper we discuss extremal problems in the general setting of LCA (locally compact abelian) groups, taking arbitrary regular Borel measures μ, ν of finite total variation and doubly positive functions f vanishing at infinity in place of f in (1). The question of finding conditions for (1) to hold was posed by Halász in oral communication to us. To the best of our knowledge such comparisons of weighted averages with respect to different weights were first worked out and employed in the seminal number theory paper [12] by Halász. Although our results are new in the setting of positive definite functions on the real line as well, we have decided to formulate them for general LCA groups.…”