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

## TL;DR

This paper explores extremal problems related to positive definite functions on locally compact Abelian groups, connecting Delsarte's and Turán's problems to packing densities and providing upper bounds based on set densities.

## Contribution

It introduces new bounds for the extremal constant C(X,Y) in the context of packing problems on locally compact Abelian groups, extending classical Fourier analysis methods.

## Key findings

- Derived an upper estimate for C(X,Y) using packing conditions.
- Connected extremal problems to packing densities via Fourier analysis.
- Extended Delsarte's method to more general group settings.

## Abstract

Let G be a locally compact Abelian group, and let X, Y be two open sets in G. We investigate the extremal constant C(X,Y) defined to be the supremum of integrals of functions f from the class F(X,Y), where F(X,Y) is the family of positive definite functions f on G such that f(0) = 1, the positive part of f is supported in X, and its negative part is supported in Y. In the case when X=Y, the problem is exactly the so-called Tur\'an problem for the set X. When Y= G, i.e., there is a restriction only on the set of positivity of f, we obtain the Delsarte problem. The Delsarte problem 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(X,Y) in the situation when the set X 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.