Duality for Delsarte's extremal problem on locally compact Abelian groups

TL;DR

This paper extends the Delsarte extremal problem to locally compact Abelian groups, establishing duality and strong duality results using harmonic analysis and a functional analytic approach.

Contribution

It generalizes the Delsarte problem framework, unifies previous results, and provides a new proof technique avoiding restrictive assumptions.

Findings

Established the dual problem in the generalized setting.

Proved a strong duality theorem for the problem.

Unified and extended earlier results in harmonic analysis.

Abstract

The Delsarte extremal problem for positive definite functions, originally introduced by Delsarte in coding theory to bound the size of error-correcting codes, has since found applications in diverse areas such as sphere packing, Fuglede's spectral set conjecture, and $1$-avoiding sets. Recent developments have established the existence of extremizers in fairly general settings and identified precise linear programming dual formulations, together with strong duality results, in several important cases including finite groups and $\mathbb{R}^d$. In this paper, we consider a generalized Delsarte problem on locally compact Abelian groups, providing a natural framework for harmonic analysis. We extend both the normalization and the objective functional to encompass a wide range of previously studied cases, while avoiding restrictive topological assumptions common in the literature. Within this general setting, we derive the corresponding dual problem and prove a strong duality theorem, thereby unifying and extending earlier results. Naturally, our proof uses harmonic analysis, but the key is a functional analytic approach which distinguishes our proof from existing methods.