Landau's necessary density conditions for LCA groups

TL;DR

This paper extends Landau's necessary density conditions for sampling and interpolation from classical Fourier analysis to the broader context of locally compact abelian groups, using operator theory and Fourier analysis principles.

Contribution

It generalizes Landau's density conditions to LCA groups, adapting the proof techniques to a more abstract harmonic analysis setting.

Findings

Extended Landau's conditions to LCA groups

Utilized the comparison principle of Ramanathan and Steger

Provided a framework for density analysis in abstract harmonic analysis

Abstract

H. Landau's necessary density conditions for sampling and interpolation may be viewed as a general principle resting on a basic fact of Fourier analysis: The complex exponentials $e^{i kx}$ ($k$ in $\mathbb{Z}$) constitute an orthogonal basis for $L^2([-\pi,\pi])$. The present paper extends Landau's conditions to the setting of locally compact abelian (LCA) groups, relying in an analogous way on the basics of Fourier analysis. The technicalities--in either case of an operator theoretic nature--are however quite different. We will base our proofs on the comparison principle of J. Ramanathan and T. Steger.