Beurling-Landau's density on compact manifolds

TL;DR

This paper studies the properties of eigenfunction spaces on compact manifolds, analyzing sampling and interpolation conditions, and demonstrating the equidistribution of Fekete arrays using Beurling-Landau densities.

Contribution

It introduces an analogy between eigenfunction spaces and polynomial spaces, providing necessary density conditions for sampling and interpolation on manifolds.

Findings

Necessary conditions for sampling and interpolation are established.

Fekete arrays are shown to be equidistributed on certain manifolds.

An analogy with Paley-Wiener spaces is developed for eigenfunction spaces.

Abstract

Given a compact Riemannian manifold $M$, we consider the subspace of $L^2(M)$ generated by the eigenfunctions of the Laplacian of eigenvalue less than $L\geq 1$. This space behaves like a space of polynomials and we have an analogy with the Paley-Wiener spaces. We study the interpolating and Marcienkiewicz-Zygmund (M-Z) families and provide necessary conditions for sampling and interpolation in terms of the Beurling-Landau densities. As an application, we prove the equidistribution of the Fekete arrays on some compact manifolds.