Shannon sampling and Weak Weyl's Law on compact Riemannian manifolds

Abstract

The well known Weyl's asymptotic formula gives an approximation to the number $\mathcal{N}_{\omega}$ of eigenvalues (counted with multiplicities) on an interval $[0,\>\omega]$ of the Laplace-Beltrami operator on a compact Riemannian manifold ${\bf M}$. In this paper we approach this question from the point of view of Shannon-type sampling on compact Riemannian manifolds. Namely, we give a direct proof that $\mathcal{N}_{\omega}$ is comparable to cardinality of certain sampling sets for the subspace of $\omega$-bandlimited functions on ${\bf M}$.