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

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 $\mathbf{M}$. 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 $\mathbf{M}$.