Sampling and Weyl's Law on compact Riemannian manifolds

The well known Weyl's asymptotic formula gives an approximation to the number N_ω of eigenvalues (counted with multiplicities) on an interval [0, ω] of the Laplace-Beltrami operator on a compact Riemannian manifold M. In this paper we approach this question from the point of view of Shannon-type sampling on compact Riemannian manifolds. Namely, we show that N_ω is comparable to cardinality of certain sampling sets for the subspace of ω-bandlimited functions on M.