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 an elliptic second-order differential self-adjoint non-negative 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 $\mathbf{M}$.