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

## TL;DR

This paper connects Weyl's asymptotic formula with Shannon sampling theory on compact Riemannian manifolds, showing that eigenvalue counts are comparable to sampling set cardinalities for bandlimited functions.

## Contribution

It provides a direct proof linking eigenvalue distribution to sampling set cardinality on Riemannian manifolds, bridging spectral theory and sampling theory.

## Key findings

- Eigenvalue count $\\mathcal{N}_{\omega}$ is comparable to sampling set cardinality.
- Establishes a Shannon-type sampling framework on Riemannian manifolds.
- Provides a new perspective on Weyl's law via sampling theory.

## 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 ${\bf M}$.