Shannon sampling and weak Weyl's law on compact Riemannian manifolds (Q2417884)

Let $\mathcal{M}=(M,g)$ be a closed Riemannian manifold of dimension $m$. Let $L$ be an elliptic self-adjoint non-negative second order partial differential operator. There exists a spectral resolution $\{\phi_n,\lambda_n\}$ for $L$ where the $\phi_n$ form a complete orthonormal basis for $L^2(M)$ with $L\phi_n=\lambda_n\phi_n$, $0\le\lambda_1\le\dots\le\lambda_n$ and $\lambda_n\rightarrow\infty$. Let $E_\omega(L):=\operatorname{Span}\{\phi_n:\lambda_n\le\omega\}$ be the $\omega$-bandlimited functions. Set $n(\omega):=\dim\{E_\omega(L)\}$; this is the number of eigenvalues (counted with multiplicity) that do not exceed $\omega$. The Weyl asymptotic formula yields $n(\omega)\sim c(m)\text{Vol}(M)\omega^{m/2}$. The author gives a proof of the weak Weyl's law without using the Weyl asymptotic formula, i.e. $n(\omega)$ is comparable to the cardinality of certain sampling sets for the $\omega$-bandlimited functions on $M$. For the entire collection see [Zbl 1409.35008].