  Let $\left( \mathcal{M},g\right) $ be a $d$-dimensional compact connected
Riemannian manifold and let $\left\{ \varphi_{m}\right\}_{m=0}^{+\infty}$ be a
complete sequence of orthonormal eigenfunctions of the Laplace-Beltrami
operator on $\mathcal{M}$. We show that there exists a positive constant $C$
such that for all integers $N$ and $X$ and for all finite sequences of $N$
points in $\mathcal{M}$, $\left\{ x\left( j\right) \right\}_{j=1}^{N}$, and
positive weights $\left\{ a_{j}\right\}_{j=1}^{N}$ we have \[ \sum_{m=0}^{X} |
\sum_{j=1}^{N} a_{j} \varphi_{m} ( x( j) ) | ^{2}\geq \max \{
CX\sum_{j=1}^{N}a_{j}^{2},( \sum_{j=1}^{N}a_{j}) ^{2}\}.\]

Analysis of PDEs