Spectral Limitations of Quadrature Rules and Generalized Spherical Designs

We study manifolds $M$ equipped with a quadrature rule $${\int }_M\varphi (x)dx\simeq \sum _{i=1}^n{a}_i\varphi (x_i)\mathrm{.}$$ We show that $n-$point quadrature rules with nonnegative weights on a compact $d-$dimensional manifold cannot integrate more than at most the first $c_{d}n+o(n)$ Laplacian eigenfunctions exactly. The constants $c_d$ are explicitly computed and $c_2=4$. The result is new even on ${\mathbb{S}}^2$ where it generalizes results on spherical designs.