Spectral Limitations of Quadrature Rules and Generalized Spherical Designs

Abstract

Abstract We study manifolds $M$ equipped with a quadrature rule $$\begin{equation} \int_{M}{\phi(x)\,\mathrm{d}x} \simeq \sum_{i=1}^{n}{a_i \phi(x_i)}.\end{equation*}$$We show that $n$-point quadrature rules with nonnegative weights on a compact $d$-dimensional manifold cannot integrate more than at most the 1st $c_{d}n + o(n)$ 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.