# Spectral Limitations of Quadrature Rules and Generalized Spherical Designs.

@article{Steinerberger2017SpectralLO,
title={Spectral Limitations of Quadrature Rules and Generalized Spherical Designs.},
author={S. Steinerberger},
journal={arXiv: Spectral Theory},
year={2017}
}
• S. Steinerberger
• Published 2017
• Mathematics
• arXiv: Spectral Theory
• We study manifolds $M$ equipped with a quadrature rule $$\int_{M}{\phi(x) dx} \simeq \sum_{i=1}^{n}{a_i \phi(x_i)}.$$ 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.