Spectral Limitations of Quadrature Rules and Generalized Spherical Designs
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 arXiv:
 arXiv:1708.08736
 Bibcode:
 2017arXiv170808736S
 Keywords:

 Mathematics  Spectral Theory;
 Mathematics  Analysis of PDEs;
 Mathematics  Combinatorics;
 Mathematics  Numerical Analysis
 EPrint:
 to appear in IMRN