Spectral Limitations of Quadrature Rules and Generalized Spherical Designs

TL;DR

This paper establishes fundamental spectral limitations of quadrature rules with nonnegative weights on manifolds, showing they cannot exactly integrate more than a linear number of Laplacian eigenfunctions relative to the number of points.

Contribution

It provides explicit bounds on the number of Laplacian eigenfunctions exactly integrated by quadrature rules on manifolds, generalizing known results on spherical designs.

Findings

Quadrature rules with nonnegative weights on manifolds are limited to integrating a linear number of eigenfunctions.

Explicit constants for the bounds are computed, with c_2=4 on the 2-sphere.

The results extend and generalize previous findings on 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.