Spectral Limitations of Quadrature Rules and Generalized Spherical Designs

Abstract

We study manifolds M equipped with a quadrature ruleWe 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 S 2 where it generalizes results on spherical designs.2010 Mathematics Subject Classification. 35J05, 35J20, 35Q82, 52A37, 65D32.

Generalized designs on graphs: Sampling, spectra, symmetries

“…Taking the limit suggests that the behavior in the continuous setting should be given by Brownian motion and can be rephrased as a packing problem for heat kernels. This is exactly the heuristic derived in [19].…”

“…A lemma of Montgomery (see also ) may be understood as the study of the analogue of spherical designs on , where polynomials are replaced by trigonometric polynomials—this result does not seem to be very well known in this community since the relevant statement appears as a lemma and is used for a very different purpose. To the best of our knowledge, the first upper bound for weighted spherical designs on Riemannian manifolds is due to the author ; the eigenfunctions of the Laplacian on are polynomials, the classical setting is therefore included as a special case. We briefly remark that there is also the study of combinatorial designs (see, e.g., ): these are families of a subset with highly structured intersection patterns.…”

Quadrature Points via Heat Kernel Repulsion

“…Related results. Our approach here is inspired by the recent paper by one of the authors [51]. We start by describing [51] which combines the two approaches mentioned in the introduction: clearly, on any given manifold, a natural set of low-frequency orthogonal functions is given by the eigenfunctions of the Laplacian −∆ g .…”

“…Our approach here is inspired by the recent paper by one of the authors [51]. We start by describing [51] which combines the two approaches mentioned in the introduction: clearly, on any given manifold, a natural set of low-frequency orthogonal functions is given by the eigenfunctions of the Laplacian −∆ g . Indeed, by the Courant-Fischer-Weyl minimax principle, these can be said to be uniquely characterized by orthogonality and the requirement of minimizing the Dirichlet functional.…”

A nonlocal functional promoting low-discrepancy point sets

“…This paper is partially motivated by earlier results about how to distribute points on a manifold in a regular way. One idea (from [29,35]) is to not construct these points a priori but instead use (local) minimizers of an energy functional. For example, suppose we want to distribute N points on the two-dimensional torus T 2 in a way that is good for numerical integration.…”