A walk through energy, discrepancy, numerical integration and group invariant measures on measurable subsets of euclidean space

Abstract

(A) The celebrated Gaussian quadrature formula on finite intervals tells us that the Gauss nodes are the zeros of the unique solution of an extremal problem. We announce recent results of Damelin, Grabner, Levesley, Ragozin and Sun which derive quadrature estimates on compact, homogenous manifolds embedded in Euclidean spaces, via energy functionals associated with a class of group-invariant kernels which are generalizations of zonal kernels on the spheres or radial kernels in euclidean spaces. Our results apply, in particular, to weighted Riesz kernels defined on spheres and certain projective spaces. Our energy functionals describe both uniform and perturbed uniform distribution of quadrature point sets. (B) Given $\mathcal{X}$, some measurable subset of Euclidean space, one sometimes wants to construct, a design, a finite set of points, $\mathcal{P} \subset \mathcal{X}$, with a small energy or discrepancy. We announce recent results of Damelin, Hickernell, Ragozin and Zeng which show that these two measures of quality are equivalent when they are defined via positive definite kernels $K:\mathcal{X}^2(=\mathcal{X}\times\mathcal{X}) \to \mathbb{R}$. The error of approximating the integral $\int_{\mathcal{X}} f(\mathbf{\mathit{x}}) \, {\rm d} \mu(\mathbf{\mathit{x}})$ by the sample average of f over $\mathcal{P}$ has a tight upper bound in terms the energy or discrepancy of $\mathcal{P}$. The tightness of this error bound follows by requiring f to lie in the Hilbert space with reproducing kernel K. The theory presented here provides an interpretation of the best design for numerical integration as one with minimum energy, provided that the μ defining the integration problem is the equilibrium measure or charge distribution corresponding to the energy kernel, K. (C) Let $\mathcal{X}$ be the orbit of a compact, possibly non Abelian group, $\mathcal{G}$, acting as measurable transformations of $\mathcal{X}$ and the kernel K is invariant under the group action. We announce recent results of Damelin, Hickernell, Ragozin and Zeng which show that the equilibrium measure is the normalized measure on $\mathcal{X}$ induced by Haar measure on $\mathcal{G}$. This allows us to calculate explicit representations of equilibrium measures. There is an extensive literature on the topics (A–C). We emphasize that this paper surveys recent work of Damelin, Grabner, Levesley, Hickernell, Ragozin, Sun and Zeng and does not mean to serve as a comprehensive survey of all recent work covered by the topics (A–C).