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

(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 X, some measurable subset of Euclidean space, one sometimes wants to construct, a design, a finite set of points, P 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...

• Type: Journal
• Year: 2008
• Where: NA
• Authors: S. B. Damelin
• Keywords: Energy Functionals, Euclidean Space, Gaussian Quadrature Formula, NA 2008