Energies, group-invariant kernels and numerical integration on compact manifolds

S.B Damelin,J Levesley,D.L Ragozin,X Sun

doi:10.1016/j.jco.2008.09.001

Abstract

The purpose of this paper is to derive quadrature estimates on compact, homogeneous 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.

Highlights

Let M be a compact d-dimensional manifold (d ≥ 1) embedded in the Euclidean space IRd+k for some k ≥ 0
In [3], this idea was extended to spheres whereby the authors related numerical integration to an extremal problem using Riesz energy and a class of smooth kernels defined on the spheres which are called zonal in what follows
The purpose of this paper is to derive quadrature estimates on compact and homogeneous manifolds M embedded in Euclidean space, via energy functionals associated with a class of group-invariant kernels that are generalizations of zonal kernels on the sphere and radial kernels in euclidean spaces

Summary

Introduction

Let M be a compact d-dimensional manifold (d ≥ 1) embedded in the Euclidean space IRd+k for some k ≥ 0. In [3], this idea was extended to spheres whereby the authors related numerical integration to an extremal problem using Riesz energy and a class of smooth kernels defined on the spheres which are called zonal in what follows. The purpose of this paper is to derive quadrature estimates on compact and homogeneous manifolds M embedded in Euclidean space, via energy functionals associated with a class of group-invariant kernels that are generalizations of zonal kernels on the sphere and radial kernels in euclidean spaces. In particular to kernels such as weighted Riesz kernels and classes of smooth functions defined on spheres and certain projective spaces.

Harmonic analysis on compact homogeneous manifolds

Group invariant kernels and smooth convolution

N -point discrete κ energy

Quadrature for smooth functions on the sphere and projective spaces