Spherical Tε-designs for approximations on the sphere

Abstract

A spherical t-design is a set of points on the unit sphere that are nodes of a quadrature rule with positive equal weights that is exact for all spherical polynomials of degree <= t. The existence of a spherical t-design with (t+1)(2) points in a set of interval enclosures on the unit sphere S-2 subset of R-3 for any 0 <= t <= 100 is proved by Chen, Frommer, and Lang (2011). However, how to choose a set of points from the set of interval enclosures to obtain a spherical t-design with (t + 1) 2 points is not given in loc. cit. It is known that (t + 1)(2) is the dimension of the space of spherical polynomials of degree at most t in 3 variables on S-2. In this paper we investigate a new concept of point sets on the sphere named spherical t(epsilon)-design (0 <= epsilon < 1), which are nodes of a positive, but not necessarily equal, weight quadrature rule exact for polynomials of degree <= t. The parameter epsilon is used to control the variation of the weights, while the sum of the weights is equal to the area of the sphere. A spherical t(epsilon)-design is a spherical t-design when epsilon = 0, and a spherical t-design is a spherical t,design for any 0 < epsilon < 1. We show that any point set chosen from the set of interval enclosures (loc. cit.) is a spherical t(epsilon)-design. We then study the worst-case error in a Sobolev space for quadrature rules using spherical t(epsilon)-designs, and investigate a model of polynomial approximation with l(1)-regularization using spherical t(epsilon)-designs. Numerical results illustrate the good performance of spherical t(epsilon)-designs for numerical integration and function approximation on the sphere.