Spherical $t_\epsilon$-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 $\le t$. The existence of a spherical $t$-design with $(t+1)^2$ points in a set of interval enclosures on the unit sphere $\mathbb {S}^2 \subset \mathbb {R}^3$ for any $0\le t \le 100$ 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 $\mathbb {S}^2$. In this paper we investigate a new concept of point sets on the sphere named spherical $t_\epsilon$-design ($0 \le \epsilon <1$), which are nodes of a positive, but not necessarily equal, weight quadrature rule exact for polynomials of degree $\le t$. 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 $t_\epsilon$-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.