Spherical T_∈-Designs for Approximations on the Sphere

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 S² ⊂ ℝ³ 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 S2. In this paper we investigate a new concept of point sets on the sphere named spherical t_∈-design (0 ≤ ∈ < 1), which are nodes of a positive, but not necessarily equal, weight quadrature rule exact for polynomials of degree ≤ t. The parameter ∈ 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_∈-design is a spherical t-design when ∈ = 0, and a spherical t-design is a spherical t_∈- design for any 0 < ∈ < 1. We show that any point set chosen from the set of interval enclosures (loc. cit.) is a spherical t_∈-design. We then study the worstcase error in a Sobolev space for quadrature rules using spherical t_∈-designs, and investigate a model of polynomial approximation with l¹-regularization using spherical t_∈-designs. Numerical results illustrate the good performance of spherical t_∈-designs for numerical integration and function approximation on the sphere.