SPHERICAL tε-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 in [17, Chen, Frommer, Lang, Computational existence proofs for spherical t-designs, Numer. Math., 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 [17]. 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 tε-design (0 ≤ ε < 1), which are nodes of a positive, but not necessarily equal, weight quadrature rule exact for polynomials of degree ≤ t. The parameter ε 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 tεdesign is a spherical t-design when ε = 0, and a spherical t-design is a spherical tε-design for any 0 < ε < 1. We show that any point set chosen from the set of interval enclosures [17] is a spherical tε-design. We then study the worstcase error in a Sobolev space for quadrature rules using spherical tε-designs, and investigate a model of polynomial approximation with l1-regularization using spherical tε-designs. Numerical results illustrate the good performance of spherical tε-designs for numerical integration and function approximation on the sphere.