## 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} ⊂ 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 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 l^{1}-regularization using spherical t_{∈}-designs. Numerical results illustrate the good performance of spherical t_{∈}-designs for numerical integration and function approximation on the sphere.

Original language | English |
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Pages (from-to) | 2831-2855 |

Number of pages | 25 |

Journal | Mathematics of Computation |

Volume | 87 |

Issue number | 314 |

DOIs | |

Publication status | Published - 5 Feb 2018 |

## Keywords

- Interval analysis
- L-regularization
- Numerical integration
- Polynomial approximation
- Spherical t-designs

## ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics