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Mathematics of Computation

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Spherical $ t_\epsilon$-designs for approximations on the sphere


Authors: Yang Zhou and Xiaojun Chen
Journal: Math. Comp.
MSC (2010): Primary 65D30, 41A10, 65G30
DOI: https://doi.org/10.1090/mcom/3306
Published electronically: February 5, 2018
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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.


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Additional Information

Yang Zhou
Affiliation: School of Mathematics and Statistics, Shandong Normal University, Jinan, Shangdong, China 250000
Email: andres.zhou@connect.polyu.hk

Xiaojun Chen
Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
Email: maxjchen@polyu.edu.hk

DOI: https://doi.org/10.1090/mcom/3306
Keywords: Spherical $t$-designs, polynomial approximation, interval analysis, numerical integration, $l_1$-regularization
Received by editor(s): December 28, 2015
Received by editor(s) in revised form: November 9, 2016, April 26, 2017, and June 14, 2017
Published electronically: February 5, 2018
Additional Notes: The first author’s work was supported in part by Department of Applied Mathematics, The Hong Kong Polytechnic University and Hong Kong Research Council Grant PolyU5002/13p and in part by NSFC grant No. 11626147.
The second author’s work was supported in part by Hong Kong Research Council Grant PolyU153001/14p.
Article copyright: © Copyright 2018 American Mathematical Society

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