Spherical $t_\epsilon$-Designs for Approximations on the Sphere

TL;DR

This paper introduces spherical $t_$-designs, a new concept generalizing spherical $t$-designs, demonstrating their effectiveness for numerical integration and approximation on the sphere, with theoretical and numerical validation.

Contribution

The paper proposes spherical $t_$-designs, extending spherical $t$-designs, and shows their existence from interval enclosures, along with analysis of their error properties and practical applications.

Findings

Spherical $t_$-designs include all spherical $t$-designs as special cases.

Point sets from interval enclosures are spherical $t_$-designs.

Numerical experiments show good performance in integration and approximation.

Abstract

A spherical $t$-design is a set of points on the sphere that are nodes of a positive equal weight quadrature rule having algebraic accuracy $t$ for all spherical polynomials with degrees $\le t$. Spherical $t$-designs have many distinguished properties in approximations on the sphere and receive remarkable attention. Although the existence of a spherical $t$-design is known for any $t\ge 0$, a spherical design is only known in a set of interval enclosures on the sphere \cite{chen2011computational} for $t\le 100$. It is unknown how to choose a set of points from the set of interval enclosures to obtain a spherical $t$-design. In this paper we investigate a new concept of point sets on the sphere named spherical $t_\epsilon$-design ($0<\epsilon<1$), which are nodes of a positive weight quadrature rule with algebraic accuracy $t$. The sum of the weights is equal to the area of the sphere and the mean value of the weights is equal to the weight of the quadrature rule defined by the spherical $t$-design. 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 \cite{chen2011computational} is a spherical $t_\epsilon$-design. We then study the worst-case errors of quadrature rules using spherical $t_\epsilon$-designs in a Sobolev space, and investigate a model of polynomial approximation with the $l_1$-regularization using spherical $t_\epsilon$-designs. Numerical results illustrate good performance of spherical $t_\epsilon$-designs for numerical integration and function approximation on the sphere.