球面 $t_ε$-设计用于球面上的近似
Spherical $t_ε$-Designs for Approximations on the Sphere
AI总结：
本文提出球面 $t_ε$-设计，用于改进球面上的数值积分和函数逼近，通过引入权重和均值条件，扩展了传统球面 $t$-设计的应用范围。
摘要：
一个球面 $t$-设计是球面上的一组点，作为具有正权重的正则规则的节点，具有代数精度 $t$，对所有度数不超过 $t$ 的球面多项式都具有正准确性。球面 $t$-设计在球面上的近似中具有许多显著性质，并受到广泛关注。尽管已知对于任何 $t\ge 0$ 都存在球面 $t$-设计，但仅在球面上的区间封套中已知对于 $t\le 100$。不清楚如何从区间封套中选择点集以获得球面 $t$-设计。本文研究了一种新的名为球面 $t_ε$-设计（$0<ε<1$）的点集，这些点是具有代数精度 $t$ 的正权重求积规则的节点。权重之和等于球面的面积，权重的平均值等于由球面 $t$-设计定义的求积规则的权重。当 $ε=0$ 时，球面 $t_ε$-设计是球面 $t$-设计，而球面 $t$-设计对于任何 $0<ε<1$ 都是球面 $t_ε$-设计。我们证明了任何从区间封套中选择的点集都是球面 $t_ε$-设计。然后我们研究了使用球面 $t_ε$-设计在Sobolev空间中的最坏情况误差，并研究了使用球面 $t_ε$-设计的多项式逼近模型，采用 $l_1$-正则化。数值结果展示了球面 $t_ε$-设计在球面上的数值积分和函数逼近中的良好性能。
英文摘要：
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 $t_ε$-design ($0<ε<1$), 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 $t_ε$-design is a spherical $t$-design when $ε=0,$ and a spherical $t$-design is a spherical $t_ε$-design for any $0<ε<1$. We show that any point set chosen from the set of interval enclosures \cite{chen2011computational} is a spherical $t_ε$-design. We then study the worst-case errors of quadrature rules using spherical $t_ε$-designs in a Sobolev space, and investigate a model of polynomial approximation with the $l_1$-regularization using spherical $t_ε$-designs. Numerical results illustrate good performance of spherical $t_ε$-designs for numerical integration and function approximation on the sphere.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
