哈尔滨工业大学（深圳）学术讲座

演讲人Speaker：安聪沛     

题目Title: Can hyperinterpolation part with quadrature exactness?

时间Date： 2025年3月25日       Time：10:00-11:00

地点Venue： H 栋 404 室

内容摘要Abstract： 

We discuss the approximation of continuous functions on the unit sphere by spherical polynomials of degree n via hyperinterpolation. Hyperinterpolation of degree n is a discrete approximation of the L2-orthogonal projection of degree n with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most 2n. This talk aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz--Zygmund property. Consequently, hyperinterpolation can be constructed by a positive-weight quadrature rule--not necessarily with quadrature exactness. This scheme is called unfettered hyperinterpolation. We provide a reasonable error estimate for unfettered hyperinterpolation. The error estimate generally consists of two terms: a term representing the error estimate of the original hyperinterpolation of full quadrature exactness and another introduced as compensation for the loss of exactness degrees. A guide to controlling the newly introduced term in practice is provided. In particular, if the quadrature points form a quasi-Monte Carlo design, then there is a refined error estimate. Numerical experiments verify the error estimates and the practical guide.

讲座讨论通过超插值的方式，用 n 阶球面多项式逼近单位球面上的连续函数。度数为 n 的超插值是度数为 n 的 L2 正交投影的离散近似，其傅里叶系数通过正权求积规则得到，该方法可精确积分所有度数至多为 2n 的球面多项式。本讲座旨在用 Marcinkiewicz--Zygmund 性质代替积分精确性假设，从而绕过这一假设。因此，超插值可以通过正权求积规则来构造——不一定具有积分精确性。这种方案被称为无约束超插值。我们为无约束超插值提供了合理的误差估计。误差估计一般由两个项组成：一个项代表原始超插值求积精度的误差估计，另一个项则是对求积精度损失的补偿。本文提供了在实践中控制新引入项的指导。特别是，如果求积节点形成了拟蒙特卡罗设计，那么就会有一个细化的误差估计值。数值实验验证了误差估计和实践指导。

个人简介(About the speaker):

安聪沛，本科、硕士毕业于中南大学，师从向淑晃，博士毕业于香港理工大学,师从 Chen Xiaojun (AMS Fellow, SIAM Fellow )和Ian H Sloan (AMSFellow, SIAM Fellow, Fellow ofthe Australian Academy of Science),现为贵州大学一流学科特聘教授。美国《数学评论》评论员。研究成果在球 t-设计，振荡积分的近似计算，多项式构造逼近，反问题计算上取得不少同行关注的结果。例如 2022年菲尔兹奖得主 Mayna Vasovska,就证明过安聪沛与和作者提出的关于球 t-设计猜想。