基于球面t-设计的球面多项式逼近研究

项目名称： 基于球面t-设计的球面多项式逼近研究

项目编号： No.11301222

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 安聪沛

作者单位： 暨南大学

项目金额： 22万元

中文摘要： 球面上函数逼近研究具有重要的理论和广泛的应用背景。球面多项式是球面上函数逼近的一个重要而基本的工具。本项目首先研究球面上的多项式逼近格式：插值，超插值，过滤超插值， Newman-Shapiro算子逼近，广义 de la Vallée-Poussin逼近，正则化逼近等。从而提出新的多项式逼近格式。并且在不同的函数空间意义下研究逼近格式的算子范数的大小、误差估计以及逼近的收敛性。从实际计算的角度，利用球面t-设计作为数值积分结点，数值实现逼近。更进一步，本项目将研究正则化参数选取对逼近质量的影响。最后，本项目将理论研究的成果应用球体函数逼近、医疗图像恢复、大气模拟、卫星数据等实际问题中。我们期待该课题的立项，以便开展理论研究和实际应用。

中文关键词： 球面t-设计；逼近；正则化；特殊函数；

英文摘要： The study of approximation of functions over the unit sphere has great theoretic and practical implications. Spherical polynomials are an important and basic tool in the approximation of functions. In this project, we first study polynomial approximation schemes on the unit sphere: interpolation, hyperinterpolation, filtered hyperinterpolation, Newman-Shapiro operator approximation, generalizing de la Vallée-Poussin approximation, regularized approximation and so on. Therefore, we propose new approximation schemes. Meanwhile, we study the operator norm and convergence in a variety of function spaces. In practical computation, we employ spherical t-designs as the quadrature nodes to realize the numerical approximation. Moreover, we study the relationship between the approximation quality and the choice of point sets on the unit sphere via the comparison of geometric distribution of point sets on the unit sphere. Last but far from the least, we will apply the obtained theoretical results into approximation method for the balls and practical problems: medical imaging, atomspheric science, satellite data and so on. The spherical approximation problems have been of great concern by international experts. We look forward to establishing the project with the support from NNSF, to carry more favorable theoretic research

英文关键词： spherical t-design；approximation；regularization；special function；