基于球面t-设计的球面多项式逼近研究

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 and practical applications.

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。

本项目围绕球面上的逼近问题展开探索，以球面t-设计为节点，对遇到的新问题进行了研究。取得以下结果。.1研究了径向基函数和多项式混合逼近，在l2-l1范数意义下。取得了较以往更好的球面逼近效果，特别对于边缘和无光滑连接处。.2对于不同的逼近格式，选取了不同的正则化算子，得出了满意的逼近效果。.3研究了超几何函数的计算，对于展开和逼近有新的结果，应用到球面逼近问题。.4提出了代数奇异贝塞尔变换的新数值方法和误差估计。.5计算了好条件的球面t-设计，t达到160，并且提出了epsilon-t设计的概念和计算。