Extremal Problems, Orthogonal Polynomials and Approximation Theory

主要调查者：Edward B.Saff联合首席调查者：E.A.Rakhmanov该项目涉及使用位势理论和算子理论来研究三个主要问题领域：(1)N维单位球面的离散化，包括最小能量排列的渐近性和构造近最优点的算法；(2)关于连续和离散权的正交多项式的弱和强渐近性；函数空间中的最佳有理逼近和亚纯逼近，包括Adamyan、Arov和Krein理论的推广。要研究的主要问题是找到球体表面上大量(通常在数百或数千个)点的近乎最优的布置，使得这些点相对于球体上的面积均匀分散(即，位于任意帽中的点的比例接近于通过将帽的面积除以球体的总面积而获得的比率)。为此，研究人员将使用之前NSF拨款中引入的最低能源标准。通过分析渐近行为(随着点的数量增加)，研究人员的目标是设计出简单的方案，以接近最小的能量生成配置。这一问题的重要性在S.Smer最近的一篇文章中得到了强调，他将这种球点格式的研究列为下个世纪的18个主要数学问题领域之一。事实上，这项研究具有广泛的实际应用，其中包括：(1)大的、稳定的碳分子(称为富勒烯)的结构分析；(2)为轨道卫星收集的数据确定地球表面的最佳采样点；(3)当飞机周围的球面覆盖是必要的时，测试飞机的雷达传感器。

Grant Title: Extremal Problems, Orthogonal Polynomials, and Approximation Theory Principal Investigator: Edward B. Saff Co-Principal Investigator: E. A. Rakhmanov The project involves the use of potential theory and operator theory to investigate three main problem areas: (1) discretization of the surface of the unit sphere in N-dimensions, including asymptotics for minimal energy arrangements and algorithms for constructing near optimal points; (2) weak and strong asymptotics for orthogonal polynomials with respect to continuous and discrete weights; and (3) best rational and meromorphic approximation in function spaces, including generalizations of the theory of Adamyan, Arov, and Krein. The main problem to be investigated concerns finding nearly optimal arrangements of large numbers of points (typically in the hundreds or thousands) on the surface of a sphere so that these points are uniformly dispersed with respect to area on the sphere (i.e., the proportion of points lying in an arbitrary cap is close to the ratio obtained by dividing the area of the cap by the total area of the sphere). For this purpose the investigators will use minimum energy criteria which were introduced in a previous NSF grant. By analyzing the asymptotic behavior (as the number of points grows large), the investigators aim to devise simple schemes that generate configurations with near minimal energy. The importance of this problem is underscored in a recent article by S.Smale, who lists the study of such spherical point schemes as one of the eighteen main mathematical problem areas for the next century. Indeed this study has a wide variety of practical applications, among which are the following: (1) the structural analysis of large, stable, carbon molecules (known as fullerenes); (2) the determination of optimal sampling points on the Earth's surface for data gathered by orbiting satellites; and (3) the testing of radar sensors for aircraft when spherical coverage around the aircraft is essential.