Extremal Problems, Orthogonal Polynomials and Approximation Theory

极值问题，正交多项式，和 逼近理论 主要研究者：Edward B。SAFF 共同主要研究者：E. A.拉赫莫诺夫 该项目涉及使用势理论和算子理论来研究三个主要问题领域：（1）曲面的离散化 的单位球在N维，包括渐近极小 能量安排和构造近最优点的算法; (2)关于连续权和离散权的正交多项式的弱渐近性和强渐近性;（3）最佳有理亚纯 函数空间中的近似，包括 阿达米扬、阿罗夫和克莱因的理论。 要研究的主要问题是找到接近最优的 大量点的排列（通常以数百或更少的数量）， 在一个球体的表面上，使这些点是一致的 相对于球体上的区域分散（即，的比例 位于任意帽中的点接近于通过以下方式获得的比率： 将帽的面积除以球体的总面积）。为此 研究人员将使用最低能量标准， 在以前的NSF补助金中引入。通过分析渐近行为 (as点的数量越来越大），调查人员的目标是设计 简单的计划，产生配置接近最小的能量。的 这一问题的重要性在S.Smale最近的一篇文章中得到了强调， 他把对这种球面点方案的研究列为十八大 下个世纪的主要数学问题领域。事实上，这项研究 具有广泛的实际应用，其中包括 以下内容：（1）大的、稳定的碳分子的结构分析 （称为富勒烯）;（2）确定最佳采样点， 地球表面的轨道卫星收集的数据;及（3） 当飞机周围的球面覆盖是必要的时，对飞机的雷达传感器进行测试。

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.