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Multipoint Pade Approximation, Orthogonal Polynomials, and Random Matrices

多点 Pade 近似、正交多项式和随机矩阵

基本信息

批准号：
1800251
负责人：
Doron Lubinsky
金额：
$ 25.92万
依托单位：
Georgia Tech Research Corporation
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-05-01 至 2023-04-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800251&HistoricalAwards=false
关键词：
Multipoint Pade Approximation Orthogonal Polynomials

项目摘要

It was the French mathematician Charles Hermite who in 1873 proved that the constant e is a transcendental irrational, that is, e is not the root of a polynomial with integer coefficients. Nine years later, Ferdinand von Lindemann use Hermite's method to resolve the old problem that the number pi is transcendental. Hermite's tool was something we now call the Hermite-Padé approximant. His doctoral student Henri Padé studied a special case, a type of rational function that is now called the Padé approximant. These have turned out be a useful tool in problems as varied as scattering physics, numerical solution of differential equations, and rational approximation. Indeed, calculators still use rational approximations related to Padé approximants for calculating special functions. There are a great many unsolved problems about convergence of sequences of Padé approximants. One of the project goals is to resolve some of these problems. Orthogonal polynomials turn out to be the denominator polynomials in certain Padé approximants, but are an even more important topic in their own right. They have applications in areas ranging from statistics to mathematical physics. The specific goals of the project include investigating the notion of 'exact interpolation' for sequences of multipoint Padé approximants. The PI recently proved that when there is exact interpolation, namely no extra interpolation points, then subsequences or full sequences of Padé approximants converge uniformly in compact sets. The PI intends to establish explicit conditions for exact interpolation. On orthogonal polynomials, the PI recently discovered that universality limits for random matrices can be turned into pointwise asymptotics for orthogonal polynomials at the endpoints of the interval of orthogonality. The PI intends to explore this in the bulk. On the flip side, the PI intends to use recent asymptotics of Eli Levin and the PI to establish universality limits for eigenvalues, as well as distribution of spacings of successive eigenvalues. Additional goals include establishing explicit formulae for Dirichlet orthogonal polynomials. The results will be disseminated in papers and at conferences. The PI recently co-organized a Computational Methods and Function Theory conference in Lublin, and will help co-organize the next one, most probably in Chile. The PI is also hoping to train another graduate student.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

1873年，法国数学家夏尔·埃尔米特证明了常数e是超越无理数，也就是说，e不是整系数多项式的根。九年后，费迪南·冯·林德曼用厄米的方法解决了圆周率是超越数的老问题。厄米的工具是我们现在称之为厄米-帕德逼近。他的博士生Henri Padé研究了一种特殊的情况，一种现在被称为Padé逼近的有理函数。这些已被证明是一个有用的工具，在不同的散射物理问题，微分方程的数值解，和合理的近似。事实上，计算器仍然使用与Padé近似有关的有理近似来计算特殊函数。关于Padé逼近序列的收敛性问题有很多未解决的问题。该项目的目标之一是解决其中的一些问题。正交多项式在某些Padé逼近中是分母多项式，但它们本身是一个更重要的主题。它们在从统计学到数学物理的各个领域都有应用。该项目的具体目标包括研究多点Padé逼近序列的“精确插值”概念。PI最近证明了，当有精确插值，即没有额外的插值点，那么连续或Padé逼近的全序列一致收敛于紧集。PI旨在建立精确插值的显式条件。关于正交多项式，PI最近发现随机矩阵的普适性极限可以转化为正交多项式在正交区间端点处的逐点渐近。PI打算大量探索这一点。另一方面，PI打算使用Eli Levin和PI最近的渐近性来建立特征值的普适性极限，以及连续特征值的间距分布。其他目标包括建立狄利克雷正交多项式的显式公式。调查结果将在文件和会议上散发。PI最近在卢布林共同组织了一次计算方法和函数理论会议，并将帮助共同组织下一次会议，最有可能在智利。PI还希望培养另一名研究生。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。