Multipoint Pade Approximation, Orthogonal Polynomials, and Random Matrices

法国数学家查尔斯·埃尔米特在1873年证明了常数e是一个超越无理数，也就是说，e不是整数系数多项式的根。9年后，费迪南德·冯·林德曼用埃尔米特的方法解决了圆周率是超越数的老问题。埃尔米特的工具就是我们现在所说的埃尔米特-帕德帕尔近似法。他的博士生Henri pad<s:1>研究了一种特殊情况，一种有理函数，现在被称为pad<s:1>近似函数。在散射物理、微分方程的数值解和有理近似等各种问题中，这些都是有用的工具。实际上，计算器仍然使用与帕岱尔近似相关的有理近似来计算特殊函数。关于pad<s:1>近似数列的收敛性有许多尚未解决的问题。该项目的目标之一就是解决其中的一些问题。正交多项式在某些帕岱格近似中是分母多项式，但它本身是一个更重要的话题。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pad<s:1>近似序列的“精确插值”概念。PI最近证明了当存在精确插值时，即没有额外的插值点，那么pad<s:1>近似的子序列或满序列在紧集中一致收敛。PI旨在为精确插值建立明确的条件。在正交多项式上，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点渐近性。PI打算对这一问题进行大规模探索。另一方面，PI打算利用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隔分布。其他目标包括建立狄利克雷正交多项式的显式公式。研究结果将在论文和会议上散发。PI最近在卢布林联合组织了一次计算方法和函数理论会议，并将帮助联合组织下一次会议，最有可能在智利举行。PI还希望再培养一名研究生。该奖项反映了美国国家科学基金会的法定使命，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。

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.