Multipoint Pade Approximation, Orthogonal Polynomials, and Random Matrices

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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.