Supercritical regime for the kissing polynomials

Author(s)

Celsus, Andrew F. ^[1] ; Silva, Guilherme L. F. ^[2]

Affiliation

^[1] Univ Cambridge, Cambridge - England

^[2] Univ Sao Paulo ICMC USP, Inst Ciencias Matemat & Comp, Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: Journal of Approximation Theory; v. 255, JUL 2020.
• Web of Science Citations: 0

Abstract

We study a family of polynomials which are orthogonal with respect to the varying, highly oscillatory complex weight function e(ni lambda z) on {[}-1, 1], where lambda is a positive parameter. This family of polynomials has appeared in the literature recently in connection with complex quadrature rules, and their asymptotics have been previously studied when lambda is smaller than a certain critical value, lambda(c). Our main goal is to compute their asymptotics when lambda > lambda(c). We first provide a geometric description, based on the theory of quadratic differentials, of the curves in the complex plane which will eventually support the asymptotic zero distribution of these polynomials. Next, using the powerful Riemann-Hilbert formulation of the orthogonal polynomials due to Fokas, Its, and Kitaev, along with its method of asymptotic solution via Deift-Zhou nonlinear steepest descent, we provide uniform asymptotics of the polynomials throughout the complex plane. Although much of this asymptotic analysis follows along the lines of previous works in the literature, the main obstacle appears in the construction of the so-called global parametrix. This construction is carried out in an explicit way with the help of certain integrals of elliptic type. In stark contrast to the situation one typically encounters in the presence of real orthogonality, an interesting byproduct of this construction is that there is a discrete set of values of lambda for which one cannot solve the model Riemann-Hilbert problem, and as such the corresponding polynomials fail to exist. (C) 2020 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 19/16062-1 - Asymptotic analysis of interacting particle systems and random matrix theory
• Grantee: Guilherme Lima Ferreira da Silva
• Support Opportunities: Research Grants - Young Investigators Grants