A Riemann-Hilbert Approach to the Kissing Polynomials

Abstract

Motivated by the numerical treatment of highly oscillatory integrals, this thesis studies a family of polynomials known as the Kissing Polynomials through Riemann-Hilbert techniques. The Kissing Polynomials are a family of non-Hermitian orthogonal polynomials, which are orthogonal with respect to the complex weight function $\exp(i\omega z)$ over the interval $[-1,1]$, where $\omega>0$. Although they have already been used to derive complex quadrature rules, there remain two main questions which this thesis addresses. The first is the existence of such polynomials; the second is the behavior of these polynomials throughout the complex plane. The first two chapters of this thesis provide the necessary background needed for the main results presented in the later chapters. In the first chapter, the connection between the numerical integration of highly oscillatory integrals and the Kissing Polynomials is established. Furthermore, we present the theory of non-Hermitian orthogonal polynomials and provide a more detailed description of the results in this thesis. The second chapter is a review on the formulation of the Kissing Polynomials as a solution to a matrix valued Riemann-Hilbert problem. This formulation is crucial to establishing both the existence of the Kissing Polynomials and its properties throughout the complex plane. Moreover, we also provide an overview of the powerful non-commutative steepest descent technique developed by Deift and Zhou in the mid 1990s used to compute the asymptotics for oscillatory Riemann-Hilbert problems, which will be used extensively in Chapters 4 and 5. In Chapter 3, we utilize the Riemann-Hilbert approach of Fokas, Its, and Kitaev to establish our first main result: the existence of the even degree Kissing polynomials for all values of $\omega>0$. First, we use the Riemann-Hilbert problem to show that the Kissing Polynomials satisfy a certain linear ordinary differential equation. Then, using standard results on differential equations, along with previous results on the Kissing Polynomials found in the literature, we are able to provide the desired result. In Chapter 4, we turn our attention to the behavior of the Kissing Polynomials as both the degree $n$ and parameter $\omega$ become large. To achieve this, we formulate this problem in terms of varying-weight Kissing polynomials, whose asymptotics can be handled with the Deift-Zhou steepest descent analysis. Now, the weight function depends now on $n$, the degree of the underlying polynomial. We are able to provide uniform asymptotics of the Kissing Polynomials as both $n$ and $\omega$ go to infinity at a linear rate such that the ratio $\omega/n >t_c$, where $t_c$ is a to be specified critical value. In Chapter 5, we generalize the results of Chapter 4 and study polynomials which are orthogonal with respect to the varying, complex weight, $\exp(n s z)$, over the interval $[-1,1]$, where now $s\in\mathbb{C}$. We will see that there are curves in the $s$-plane, called breaking curves, which separate regions corresponding to differing asymptotic behavior of the polynomials. In this chapter, we provide the large $n$ behavior of the recurrence coefficients associated to these polynomials. Finally, we also study the behavior of these recurrence coefficients as the parameter $s$ approaches a breaking curve in a specified double scaling limit.PhD Studentship: Cantab Capital Institute for the Mathematics of Informatio