Recent Advances in Asymptotic Analysis

TL;DR

This survey reviews recent developments in asymptotic analysis over the past fifty years, highlighting classical methods, new theories for difference equations, and modern techniques like the nonlinear steepest descent and Wiener-Hopf methods.

Contribution

It introduces recent advancements in asymptotic theory, including the nonlinear steepest descent method and applications to orthogonal polynomials without differential equations.

Findings

Development of a uniform asymptotic treatment for Heisenberg polynomials

Introduction of the nonlinear steepest descent method for orthogonal polynomials

Application of Wiener-Hopf technique to integral equations

Abstract

This is a survey article on an old topic in classical analysis. We present some new developments in asymptotics in the last fifty years. We start with the classical method of Darboux and its generalizations, including an uniformity treatment which has a direct application to the Heisenberg polynomials. We then present the development of an asymptotic theory for difference equations, which is a major advancement since the work of Birkhoff and Trjitzinsky in 1933. A new method was introduced into this field in the nineteen nineties, which is now known as the nonlinear steepest descent method or the Riemann-Hilbert approach. The advantage of this method is that it can be applied to orthogonal polynomials which do not satisfy any differential or difference equations neither do they have any integral representations. As an example, we mention the case of orthogonal polynomials with respect to the Freud weight. Finally, we show how the Wiener-Hopf technique can be used to derive asymptotic expansions for the solutions of an integral equation on a half line.