A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants

TL;DR

This paper develops a Riemann-Hilbert framework to analyze the asymptotics of Toeplitz+Hankel determinants and associated orthogonal polynomials, without assuming specific relations between the symbols, extending existing operator-theoretic approaches.

Contribution

It formulates a novel 4x4 Riemann-Hilbert problem for Toeplitz+Hankel determinants with general symbols and develops steepest descent methods for their asymptotic analysis.

Findings

Derived asymptotics for orthogonal polynomial norms and polynomials

Introduced a solvable model case under certain symbol ratio conditions

Linked the Riemann-Hilbert approach to recent operator-theoretic methods

Abstract

In this paper we will formulate $4\times4$ Riemann-Hilbert problems for Toeplitz+Hankel determinants and the associated system of orthogonal polynomials, when the Hankel symbol is supported on the unit circle and also when it is supported on an interval $[a,b]$, $0<a<b<1$. The distinguishing feature of this work is that in the formulation of the Riemann-Hilbert problem no specific relationship is assumed between the Toeplitz and Hankel symbols. We will develop nonlinear steepest descent methods for analysing these problems in the case where the symbols are smooth (i.e., in the absence of Fisher-Hartwig singularities) and admit an analytic continuation in a neighborhood of the unit circle (if the symbol's support is the unit circle). We will finally introduce a model problem and will present its solution requiring certain conditions on the ratio of Hankel and Toeplitz symbols. This in turn will allow us to find the asymptotics of the norms $h_n$ of the corresponding orthogonal polynomials and, in fact, the large $n$ asymptotics of the polynomials themselves. We will explain how this solvable case is related to the recent operator-theoretic approach in [Basor E., Ehrhardt T., in Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics, Oper. Theory Adv. Appl., Vol. 259, Birkh\"auser/Springer, Cham, 2017, 125-154, arXiv:1603.00506] to Toeplitz+Hankel determinants. At the end we will discuss the prospects of future work and outline several technical, as well as conceptual, issues which we are going to address next within the $4\times 4$ Riemann-Hilbert framework introduced in this paper.