On Smooth Perturbations of Chebysh\"ev Polynomials and $\bar\partial$-Riemann-Hilbert Method

TL;DR

This paper develops a $ar ext{ extdegree}$-Riemann-Hilbert approach to analyze the asymptotic behavior of orthogonal polynomials with smooth weight functions, extending classical methods to handle perturbations.

Contribution

It introduces a $ar ext{ extdegree}$-extension of the Riemann-Hilbert method for studying asymptotics of orthogonal polynomials with smooth weights, advancing analytical techniques.

Findings

Derived asymptotic formulas for orthogonal polynomials with smooth weights.

Extended Riemann-Hilbert method to include $ar ext{ extdegree}$-extensions.

Provided new insights into polynomial behavior under smooth perturbations.

Abstract

$\bar\partial$-extension of the matrix Riemann-Hilbert method is used to study asymptotics of the polynomials $P_n(z)$ satisfying orthogonality relations \[ \int_{-1}^1 x^lP_n(x)\frac{\rho(x)dx}{\sqrt{1-x^2}}=0, \quad l\in\{0,\ldots,n-1\}, \] where $\rho(x)$ is a positive $m$ times continuously differentiable function on $[-1,1]$, $m\geq3$.