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

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 $[-1,1]$, $m\geq3$.