On smooth perturbations of Chebyshëv polynomials and -Riemann–Hilbert method

Abstract

Abstract $\bar {\partial } $ -extension of the matrix Riemann–Hilbert method is used to study asymptotics of the polynomials $ P_n(z) $ satisfying orthogonality relations $$ \begin{align*} \int_{-1}^1 x^lP_n(x)\frac{\rho(x)dx}{\sqrt{1-x^2}}=0, \quad l\in\{0,\ldots,n-1\}, \end{align*} $$ where $ \rho (x) $ is a positive $ m $ times continuously differentiable function on $ [-1,1] $ , $ m\geq 3 $ .