Convergent interpolatory quadrature rules and orthogonal polynomials of varying measures

Let (Pn) be a sequence of polynomials such that Pn(x) > 0 for x ∈ [− 1, 1] and limn→∞deg(Pn)/n=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lim \limits _{n\to \infty }\text {deg}(P_{n})/n = 1$\end{document}. Let qn be the nth monic orthogonal polynomial with respect to Pn-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {P}_{n}^{-1} $\end{document}dμ, where μ is a measure on [− 1, 1] that is regular in the sense of Stahl and Totik. We prove that the interpolatory quadrature rule with nodes at the zeros of qn is convergent with respect to μ provided that the zeros of Pn lie outside a certain curve surrounding [− 1, 1].