Asymptotics of orthogonal polynomials beyond the scope of Szego's theorem

Abstract

First we give here a simple proof of a remarkable result of Videnskii and Shirokov: let $B$ be a Blaschke product with $n$ zeros, then there exists an outer function $\varphi ,\varphi (0)=1$, such that $\Vert (B\varphi {)}^{\prime }\Vert \le Cn$, where $C$ is an absolute constant. Then we apply this result to a certain problem of finding the asymptotic of orthogonal polynomials.