On zeros, bounds, and asymptotics for orthogonal polynomials on the unit circle

Abstract

Let $\mu$ be a measure on the unit circle that is regular in the sense of Stahl, Totik and Ullmann. Let $\{\varphi_{n}\}$ be the orthonormal polynomials for $\mu$ and $z_{jn}\}$ their zeros. Let $\mu$ be absolutely continuous in an arc $\Delta$ of the unit circle, with $\mu'$ positive and continuous there. We show that uniform boundedness of the orthonormal polynomials in subarcs $\Gamma$ of $\Delta$ is equivalent to certain asymptotic behaviour of their zeros inside sectors that rest on $\Gamma$. Similarly the uniform limit $\lim_{n\to \infty}|\varphi_{n}(z)|^{2}\mu'(z)=1$ is equivalent to related asymptotics for the zeros in such sectors. Bibliography: 27 titles.