Bounds on orthogonal polynomials and separation of their zeros

Abstract

Let $\{p_n\}$ denote the orthonormal polynomials associated with a measure $\mu$ with compact support on the real line. Let $\mu$ be regular in the sense of Stahl, Totik, and Ullmann, and $I$ be a subinterval of the support in which $\mu$ is absolutely continuous, while ${\mu }^{\prime }$ is positive and continuous there. We show that boundedness of the $\{p_n\}$ in that subinterval is closely related to the spacing of zeros of $p_n$ and $p_{n-1}$ in that interval. One ingredient is proving that “local limits” imply universality limits.