Bounds on orthogonal polynomials and separation of their zeros

Let {p_n}denote the orthonormal polynomials associated with a measure μ with compact support on the real line. Let μ be regular in the sense of Stahl, Totik, and Ullmann, and I be a subinterval of the support in which μ is absolutely continuous, while μ′ 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.