Local asymptotics for orthonormal polynomials in the interior of the support via universality

Abstract:

We establish local pointwise asymptotics for orthonormal polynomials inside the support of the measure using universality limits. For example, if a measure $\mu$ has compact support, is regular in the sense of Stahl, Totik, and Ullmann, and in some subinterval $I$, $\mu$ is absolutely continuous and $\mu ^{\prime }$ is positive and continuous, we prove that for $y_{jn}$ in a compact subset of $I^{o}$ with $p_{n}^{\prime }\left ( y_{jn}\right ) =0$, we have \begin{equation*} \lim _{n\rightarrow \infty }\frac {p_{n}\left ( y_{jn}+\frac {z}{n\omega \left ( y_{jn}\right ) }\right ) }{p_{n}\left ( y_{jn}\right ) }=\cos \pi z \end{equation*} uniformly in $y_{jn}$ and for $z$ in compact subsets of the plane. Here $\omega$ is the equilibrium density for the support of $\mu$.