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

We establish local pointwise asymptotics for orthonormal polynomials inside the support of the measure using universality limits. For example, if a measure μ\mu has compact support, is regular in the sense of Stahl, Totik, and Ullmann, and in some subinterval II, μ\mu is absolutely continuous and μ′\mu ^{\prime } is positive and continuous, we prove that for yjny_{jn} in a compact subset of IoI^{o} with pn′(yjn)=0p_{n}^{\prime }\left ( y_{jn}\right ) =0, we have limn→∞pn(yjn+znω(yjn))pn(yjn)=cos⁡πz\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 yjny_{jn} and for zz in compact subsets of the plane. Here ω\omega is the equilibrium density for the support of μ\mu.