Universality in measure in the bulk for varying weights (Q516605)

The authors deal with universality results in measure for sequences of varying weights \(\{g_{n} \exp (-2nQ)\},\) where \(\exp(-Q)\) is an exponential weight and \(g_{n}\) are functions which admit suitable polynomial approximations. It is assumed that \(Q\) is continuous while \(Q'\) is continuous up to a finitely number of points of the support of the measure that is constituted by finitely many compact intervals. It is also assumed that the equilibrium measure \(\omega_{Q}\) is absolutely continuous and \(\omega'_{Q}\) is positive and continuous in the interior of \(S_{Q},\) the support of \(\omega_{Q},\) up to at finitely many points. They extend a previous result by the authors [Adv. Math. 219, No. 3, 743--779 (2008; Zbl 1176.28014)] to varying weights whose proof depends heavily on asymptotics of Christoffel functions established in [\textit{V. Totik}, Adv. Appl. Math. 25, No. 4, 322--351 (2000; Zbl 0970.42016)]. The authors prove that universality holds in measure for weights under more general hypotheses than those of their previous contribution. First, they obtain a equiconvergence result (Theorem 1.1) and, as a consequence, the universality in measure to the sinc kernel follows. Potential theoretic estimates are used together with pointwise estimates in order to get Theorem 1.1.