A New Approach to Universality at the Edge of the Spectrum

Abstract

We show how localization and smoothing techniques can be used to establish universality at the edge of the spectrum for a fixed positive measure on [-1,1]. Assume that the measure is a regular measure, and is absolutely continuous in some closed neighborhood J of 1. Assume that in J, w = h * Jacobi, where h(1)>0 and h is continuous at 1. Then universality at 1 for ourgiven measure follows from universality at 1 for the classical Jacobi weight. Note that no smoothness is required of h, only continuity.