The Formula steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights

Abstract

We develop a new asymptotic method for the analysis of matrix Riemann-Hilbert problems. Our method is a generalization of the steepest descent method first proposed by Deift and Zhou; however our method systematically handles jump matrices that need not be analytic. The essential technique is to introduce nonanalytic extensions of certain functions appearing in the jump matrix, and to therefore convert the Riemann-Hilbert problem into a δ problem. We use our method to study several asymptotic problems of polynomials orthogonal with respect to a measure given on the unit circle, obtaining new detailed uniform convergence results, and for some classes of nonanalytic weights, complete information about the asymptotic behavior of the individual zeros.