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

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; how ever 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 ∂̄ problem. We use our method to study several asymptotic problemsof polynomials orthogonal with respect to a measure given on theunit circle, obtaining new detailed uniform convergence results,and for some classes of nonanalytic weights, complete information about the asymptotic behavior of the individual zeros.