Strong asymptotic behavior and weak convergence of polynomials orthogonal on an arc of the unit circle

Let sigma be a Finite positive Borel measure supported on an are gamma of the unit circle, such that sigma' > 0 a.e. on gamma. We obtain a theorem about the weal, convergence of the corresponding sequence of orthonormal polynomials. Moreover, we prove an analogue of the Szego-Geronimus theorem on strong asymptotics of the orthogonal polynomials on the complement of gamma, which completes to its full extent a result of N. 1. Akhiezer, The key tool in the proofs is the use of orthogonality with respect to varying measures. (C) 2001 Academic Press.