Strong Asymptotic Behavior and Weak Convergence of Polynomials Orthogonal on an Arc of the Unit Circle

Abstract

Let @s be a finite positive Borel measure supported on an arc @c of the unit circle, such that @s'>0 a.e. on @c. We obtain a theorem about the weak convergence of the corresponding sequence of orthonormal polynomials. Moreover, we prove an analogue of the [email protected]?-Geronimus theorem on strong asymptotics of the orthogonal polynomials on the complement of @c, which completes to its full extent a result of N. I. Akhiezer. The key tool in the proofs is the use of orthogonality with respect to varying measures.