Fine asymptotics of Christoffel functions for general measures

Abstract

Let u be a measure on the unit circle satisfying Szegő's condition. In 1991, A. Mate calculated precisely the first-order asymptotic behavior of the sequence of Christoffel functions associated with u when the point of evaluation lies on the circle, resolving a long-standing open problem. We extend his results to measures supported on smooth curves in the plane. In the process, we derive new asymptotic estimates for the Cesaro means of the corresponding 1-Faber polynomials and investigate some applications to orthogonal polynomials, linear ill-posed problems and the mean ergodic theorem.