Graduation Year

2009

Document Type

Dissertation

Degree

Ph.D.

Degree Granting Department

Mathematics and Statistics

Major Professor

Vilmos Totik, Ph.D.

Keywords

Faber, Orthogonal polynomials, Zero-spacing, Ill-posed problems, Potential theory

Abstract

In 1991, A. Máté precisely calculated the first-order asymptotic behavior of the sequence of Christoffel functions associated with Szego measures on the unit circle. Our principal goal is the abstraction of his result in two directions. We compute the translated asymptotics, the limit as n proceeds to infinity of "n" times the order-n Christoffel function evaluated at "x" plus "a divided by n", and obtain, as a corollary, a universality limit for the fairly broad class of Szego weights. Finally, we prove Máté's result for measures supported on smooth curves in the plane. Our proof of the latter derives, in part, from a precise estimate of certain weighted means of the Faber polynomials associated with the support of the measure. Finally, we investigate a variety of applications, including two novel applications to ill-posed problems in Hilbert space and the mean ergodic theorem.

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