Asymptotic Properties of Orthogonal and Extremal Polynomials

Citation

Simanek, Brian Zachary (2012) Asymptotic Properties of Orthogonal and Extremal Polynomials. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/B44M-XJ50. https://resolver.caltech.edu/CaltechTHESIS:05222012-113808604

Abstract

This thesis is devoted to asymptotic properties of extremal polynomials in a variety of settings. Special attention is given to the orthonormal and monic orthogonal polynomials. Given a positive real number q and a measure with compact and infinite support in the complex plane, one can define - for every natural number n - a monic polynomial of degree n having minimal L^q-norm with respect to the given measure among all monic polynomials of the same degree. Dividing this polynomial by its norm produces a normalized extremal polynomial. We will study the asymptotic behavior of these extremal polynomials when the given measure is of a certain very general form. Our results concerning extremal polynomial asymptotics will include Szego asymptotics, ratio asymptotics, and relative asymptotics. We will also study the associated Christoffel functions and the weak asymptotic behavior of sequences of measures derived from the normalized extremal polynomials.

Thesis Files