Asymptotic Properties of Orthogonal and Extremal Polynomials

Author: Simanek, Brian Zachary

Year: 2012

Degree: Dissertation (Ph.D.)

Advisor: Simon, Barry M.

Committee Members: Simon, Barry M.; Wilson, Richard M.; Krueger, Helge; Rains, Eric M.

Option: Mathematics

DOI: 10.7907/B44M-XJ50

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 Lq-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.

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