Simanek, Brian Zachary (2012) Asymptotic properties of orthogonal and extremal polynomials. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:05222012-113808604
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.
|Item Type:||Thesis (Dissertation (Ph.D.))|
|Subject Keywords:||Orthogonal Polynomials; Extremal Polynomials; Szego Asymptotics; Ratio Asymptotics; Relative Asymptotics; Weak Asymptotic Measures|
|Degree Grantor:||California Institute of Technology|
|Division:||Physics, Mathematics and Astronomy|
|Thesis Availability:||Public (worldwide access)|
|Defense Date:||16 May 2012|
|Non-Caltech Author Email:||briansimanek7 (AT) gmail.com|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Brian Simanek|
|Deposited On:||23 May 2012 18:11|
|Last Modified:||26 Dec 2012 04:43|
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