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Asymptotic properties of orthogonal and extremal polynomials


Simanek, Brian Zachary (2012) Asymptotic properties of orthogonal and extremal polynomials. Dissertation (Ph.D.), California Institute of Technology.


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
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Simon, Barry M.
Thesis Committee:
  • Simon, Barry M. (chair)
  • Wilson, Richard M.
  • Krueger, Helge
  • Rains, Eric M.
Defense Date:16 May 2012
Author Email:briansimanek7 (AT)
Funding AgencyGrant Number
NSF Graduate Research FellowshipDGE-1144469
NSF Graduate Research FellowshipDGE-0703267
Record Number:CaltechTHESIS:05222012-113808604
Persistent URL:
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Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7061
Deposited By: Brian Simanek
Deposited On:23 May 2012 18:11
Last Modified:26 Dec 2012 04:43

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