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Orthogonal Polynomials, Paraorthogonal Polynomials, and Point Perturbation


Wong, Manwah Lilian (2009) Orthogonal Polynomials, Paraorthogonal Polynomials, and Point Perturbation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/E55S-4A69.


This thesis consists of three parts.

Part 1 starts with an introduction to orthogonal polynomials, to be followed by some well-known theorems pertinent to the results we shall discuss. It also states the new results that are going to be proven in Parts 2 and 3.

In Part 2, we consider a sequence of paraorthogonal polynomials and investigate their zeros. Then we introduce paraorthogonal polynomials of the second kind and prove that zeros of first and second kind paraorthogonal polynomials interlace.

In Part 3, we consider the point mass problem. First, we give the point mass formula for the perturbed Verblunsky coefficients. Then we investigate the asymptotics of orthogonal polynomials on the unit circle and apply the results to the point mass formula to compute the perturbed Verblunsky coefficients. Finally, we present two examples, one on the unit circle and one on the real line, such that adding a point mass will generate non-exponential perturbations of the recursion coefficients.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:paraorthogonal polynomials; point perturbation
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)
  • Borodin, Alexei
  • Martinez-Finkelshtein, Andrei
  • Zinchenko, Maxim
Defense Date:27 May 2009
Non-Caltech Author Email:wongmw (AT)
Record Number:CaltechETD:etd-06092009-004344
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:5229
Deposited By: Imported from ETD-db
Deposited On:18 Jun 2009
Last Modified:26 Nov 2019 20:24

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