Point Perturbation and Asymptotics Orthogonal Polynomials

Series: 
Analysis Seminar
Thursday, October 15, 2009 - 14:00
1 hour (actually 50 minutes)
Location: 
Skiles 255 **NOTE ROOM CHANGE AND SPECIAL DAY**
,  
University of Oklahoma
Organizer: 
In this talk, I will discuss some results obtained in my Ph.D. thesis. First, the point mass formula will be introduced. Using the formula, we shall see how the asymptotics of orthogonal polynomials relate to the perturbed Verblunsky coefficients. Then I will discuss two classes of measures on the unit circle -- one with Verblunsky coefficients \alpha_n --> 0 and the other one with \alpha_n --> L (non-zero) -- and explain the methods I used to tackle the point mass problem involving these measures. Finally, I will discuss the point mass problem on the real line. For a long time it was believed that point mass perturbation will generate exponentially small perturbation on the recursion coefficients. I will demonstrate that indeed there is a large class of measures such that that proposition is false.