Math 521: Advanced Topics in Real Analysis
In this course we will discuss the theory of orthogonal polynomials. One starts with a probability measure with infinite support in the complex plane and applies the Gram-Schmidt procedure to the sequence of monomials. If the measure is supported by the real line or the unit circle, the resulting sequence of polynomials is known to obey recursion relations. Of central importance in the field is a study of the map from the measure to the coefficients appearing in the recursion relations and the inverse of this map. A closely related correspondence, which is of similar importance, is between the measure and matrix representations of the operator given by multiplication with the independent variable. This course will cover the general theory in these two cases, OPRL and OPUC, as well as a connection between the two, known as the Geronimus relations. We will also cover selected topics for specific classes of OPRL or OPUC.
Prerequisites: Real and complex analysis.
Meeting Time: TTh 1:00 PM -- 2:15 PM
Location: Herman Brown Hall 427
Email: [my last name] at rice dot edu
Office: Herman Brown Hall 434
Office Hour: Th 2:15 PM -- 3:15 PM and by appointment
B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory. American Mathematical Society Colloquium Publications, 54, Part 1. American Mathematical Society, Providence, RI, 2005.
B. Simon, Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory. American Mathematical Society Colloquium Publications, 54, Part 2. American Mathematical Society, Providence, RI, 2005.
B. Simon, Szego's Theorem and Its Descendants: Spectral Theory for L2
Perturbations of Orthogonal Polynomials. Princeton University Press, Princeton, NJ, to appear in Fall 2010.
The final grade for the course will be based on attendance, class participation, and the presentation of a recent research paper on orthogonal polynomials. The instructor will suggest a number of suitable papers.