PDEA Seminar: What are orthogonal, paraorthogonal, and bi-orthogonal polynomials on the unit circle? A spectral and dynamical introduction

Abstract: In this talk, I will give an accessible introduction to orthogonal, paraorthogonal, and bi-orthogonal (Baxter) polynomials on the unit circle. We will explore these objects from two complementary perspectives: the spectral point of view (using unitary pentadiagonal CMV matrices) and the dynamical point of view (via Szegő and Baxter difference equations). I will show how this theory naturally applies to approximation theory (Gaussian quadrature) and to the asymptotics of Toeplitz determinants (Szegő's limit theorems). The talk is intended as an introduction for a general mathematical audience, concluding with a brief look at some recent results regarding the non-Hermitian Baxter setting. No prior background is assumed.

This is a seminar in the PDEs and Applications seminar series.