Zero Spacings of Paraorthogonal Polynomials on the Unit Circle

Abstract

We prove some new results about the spacing between neighboring zeros of paraorthogonal polynomials on the unit circle. Our methods also provide new proofs of some existing results. The main tool we will use is a formula for the phase of the appropriate Blaschke product at points on the unit circle.

Markov theorem for weight functions on the unit circle

“…Over the years there were many extensions to the classical theory of orthogonal polynomials on the real line (OPRL). After the influential works by Delsarte and Genin [14,15,16] and Jones et al [31] about the nowadays called paraorthogonal polynomials on the unit circle (POPUC) -in many senses the appropriate complex analog of OPRL-, this collection of polynomials and their zeros have received considerable attention from two disparate audiences, namely researchers in orthogonal polynomials and researchers in numerical linear algebra (see for instance [27,25,26,1,14,28,15,16,49,6,2,8,42,32,44,43,50,45,39,40,11,12,38,10,41,7]). It must be said that rarely in the numerical linear algebra context the name POPUC is used; however, the reader has to proceed with caution in the literature because many results on POPUC were first discovered in this framework.…”