On the spacing of zeros of paraorthogonal polynomials for singular measures

Abstract

We prove a lower bound on the spacing of zeros of paraorthogonal polynomials on the unit circle, based on continuity of the underlying measure as measured by Hausdorff dimensions. We complement this with the analog of the result from [2] showing that clock spacing holds even for certain singular continuous measures.

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.…”

“…As far as we know the dependence of the zeros of f n (•; r, s) on r has been studied only when s = 0 (see [19,Theorem 2]). However, the case dµ (r) = dµ (r,0) (see [30,Example 8.2.5]) is especially simple because there is a direct connection with the ultrashperical polynomials 7 . Indeed, by (27), we have…”