Asymptotic behavior of the Verblunsky coefficients for the OPUC with a varying weight

TL;DR

This paper analyzes the asymptotic behavior of Verblunsky coefficients for orthogonal polynomials on the unit circle with a varying weight, using potential theory and string equations, under specific smoothness and support conditions.

Contribution

It provides the first detailed asymptotic analysis of Verblunsky coefficients for OPUC with a varying exponential weight and equilibrium measure supported on a single interval.

Findings

Asymptotic formulas for Verblunsky coefficients derived

Connection established between coefficients and string equations

Results applicable under smoothness and support assumptions

Abstract

We present an asymptotic analysis of the Verblunsky coefficients for the polynomials orthogonal on the unit circle with the varying weight $e^{-nV(\cos x)}$, assuming that the potential $V$ has four bounded derivatives on $[-1,1]$ and the equilibrium measure has a one interval support. We obtain the asymptotics as a solution of the system of "string" equations.