Relative Szeg\H{o} asymptotics for Toeplitz determinants

TL;DR

This paper investigates the asymptotic behavior of ratios of Toeplitz determinants associated with measures on the unit circle, revealing that second order asymptotics depend on a smooth function and specific Verblunsky coefficients, extending Szeg\

Contribution

It introduces a relative Szeg\

Findings

Second order asymptotics depend on a few Verblunsky coefficients.

Established a relative Strong Szeg\

Applicable to measures with singular components on a single arc.

Abstract

We study the asymptotic behavior, as $n\to\infty$, of ratios of Toeplitz determinants $D_n(e^h d\mu)/D_n(d\mu)$ defined by a measure $\mu$ on the unit circle and a sufficiently smooth function $h$. The approach we follow is based on the theory of orthogonal polynomials. We prove that the second order asymptotics depends on $h$ and only a few Verblunsky coefficients associated to $\mu$. As a result, we establish a relative version of the Strong Szeg\H{o} Limit Theorem for a wide class of measures $\mu$ with essential support on a single arc. In particular, this allows the measure to have a singular component within or outside of the arc.