Relative Szegő asymptotics for Toeplitz determinants

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 Szegő 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.

arXiv comment43 pages