Dynamics in the Szeg\"o class and polynomial asymptotics
Classical Analysis and ODEs
2019-10-29 v1 Spectral Theory
Abstract
We introduce the Szeg\"o class, Sz(E), for an arbitrary Parreau-Widom set E in R and study the dynamics of its elements under the left shift. When the direct Cauchy theorem holds on C\E, we show that to each J in Sz(E) there is a unique element J' in the isospectral torus, T_E, so that the left-shifts of J are asymptotic to the orbit {J'_m} on T_E. Moreover, we show that the ratio of the associated orthogonal polynomials has a limit, expressible in terms of Jost functions, as the degree n tends to infinity. This enables us to describe the large n behaviour of the orthogonal polynomials for every J in the Szeg\"o class.
Keywords
Cite
@article{arxiv.1602.04033,
title = {Dynamics in the Szeg\"o class and polynomial asymptotics},
author = {Jacob S. Christiansen},
journal= {arXiv preprint arXiv:1602.04033},
year = {2019}
}