Dynamics in the Szegő class and polynomial asymptotics

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Abstract

We introduce the Szegő class, Sz(E), for an arbitrary Parreau–Widom set E ⊂ 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 ∈ 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 ∞. This enables us to describe the large n behaviour of the orthogonal polynomials for every J in the Szegő class.

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  • Mathematics
Original languageEnglish
JournalJournal d'Analyse Mathematique
Publication statusE-pub ahead of print - 2019 Mar 19
Publication categoryResearch
Peer-reviewedYes