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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, TE, so that the left-shifts of J are asymptotic to the orbit {J′m} on TE. 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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  1. Centre for Mathematical Sciences, Lund University, Box 118, 22100, Lund, Sweden

    Jacob S. Christiansen

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  1. Jacob S. Christiansen
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Correspondence to Jacob S. Christiansen.

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Christiansen, J.S. Dynamics in the Szegő class and polynomial asymptotics. JAMA 137, 723–749 (2019). https://doi.org/10.1007/s11854-019-0013-y

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