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Finite gap Jacobi matrices. II: The Szegő class. (English) Zbl 1236.42021
The authors study Jacobi matrices \(J\) and asymptotics of the associated orthogonal polynomials, where \(\sigma_{\text{ess}}(J)\) is a finite union of disjoint closed intervals. They study Szegő’s theorem for the general finite gap case. They use Remling’s theorem about the approach to the isospectral torus together with an analysis of Jost functions to provide a new proof of Szegő asymptotics including \(L^{2}\) Szegő asymptotics on the spectrum.
For Part I, see ibid. 32, No. 1, 1–65 (2010; Zbl 1200.42012).

MSC:
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
58J53 Isospectrality
14H30 Coverings of curves, fundamental group
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
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