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Discrete Self-adjoint Dirac Systems: Asymptotic Relations, Weyl Functions and Toeplitz Matrices

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Abstract

We consider discrete Dirac systems as an approach to the study of the corresponding block Toeplitz matrices, which in many ways completes the famous approach via Szegő recurrences and matrix orthogonal polynomials. We prove an analog of the Christoffel–Darboux formula and derive the asymptotic relations for the analog of reproducing kernel (using Weyl–Titchmarsh functions of discrete Dirac systems). These asymptotic relations are expressed also in terms of block Toeplitz matrices. We study the case of rational Weyl–Titchmarsh functions (and GBDT version of the Bäcklund–Darboux transformation of the trivial discrete Dirac system) as well. It is shown that block diagonal plus block semi-separable Toeplitz matrices (which are easily inverted) appear in this case.

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Acknowledgements

This research was supported by the Austrian Science Fund (FWF) under Grant No. P29177.

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  1. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Alexander Sakhnovich

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  1. Alexander Sakhnovich
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Correspondence to Alexander Sakhnovich.

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Sakhnovich, A. Discrete Self-adjoint Dirac Systems: Asymptotic Relations, Weyl Functions and Toeplitz Matrices. Constr Approx 55, 641–659 (2022). https://doi.org/10.1007/s00365-021-09530-9

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