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arXiv · 1912.05213

Discrete self-adjoint Dirac systems: asymptotic relations, Weyl functions and Toeplitz matrices

Alexander Sakhnovich
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Abstract

We consider discrete Dirac systems as an alternative (to the famous Szeg\H{o} recurrencies and matrix orthogonal polynomials) approach to the study of the corresponding block Toeplitz matrices. 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). We study also the case of rational Weyl--Titchmarsh functions (and GBDT version of the B\"acklund-Darboux transformation of the trivial discrete Dirac system). We show that block diagonal plus block semi-separable Toeplitz matrices appear in this case.

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Alexander Sakhnovich. 2019-12-11. Discrete self-adjoint Dirac systems: asymptotic relations, Weyl functions and Toeplitz matrices. https://doi.org/10.1007/s00365-021-09530-9

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