Discrete Dirac systems on the semiaxis: rational reflection coefficients and Weyl functions

Fritzsche, Bernd; Kirstein, Bernd; Roitberg, Inna Ya.; Sakhnovich, Alexander

doi:10.1080/10236198.2019.1572126

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Cited by 4 publications

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“…Self-adjoint discrete Dirac systems have been introduced in [10] and studied further in [11,29,33,34] following the case of skew-self-adjoint discrete Dirac systems in [17]. In particular, the paper [33] is dedicated to the interrelations between self-adjoint discrete Dirac systems and block Toeplitz matrices, which (in the scalar case) are in many respects similar to the interrelations between the famous Szegő recurrences and Toeplitz matrices.…”

Section: Introductionmentioning

confidence: 99%

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

Sakhnovich¹

2019

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We consider discrete Dirac systems as an alternative (to the famous Szegő 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 Bäcklund-Darboux transformation of the trivial discrete Dirac system). We show that block diagonal plus semi-separable Toeplitz matrices appear in this case.

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