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Discrete self-adjoint Dirac systems: asymptotic relations, Weyl functions and Toeplitz matrices. (English) Zbl 1486.39024

Summary: 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.

MSC:

39A45 Difference equations in the complex domain
15B05 Toeplitz, Cauchy, and related matrices
30E10 Approximation in the complex plane
34B20 Weyl theory and its generalizations for ordinary differential equations
39A12 Discrete version of topics in analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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