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.

我们将离散狄拉克系统视为研究相应块Toeplitz矩阵的一种方法，该方法在许多方面通过Szegő递归和矩阵正交多项式完成了著名的方法。我们证明了Christoffel–Darboux公式的类似物，并得出了再生核类似物的渐近关系（使用离散Dirac系统的Weyl–Titchmarsh函数）。这些渐近关系也以块Toeplitz矩阵表示。我们还研究了有理Weyl-Titchmarsh函数的情况（以及琐碎的离散Dirac系统的Bäcklund-Darboux变换的GBDT版本）。结果表明，在这种情况下会出现块对角线加上块半可分离的Toeplitz矩阵（很容易反转）。