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

Abstract

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 block semi-separable Toeplitz matrices appear in this case.