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

TL;DR

This paper explores discrete Dirac systems as an alternative approach to analyzing block Toeplitz matrices, deriving asymptotic relations, and studying Weyl functions, including special cases with rational functions and transformations.

Contribution

It introduces a novel discrete Dirac system framework for Toeplitz matrices, deriving key formulas and asymptotic relations, and analyzing special cases with rational Weyl functions.

Findings

Derived an analog of the Christoffel--Darboux formula.

Established asymptotic relations for reproducing kernels.

Identified block diagonal plus semi-separable Toeplitz matrices in special cases.

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.