Discrete self-adjoint Dirac systems: asymptotic relations, Weyl functions and Toeplitz matrices
Classical Analysis and ODEs
2024-04-03 v1 Mathematical Physics
Functional Analysis
math.MP
Spectral Theory
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.
Keywords
Cite
@article{arxiv.1912.05213,
title = {Discrete self-adjoint Dirac systems: asymptotic relations, Weyl functions and Toeplitz matrices},
author = {Alexander Sakhnovich},
journal= {arXiv preprint arXiv:1912.05213},
year = {2024}
}
Comments
This work is an important development of the idea presented in our paper arXiv:1711.03064