Commentary on "Expansions for Eigenfunctions and Eigenvalues of large-$n$ Toeplitz Matrices"

Torsten Ehrhardt

doi:10.4279/pip.020004

Abstract

A Commentary on the paper by L. P. Kadanoff [Pap. Phys. 2, 020003 (2010)].Received: 29 September 2010, Accepted: 4 October 2010; Edited by: A. G. Green; DOI: 10.4279/PIP.020004

Highlights

A result of Widom [5] states that for certain symbols, the eigenvalues of Tn(φ) accumulate asymptotically along the curve described by φ(z), |z| = 1
The problem of the asymptotics of the eigenvalues of Toeplitz matrices has a long history and is a multi-faceted and difficult topic. It is closedly connected with the asymptotics of the determinants of Toeplitz matrices and with the Szego Limit
If φ is a rational function, it is proved that the eigenvalues do accumulate along arcs which lie inside the curve described by φ. This case is best understood because there is an explicit formula for the characteristic polynomial of Tn(φ)

Summary

Introduction

P. Kadanoff [1] is concerned with the problem of describing the asymptotics of the eigenvalues and eigenvectors of Toeplitz matrices, Tn(φ) = (φj−k)nj,−k=1 0, as the matrix size n goes to infinity. A result of Widom [5] states that for certain symbols, the eigenvalues of Tn(φ) accumulate asymptotically along the curve described by φ(z), |z| = 1. The result applies to the symbols considered here, where it is of importance that a(z) is non-smooth at precisely one point.

Results

Conclusion