Asymptotic spectra of dense Toeplitz matrices are unstable
type: Item
instance of: scholarly article
title: Asymptotic spectra of dense Toeplitz matrices are unstable
zbMATH Open document ID: 1038.65030
DOI: 10.1023/A:1025547501771
published in: Numerical Algorithms
publication date: 2003-10-14T00:00:00Z
review text: An infinite Toeplitz matrix is defined by \(T(a)=(a_{j-k}) _{j,k=0}^{\infty}\) where \(\{ a_{k} \} _{k\in \mathbb{Z}}\) are the Fourier coefficients of \(a(e^{i\theta}).\) It is well known that its spectrum is the range of its symbol \(a(\mathbf{T})\). The authors study the approximation of the spectra obtained from the finite section, which is to examine the convergence of the set of eigenvalues of the finite matrix \(T_{n}(a)=(a_{j-k}) _{j,k=0}^{n-1}\). To this end they consider the asymptotic spectrum of \(\Lambda \left( a\right) =\lim \sup_{n\rightarrow \infty }\sigma (T_{n}(a)) \) and show that \(\Lambda \) is discontinuous on the space of continuous symbols. A numerical example is used to illustrate that fact.
reviewed by: Amin Boumenir
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 1992774
zbMATH Keywords: infinite Toeplitz matrix
zbMATH Keywords: eigenvalue distribution
zbMATH Keywords: asymptotic spectrum
zbMATH Keywords: numerical example
author: A. Böttcher
author: S. M. Grudsky
MaRDI profile type: Publication
full work available at URL: a:1025547501771
OpenAlex ID: W1663058121
description: scientific article; zbMATH DE number 1992774