Remarks on extreme eigenvalues of Toeplitz matrices
type: Item
instance of: scholarly article
title: Remarks on extreme eigenvalues of Toeplitz matrices
zbMATH Open document ID: 0638.47023
DOI: 10.1155/S0161171288000055
author: M. Pourahmadi
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46263
review text: Let f be a nonnegative integrable function on \([-\pi,\pi]\), \(m=0\) its essential infimum matrix associated with f and \(\lambda_{1,n}\) its smallest eigenvalue. It follows from a theorem of Szegö that \(\lambda_{1,n}\to 0\) as \(n\to \infty\). The author discussed the rate of this convergence and obtained a lower bound for \(\lambda_{1,n}\) without any smoothness conditions. The main results are the following. Theorem 3.2 Let \(E=\{\theta:\) \(f(\theta)>0\}\) and \(\tau^*(E)\) be its outer measure, then \(\lim\sqrt{\lambda_{1,n}}\leq \tau^*(E)\). Therefore, if \(\tau^*(E)<1\), then \(\lambda_{1,n}\) decrease to zero at least exponentially as \(n\to \infty\). Theorem 3.6 If \(f^{-1}\in L\), then for all \(n\geq 1\), we have \(\lambda_{1,n}\geq 1/\pi^ 2C(n+1)^ 2\) where C is a constant depending on f and \(C=\int f^{-1}d\theta /2\pi\).
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47A70
zbMATH DE Number: 4040626
zbMATH Keywords: smallest eigenvalue
zbMATH Keywords: convergence rate
zbMATH Keywords: lower bound
zbMATH Keywords: extreme eigenvalues of Toeplitz matrices
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4040626