Eigenvalues of Hermitian Toeplitz matrices with polynomially increasing entries
type: Item
instance of: scholarly article
title: Eigenvalues of Hermitian Toeplitz matrices with polynomially increasing entries
zbMATH Open document ID: 1258.15011
DOI: 10.4171/JST/29
author: J. M. Bogoya
author: A. Böttcher
author: S. M. Grudsky
published in: Journal of Spectral Theory
publication date: 2012-08-11T00:00:00Z
review text: The authors consider the first order asymptotics, as \(n\to \infty\), of the extreme eigenvalues of the \(n\times n\) Hermitian Toeplitz matrices whose first rows have the form \([a_0,a_1,\dots ,a_{n-1}]\) where, for some \(\alpha >0\), \(a_k=k^{\alpha}+o(k^{\alpha})\) as \(k\to \infty\). Following an approach introduced by \textit{H. S.\ Widom} [Trans.\ Am.\ Math.\ Soc., 88, 491--522 (1958; Zbl 0101.09202)], they relate the eigenvalues of these matrices to those of an integral operator with kernel \(K(x,y)=|x-y|^{\alpha}\). More details are given for some special cases. In one of these cases, their results provide an alternative proof of a result of W. F.\ Trench.
reviewed by: Alan L. Andrew
Mathematics Subject Classification ID: 15B05
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 45C05
Mathematics Subject Classification ID: 47N40
zbMATH DE Number: 6064844
zbMATH Keywords: Hermitian Toeplitz matrix
zbMATH Keywords: integral operator
zbMATH Keywords: circulant matrix
zbMATH Keywords: extreme eigenvalues
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/jst/29
OpenAlex ID: W2048798484
description: scientific article; zbMATH DE number 6064844