Toeplitz matrices with an exponential growth of entries and the first Szegő limit theorem
type: Item
instance of: scholarly article
title: Toeplitz matrices with an exponential growth of entries and the first Szegő limit theorem
zbMATH Open document ID: 0967.47022
published in: Journal of Functional Analysis
publication date: 2001-08-27T00:00:00Z
review text: Let \(S(n)=\{s_{j-k}\}_{k,j=1}^n, n\leq\infty\), be a block Toeplitz matrix. The author studies such matrices with an exponential growth of entries and its connection with the first Szegő limit theorem. With Toeplitz matrix are connected two analytic at zero functions \(\varphi (\lambda)=\frac{s_0}{2}+\sum_{p=1}^\infty s_{-p}\lambda^p\) and \(\psi (\mu)=\frac{s_0}{2}+\sum_{p=1}^\infty s_{p}\mu^p\). The poles of \(\varphi\) and \(\psi\) inside unit circle allow an exponential growth of \(\|s_{\pm p}\|\). The main idea is based on replacing \(S(\infty)\) by a matrix \(S(\infty)=\{s_{j-k}\}_{k,j=1}^\infty\) with better behaviour of entries and using asymptotic \(\det S(r), r\to\infty\) instead of \(\det S(r)\). By this way a Szegő-type limit theoerm is also obtained.
reviewed by: N. Karapetiants
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 1437361
zbMATH Keywords: Toeplitz matrix
zbMATH Keywords: Szegő limit theorem
zbMATH Keywords: indefinite matrix
zbMATH Keywords: linear fractional transformation
author: Alexander L. Sakhnovich
MaRDI profile type: Publication
full work available at URL: jfan.1999.3543
OpenAlex ID: W2071615058
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DOI: 10.1006/JFAN.1999.3543
description: scientific article; zbMATH DE number 1437361