Some generalizations of Szegö's first limit theorem
type: Item
instance of: scholarly article
title: Some generalizations of Szegö's first limit theorem
zbMATH Open document ID: 0582.15008
DOI: 10.1007/BF01903988
author: Mark Grigorevich Kreĭn
author: Ilya M. Spitkovsky
published in: Analysis Mathematica
publication date: 1983-01-01T00:00:00Z
review text: The authors consider a sequence \(\{C_ j: -\infty <j<\infty \}\) of \(r\times r\) matrices and the corresponding truncated block Toeplitz matrices \(T_ n\). The assumption made throughout is that \(\Delta_ n=\det T_ n\neq 0\), \(n=0,1,\ldots\). The sequence \(\{C_ j\}\) is said to be \(\alpha\)-sectorial if for every \(\xi \in \mathbb{C}^{r(n+1)}\), \(| Im \xi T_ n\xi^*\!_ n| \leq Re \xi T_ n\xi^*\tan \alpha\), \(\alpha\in [0,\pi /2)\). A generalization is obtained, in this block matrix case and for \(\alpha\)-sectorial sequences, of the original theorem of Szegö concerning the limits of the ratios \(\Delta_{n+1}/\Delta_ n\) when \(\alpha =0\), \(r=1\). The generalization also includes previous results on the limits of these ratios for \(\alpha >0\), \(r=1\). A good bibliography of earlier work is given.
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 3931187
zbMATH Keywords: Szegö's first limit theorem
zbMATH Keywords: truncated block Toeplitz matrices
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01903988
OpenAlex ID: W2331095090
description: scientific article; zbMATH DE number 3931187