Uniform bounds on the 1-norm of the inverse of lower triangular Toeplitz matrices
type: Item
instance of: scholarly article
title: Uniform bounds on the 1-norm of the inverse of lower triangular Toeplitz matrices
zbMATH Open document ID: 1230.15002
published in: Linear Algebra and its Applications
publication date: 2011-07-15T00:00:00Z
full work available at URL: conjecture proved.pdf
review text: Consider the lower triangular \(\left( n+1\right) \times \left( n+1\right) \) Toeplitz matrix \[ T_{n}=\left( \begin{matrix} b_{0} & 0 & \cdots & 0 \\ b_{1} & b_{0} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ b_{n} & b_{n-1} & \cdots & b_{0} \end{matrix} \right) \] where \(b_{0}\geq b_{1}\geq \cdots \geq b_{n}\geq b\geq 0\) and \( \lim_{n\rightarrow \infty }b_{n}=b=0\). The authors provide a sharp uniform upper bound for \(\left\| T_{n}^{-1}\right\| _{\infty }\). In particular case where \[ b_{0}=\frac{1}{\sqrt{2}},b_{1}=\frac{\sqrt{3}-1}{\sqrt{2}},b_{2}=\frac{\sqrt{ 5}-\sqrt{3}}{\sqrt{2}},\dots,b_{n}=\frac{\sqrt{2n+1}-\sqrt{2n-1}}{\sqrt{2}} \] then it is proven that \[ \left\| T_{n}^{-1}\right\| _{1}\leq 2\sqrt{2}\left( 5-\sqrt{3}-\sqrt{5} \right) \] and thus a solution is given to the GKM conjecture. Furthermore, in case where \[ b_{0}=1,b_{1}=2^{1-a}-1,b_{2}=3^{1-a}-2^{1-a},\dots,b_{n}=\left( n+1\right) ^{1-a}-n^{1-a} \] with \(a\in \left( 0,1\right) \), upper bounds are given for \(\left\| T_{n}^{-1}\right\| _{1}\) under certain conditions, by providing in this way a partial answer to the Brunner's one-point collocation problem.
reviewed by: Nicholas Karampetakis
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 34K28
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 74B10
Mathematics Subject Classification ID: 34K06
Mathematics Subject Classification ID: 90C33
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 5924527
zbMATH Keywords: 1-norm uniform upper bound
zbMATH Keywords: inverse of lower triangular Toeplitz matrix
zbMATH Keywords: Brunner's conjecture
zbMATH Keywords: Abel equation
author: Xin Liu
author: S. McKee
author: Jinyun Yuan
author: Yaxiang Yuan
MaRDI profile type: Publication
OpenAlex ID: W2135672386
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DOI: 10.1016/J.LAA.2011.02.044
description: scientific article; zbMATH DE number 5924527