Almost strict total positivity and a class of Hurwitz polynomials
type: Item
instance of: scholarly article
title: Almost strict total positivity and a class of Hurwitz polynomials
zbMATH Open document ID: 1072.15028
author: Dimitar K. Dimitrov
author: J. M. Peña
published in: Journal of Approximation Theory
publication date: 2005-05-04T00:00:00Z
full work available at URL: http://hdl.handle.net/11449/21728
review text: A real \(n\times n\) matrix \(A:=[a_{ij}]\) is called totally positive (TP) if all its minors are nonnegative and strictly totally positive (STP) if they are positive. A TP matrix \(A\) is called almost strictly totally positive (ASTP) if \(A\) is nonsingular and, for each minor of \(A\) formed by consecutive rows and consecutive columns, the minor is positive if and only if its diagonal entries are positive. Let \(\delta=4.079\dots\) be the real root of \(x^{3}-5x^{2}+4x-1=0\). It has been shown by \textit{T. Craven} and \textit{G. Csordas} [Linear Multilinear Algebra 45, 19--34 (1998; Zbl 0922.15014)] that \(A\) is STP whenever its entries are positive and \(a_{ij}a_{i+1,j+1}\geq\delta a_{i,j+1}a_{i+1,j}\) for all \(i,j=1,\dots,n-1\). The main result of this paper is a generalization of the Craven-Csordas theorem in the case that not all entries of \(A\) are positive. It is known that for any nonsingular TP matrix \(A\) we have: \(a_{ii}>0\) for all \(i\); \(a_{hk}=0\) for all \(h\geq i\) and \(k\leq j\) whenever \(a_{ij}=0\) and \(i>j\); and \(a_{hk}=0\) for all \(h\leq i\) and \(k\geq j\) whenever \(a_{ij}=0\) and \(i<j\). The authors prove that if \(A\) is a nonnegative matrix which satisfies these latter conditions and \[ a_{ij}a_{i+1,j+1}\geq\delta a_{i,j+1}a_{i+1,j} \quad\text{whenever}\quad a_{ij}a_{i+1,j+1}>0, \tag{*} \] then \(A\) is TP\(.\) Moreover if the inequality (*) is always strict, then \(A\) is ASTP. They apply this to show that if the coefficients of \(f(z):=c_{n}z^{n}+c_{n-1}z^{n-1}+\dots+c_{0}\) are positive real numbers and \(c_{k}c_{k+1}\geq\delta c_{k-1}c_{k+1}\) for \(k=1,\dots,n-2\) then \(f(z)\) is Hurwitz stable (that is, all the zeros of \(f(z)\) have negative real parts). They give similar conditions for an entire function to belong to the Laguerre-Pólya class of type I. The authors also show that the \(\delta\) which appears in these theorems cannot be universally replaced by a constant smaller than \(4\), but conjecture that for a f…
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 30D15
zbMATH DE Number: 2165383
zbMATH Keywords: totally positive matrix
zbMATH Keywords: Hurwitz stable polynomial
zbMATH Keywords: Laguerre-Pólya class
zbMATH Keywords: strictly totally positive matrix
zbMATH Keywords: shadows' lemma
MaRDI profile type: Publication
OpenAlex ID: W2025313427
cites work: Totally positive matrices
cites work: On the Generating Functions of Totally Positive Sequences
cites work: On the Total Nonnegativity of the Hurwitz Matrix
cites work: Q4069194
cites work: The approximation of a totally positive band matrix by a strictly banded totally positive one
cites work: Spaces with Almost Strictly Totally Positive Bases
cites work: On some zero-increasing operators
cites work: Jensen polynomials and the Turán and Laguerre inequalities
cites work: Complex zero decreasing sequences
cites work: A Sufficient condition for strict total positivity of a matrix
cites work: Q3254327
cites work: Intervals of almost totally positive matrices
cites work: Almost strictly totally positive matrices
cites work: On the characterization of almost strictly totally positive matrices
cites work: Q5624248
cites work: On power series having sections with only real zeros
cites work: A Hurwitz Matrix is Totally Positive
cites work: A Sufficient Condition for All the Roots of a Polynomial To Be Real
cites work: Q5531193
cites work: Q4896037
cites work: Q5546070
cites work: Q5536376
cites work: Shape preserving representations in computer aided geometric design
cites work: Necessary conditions of Hurwitz polynomials
cites work: Some necessary conditions for Hurwitz stability.
DOI: 10.1016/J.JAT.2004.10.010
description: scientific article; zbMATH DE number 2165383