Critical points and error rank in best \(H_2\) matrix rational approximation of fixed McMillan degree
type: Item
instance of: scholarly article
title: Critical points and error rank in best \(H_2\) matrix rational approximation of fixed McMillan degree
zbMATH Open document ID: 0913.41009
published in: Constructive Approximation
publication date: 1998-05-25T00:00:00Z
review text: Let \(\widetilde{H}_{2,0}\) be the Hardy space of functions (defined in the complement of the unit disk) whose Fourier coefficients \((a_n)\) are real and satisfy \(a_n=0\) when \(n\geq 0\), endowed with the \(L_2\) norm. Similarly can be defined the Hardy space \(\widetilde{H}^{p\times m}_{2,0}\) of \(p\times m\) matrix-valued functions. The authors study the following best approximation problem: Given \(F\in \widetilde{H}^{p\times m}_{2,0}\) and an integer \(n \geq 1\), minimize \(\| F-H\| _2\) when \(H\) ranges over all (matrix-valued) rational functions analytic in the complement of the unit disk and having the ``McMillan degree'' at most \(n\). The topological structure of the set of best approximants is described in terms of the Douglas-Shapiro-Shields factorization of rational functions. In the case when \(F\) is a rational matrix (vanishing at infinity) of McMillan degree \(k\) and \(H\) is critical of degree \(n,k \geq n\) then one obtains that the rational matrix \(F-H\) has the rank \(\leq k-n\). In the particular case \(k=n\) it follows that \(H=F\) is the unique best approximant. A local uniqueness result is then derived for \(F\) of degree \(n\), when \(F\) is close enough to the compact subset \(K\) of rational matrices of degree \(n\). Deep technical results from different areas of mathematics are used.
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 93B30
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 1143074
zbMATH Keywords: matrix rational approximation in \(H_2\)
zbMATH Keywords: topology of rational matrices
zbMATH Keywords: identification of linear dynamical systems
author: L. Baratchart
author: M. Olivi
reviewed by: Ioan Şerb
MaRDI profile type: Publication
full work available at URL: s003659900075
OpenAlex ID: W1989932113
DOI: 10.1007/S003659900075
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description: scientific article; zbMATH DE number 1143074