Two-sided interpolation for matrix functions with entries in the Hardy space
type: Item
instance of: scholarly article
title: Two-sided interpolation for matrix functions with entries in the Hardy space
zbMATH Open document ID: 0828.15025
DOI: 10.1016/0024-3795(94)00074-N
author: Daniel Alpay
author: Vladimir Bolotnikov
published in: Linear Algebra and its Applications
publication date: 1995-07-24T00:00:00Z
review text: Consider the space \({\mathbf H}^{p\times q}_2\) of \(p\times q\) matrices with entries in \({\mathbf H}_2\), the Hardy space of the unit disk \({\mathbf D}\), endowed with the inner product \[ \langle F, G\rangle= \text{Trace } {1\over 2\pi} \int^{2\pi}_0 G(e^{it})^* F(e^{it}) dt. \] The following two sided residue interpolation problem is studied. Given complex matrices \(W\), \(Z\), \(A\), \(B\), \(C\), \(D\), \(\Gamma\) of appropriate sizes, find all functions \(H\in {\mathbf H}^{p\times q}_2\) such that \(\sum_{z\in {\mathbf D}} \text{Res}_{\lambda= z}(\lambda I- W)^{- 1} AH(\lambda)= C\), \(\sum_{z\in {\mathbf D}} \text{Res}_{\lambda= z} H(\lambda) B(\lambda I- Z)^{- 1}= D\), \(\sum_{z \in {\mathbf D}} \text{Res}_{\lambda= z}(\lambda I- W)^{- 1} (AH(\lambda)- C) B(\lambda I- Z)^{- 1}= \Gamma\). Here, \(\text{Res}_{\lambda= z} f(\lambda)\) stands for the residue of the function \(f(\lambda)\) at \(z\). An additional constraint \(\langle H, H\rangle\leq m\), where \(m\) is a fixed positive number, may be imposed. The solution of this interpolation problem (without the additional constraint) are given in terms of the double coset formula \(H(\lambda)= H_0(\lambda)+ Q_1(\lambda) h(\lambda) Q_2(\lambda)\), where \(h(\lambda)\in {\mathbf H}^{p\times q}_2\) is a free parameter, and the functions \(H_0(\lambda)\), \(Q_1(\lambda)\) and \(Q_2(\lambda)\) are determined explicitly by the given data of the interpolation problem. When the additional constraint is present, then a criterion for existence of solutions \(H(\lambda)\) is given in the form \(\text{Trace } X\leq m\), where \(X\) is a positive definite matrix explicitly constructed using the given data. In this case, the same formula describes all the solutions, but the function \(h(\lambda)\) is now restricted by the inequality \(\langle h, h\rangle\leq m- \text{Trace } X\). Analogous formulas are obtained for the one-sided interpolation problems in which only the first or the second interpolation equality is pres…
reviewed by: Leiba Rodman
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 778246
zbMATH Keywords: matrix functions
zbMATH Keywords: Hardy space
zbMATH Keywords: two sided residue interpolation problem
zbMATH Keywords: complex matrices
zbMATH Keywords: double coset formula
zbMATH Keywords: indefinite metric
zbMATH Keywords: Lagrange- Sylvester interpolation problems
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 778246