Two-sided interpolation for matrix functions with entries in the Hardy space
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 prescribed. Much of the analysis goes through when \({\mathbf H}^{p\times q}_2\) is endowed with an indefinite metric. Generalized Lagrange- Sylvester interpolation problems and the state-space method of their solution (exposed in the book ``Interpolation of rational matrix functions'' by \textit{J. A. Ball}, \textit{I. Gohberg} and the reviewer (1990; Zbl 0708.15011)) serve as the starting point of the analysis in the reviewed paper.
Lagrange- Sylvester interpolation problems
scientific article; zbMATH DE number 778246