The two-sided residue interpolation with nodes in the unit disk for matrix functions (Q1885192)

The author investigates the two-sided residue interpolation problem with interpolant points in the open unit disk \(\mathcal{D}\) of the complex plane. The problem is to find a matrix function in some subclass of Carathéodory class satisfying a set of interpolation conditions of the following form \[ \sum_{z\in\mathcal{D}}\text{Res}_{z=z_0}(zI-A_\zeta)^{-1}B_+ F(z)=-B_-, \] \[ \sum_{z\in\mathcal{D}}\text{Res}_{z=z_0}F(z)C_-(zI-A_\pi )^{-1}=C_+, \] \[ \sum_{z\in\mathcal{D}}\text{Res}_{z=z_0}(zI-A_\zeta)^{-1}B_ +F(z)C_-(zI-A_\pi )^{-1}=\Gamma, \] where \((C_+,C_-,A_\pi; A_\zeta,B_+,B_-;\Gamma)\) are given matrices that satisfy the Sylvester equation \[ \Gamma A_\pi-A_\zeta\Gamma= B_+C_++B_-C_-. \] Some necessary and sufficient conditions for this problem are obtained. As the author notes, the method of the paper [\textit{D. Alpay}, \textit{J. A. Ball}, \textit{I. Gohberg} and \textit{L. Rodman}, Linear Algebra Appl. 208--209, 485--521 (1994; Zbl 0854.15009)] is used for the investigations. The key moment of the investigations is a class \(C_l^{(s)}(\tau,2\pi-\tau)\) described in the article. The article should be interesting for the specialists in linear algebra, analysis and its applications.