Minimum \(w\)-entropy interpolants for matricial Carathéodory functions and maximum determinant completions of associated block Pick matrix (Q630537)

In this paper, the Nevanlinna-Pick type nondegenerate interpolation problem and the Carathéodory coefficient problem are analyzed, and using the so-called modified block Toeplitz vector approach, the two mentioned problems are connected for a particular class of solutions. The paper has four parts. After the introduction, in the second part an investigation on the unity of the Nevanlinna-Pick problem (NP) and the Carathéodory coefficient problem (C) associated with the block Toeplitz vector of (NP) is made. Several intrinsic relations between these problems are summarized. In the third part of the paper, an explicit formula for the \(w\)-entropy functional for each solution of the nondegenerate problem (NP) is obtained. A matricial generalization of the obtained \(w\)-entropy inequalities is also given. In the last section of the paper, the maximum determinant completion of the \(mq\times mq\) block Pick matrix \(P_\Delta\) of the nondegenerate Nevanlinna-Pick problem with the data \(\Delta\) is considered.