The authors present two basic methods for solving the classical interpolation problem and discuss various interrelationships between different approaches to that problem.

The first method, developed by Fritzsche and Kirstein, is based on an immediate matricial generalization of the classical Schur algorithm. It uses the fact that the contractive holomorphic p×q matrix-valued functions can be naturally integrated in the set of non-negative Hermitian (p+q)×(p+q) Borel measures on the unit circle. The method relies on the Potapov’s construction of all linear fractional transformations which map the matrix ball of all p×q contractive matrices into itself.

In the second method, Dubovoj makes use of the Potapov’s method of transforming a given interpolation problem into an equivalent matrix inequality. He solves then the transformed problem with the aid of matrix functions having particular J-properties (so-called J-elementary factors) and gives a characterization of all matrix-valued functions that represent resolvent matrices of the given interpolation problem.

The book is completed with an extensive bibliography.