Kheifets A. Unitary Scattering Systems and Interpolation Problems

The thesis proposes and develops methods for solving interpolation problems of analysis, based on the use of unitary scattering systems, which are naturally associated to the data of the problem and its solutions. The corresponding inverse problems are also solved. A generalization of the scheme of the Abstract Interpolation Problem is developed, which allows studying the problems, whose solutions are not analytic functions (such as the Nehari problem). The thesis studies the Commutant Lifting problem, a higher order analogue of the Julia - Caratheodory theorem on an angular boundary derivative and the related boundary interpolation problem in the Schur class, the extended Krein - Langer class, and the Nevanlinna - Pick problem. For the Commutant Lifting problem, a parametrization of all symbols of the given contraction is obtained for the most general case and a complete characterization of the coefficients of this parametrizating formula is given. As the application of these methods and results, two problems of D. Sarason are solved. A higher order analogue of the classical Julia - Caratheodory condition on the angular boundary derivatives of functions of the Schur class is obtained. It is proved that the higher order Julia - Caratheodory condition is equivalent to the coincidence of the boundary derivatives from inside and outside the circle as the function is continued by symmetry. An extended Krein - Langer class is studied, that contains not only meromorphic functions but also functions with jumps. It is shown that interpolation problems are more naturally solved in this class than in the classical one, since it is possible to eliminate the effect of extraneous solutions.