Synopses & Reviews
This volume is devoted to the use of the method of operator identities for investigating interpolation and expansion problems. A general interpolation problem comprising both classical and new elements is formulated. The solution of an abstract form of the Potapov inequality enables the description of the set of solutions of the general interpolation problem. Connections between the solved interpolation problem and important problems of analysis, occurring in, for example, spectral theory, nonlinear integrable equations, and generalised stationary processes are then considered. Audience: This book will be of interest to graduate students and researchers whose work involves approximations and expansions; operator theory; measure and integration; general mathematics systems; and Fourier analysis.
Synopsis
1. Interpolation problems play an important role both in theoretical and applied investigations. This explains the great number of works dedicated to classical and new interpolation problems ( 1)- 5], 8), 13)- 16], 26)- 30], 57]). In this book we use a method of operator identities for investigating interpo- lation problems. Following the method of operator identities we formulate a general interpolation problem containing the classical interpolation problems (Nevanlinna- Pick, Caratheodory, Schur, Humburger, Krein) as particular cases. We write down the abstract form of the Potapov inequality. By solving this inequality we give the description of the set of solutions of the general interpolation problem in the terms of the linear-fractional transformation. Then we apply the obtained general results to a number of classical and new interpolation problems. Some chapters of the book are dedicated to the application of the interpola- tion theory results to several other problems (the extension problem, generalized stationary processes, spectral theory, nonlinear integrable equations, functions with operator arguments). 2. Now we shall proceed to a more detailed description of the book contents.
Table of Contents
Introduction.
1. Operator Identities and Interpolation Problems.
2. Interpolation Problems in the Unite Circle.
3. Hermitian-Positive Functions of Several Variables.
4. De Branges Spaces of Entire Functions.
5. Degenerate Problems (Matrix Case).
6. Concrete Interpolation Problems.
7. Extremal Problems.
8. Spectral Problems for Canonical Systems of Difference Equations.
9. Integrable Nonlinear Equations (Discrete Case).
10. On Semi-Infinite Toda Chain.
11. Functions with an Operator Argument. Commentaries and Remarks. Bibliography. Index.