This book is devoted to the ubiquity of the Schur parameters. A dilation theoretic view leads to a unified perspective on several topics where Schur parameters appear as basic cells. Together with the transmission line, their physical counter- part, they appear in scattering theory, in modeling, prediction and filtering of nonstationary processes, in signal processing, geophysics and system theory. Modeling problems are considered for certain classes of operators, interpolation problems, determinental formulae, as well as connections with certain classes of graphs where, again, the Schur parameters could play a role. Some general algorithms that explore the transmission line are also presented in this book. As a whole, the text is self-contained and it is addressed to people interested in the previously mentioned topics or connections between them.

Klappentext

The subject of this book is about the ubiquity of the Schur parameters, whose introduction goes back to a paper of I. Schur in 1917 concerning an interpolation problem of C. Caratheodory. What followed there appears to be a truly fascinating story which, however, should be told by a professional historian. Here we provide the reader with a simplified version, mostly related to the contents of the book. In the twenties, thf~ theory of orthogonal polynomials on the unit circle was developed by G. Szego and the formulae relating these polynomials involved num­ bers (usually called Szego parameters) similar to the Schur parameters. Mean­ while, R. Nevanlinna and G. Pick studied the theory of another interpolation problem, known since then as the Nevanlinna-Pick problem, and an algorithm similar to Schur's one was obtained by Nevanlinna. In 1957, Z. Nehari solved OO an L problem which contained both Caratheodory-Schur and Nevannlina-Pick problems as particular cases. Apparently unrelated work of H. Weyl, J. von Neu­ mann and K. Friedericks concerning selfadjoint extensions of symmetric operators was connected to interpolation by M. A. Naimark and M. G Krein using some gen­ eral dilation theoretic ideas. Classical moment problems, like the trigonometric moment and Hamburger moment problems, were also related to these topics and a comprehensive account of what can be called the classical period has appeared in the monograph of M. G. Krein and A. A. Nudelman, [KN].

Inhalt
1 Schur Parameters and Positive Block Matrices.- 1.1 Preliminaries.- 1.2 Renorming Hilbert Spaces and Elementary Rotations.- 1.3 Kolmogorov Decompositions. I.- 1.4 Row and Column Contractions.- 1.5 The Structure of Positive Definite Kernels.- 1.6 Kolmogorov Decompositions. II.- 1.7 Notes.- 2 Models for Triangular Contractions.- 2.1 Preliminaries.- 2.2 The Structure of Triangular Contractions.- 2.3 Realization of Triangular Contractions.- 2.4 Unitary Couplings and Operator Ranges.- 2.5 Modeling Families of Contractions.- 2.6 Notes.- 3 Moment Problems and Interpolation.- 3.1 A Survey on Completion Problems.- 3.2 Extensions of Partial Isometries.- 3.3 Krein's Formula.- 3.4 Moment Problems.- 3.5 The Commutant Lifting Method.- 3.6 Notes.- 4 Displacement Structures.- 4.1 Structured Matrices.- 4.2 Generalized Schur Algorithm.- 4.3 Discrete Transmission-Line Models.- 4.4 Displacement Structure and Completion Problems.- 4.5 Other Applications.- 4.6 Notes.- 5 Factorization of Positive Definite Kernels.- 5.1 Spectral Factors.- 5.2 Examples.- 5.3 Schur's Algorithm, Szegö's Theory and Spectral Factors.- 5.4 Maximum Entropy.- 5.5 Notes.- 6 Nonstationary Processes.- 6.1 Modeling Nonstationary Processes.- 6.2 Kolmogorov-Wiener Prediction.- 6.3 Other Prediction Problems.- 6.4 Szegö's Limit Theorems.- 6.5 Notes.- 7 Graphs and Completion Problems.- 7.1 Preliminaries.- 7.2 Completing Positive Partial Matrices. I.- 7.3 Completing Positive Partial Matrices. II.- 7.4 Completing Contractive Partial Matrices.- 7.5 Notes.- 8 Determinantal Formulae and Optimization.- 8.1 Determinantal Formulae.- 8.2 Maximum Determinant Formulae.- 8.3 Maximum Determinant for Nonchordal Graphs.- 8.4 Inheritance Principles.- 8.5 Notes.- References.