The class of Schur functions consists of analytic functions on the unit disk that are bounded by \(1\). The Schur algorithm associates to any such function a sequence of complex constants, which is much more useful than the Taylor coefficients. There is a generalization to matrixvalued functions and a corresponding algorithm. These generalized Schur functions have important applications to the theory of linear operators, to signal processing and control theory, and to other areas of engineering. In this book, Alpay looks at matrixvalued Schur functions and their applications from the unifying point of view of spaces with reproducing kernels. This approach is used here to study the relationship between the modeling of timeinvariant dissipative linear systems and the theory of linear operators. The inverse scattering problem plays a key role in the exposition. The point of view also allows for a natural way to tackle more general cases, such as nonstationary systems, nonpositive metrics, and pairs of commuting nonselfadjoint operators. This is the English translation of a volume originally published in French by the Société Mathématique de France. Translated by Stephen S. Wilson. Titles in this series are copublished with Société Mathématique de France. SMF members are entitled to AMS member discounts. Readership Graduate students and pure and applied research mathematicians interested in functional analysis, systems theory, and control. Reviews From a review of the French edition: "This excellent survey showing a rich interplay between functional analysis, complex analysis and systems science is very informative and can be highly recommended to functional analysts curious about the systems science impact of their discipline or to theoretically inclined systems scientists, in particular those involved in the realization theory."  Zentralblatt MATH Table of Contents  Introduction
 Reproducing kernel spaces
 Theory of linear systems
 Schur algorithm and inverse scattering problem
 Operator models
 Interpolation
 The indefinite case
 The nonstationary case
 Riemann surfaces
 Conclusion
 Bibliography
 Index
