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The Schur Algorithm, Reproducing Kernel Spaces and System Theory
Daniel Alpay, Ben-Gurion University of the Negev, Beer-sheva, Israel
A co-publication of the AMS and Société Mathématique de France.

SMF/AMS Texts and Monographs
2001; 150 pp; softcover
Volume: 5
ISBN-10: 0-8218-2155-5
ISBN-13: 978-0-8218-2155-8
List Price: US$60
Member Price: US$48
Order Code: SMFAMS/5
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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 matrix-valued 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 matrix-valued 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 time-invariant 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, non-positive metrics, and pairs of commuting nonself-adjoint 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 co-published with Société Mathématique de France. SMF members are entitled to AMS member discounts.


Graduate students and pure and applied research mathematicians interested in functional analysis, systems theory, and control.


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 non-stationary case
  • Riemann surfaces
  • Conclusion
  • Bibliography
  • Index
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