Algorithme de Schur, Espaces a Noyau Reproduisant Et Théorie Des Systemes

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American Mathematical Soc., 2001 - Mathematics - 150 pages
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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 Societe Mathematique de France. This title was translated by Stephen S. Wilson.
 

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Contents

Introduction
1
Reproducing kernel spaces
13
Theory of linear systems
47
Schur algorithm and inverse scattering problem
65
Operator models
83
Interpolation
91
The indefinite case
99
The nonstationary case
107
Riemann surfaces
121
Conclusion
133
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Page 140 - Schur Recursions, Error Formulas and Convergence of Rational Estimators for Stationary Stochastic Sequences", IEEE Trans, on Info. Th., Vol. IT-27, No. 4, July 1981. 4. P. Dewilde and H. Dym, "Lossless Chain Scattering Matrices and Optimum Linear Prediction: the Vector Case", Circuit Theory and Applications, Vol.

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