## Algorithme de Schur, Espaces a Noyau Reproduisant Et Théorie Des SystemesThe 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 |

### Common terms and phrases

adjoint analogue analytic functions associated reproducing kernel Blaschke Chapter characterization Cnxn condition consider corresponds Cpxp decomposition definition denote elements equal equation equivalent example exists fact factorization fibre bundles finite dimensional finite number following result following theorem formula framework functions defined Hankel Hankel matrix Hardy space Hermitian Hilbert module Hilbert space inverse scattering problem isometric J-contractive J0-inner J0-unitary kernel Hilbert space Krein spaces Leech's theorem LEMMA mapping matrix meromorphic Moreover negative squares non-stationary notion operator models operator of multiplication Operator Theory orthogonal pair polynomials Pontryagin spaces positive functions proof PROPOSITION rational realization recall refer the reader reproducing kernel Hilbert reproducing kernel space satisfies scalar product Schur algorithm Schur function Section sequence set of functions solutions space H space with kernel Spxq stationary strictly positive subspace system theory tion Toeplitz Toeplitz matrix transfer function transformation unique unit circle unit disk unitary operators upper triangular operators vector

### Popular passages

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.