# The Schur Algorithm, Reproducing Kernel Spaces and System Theory

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*Daniel Alpay*

A co-publication of the AMS and Société Mathématique de France

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.

#### Readership

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

#### Reviews & Endorsements

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