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Dutch Mathematical
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Acta Applicandae

Book review

Author(s) Bakonyi, M.
Constantinescu, T.
Title Schur's algorithm and several applications
Publisher Longman Scientific & Technical
Year of publication 1992
Reviewed by Mirela Stefanescu

The authors develop some ideas coming from the Schur's work Uber Potenzreihen, die im Innern des Einheitkreises Beschrankt sind, published in 1917. The Schur's algorithm (initially viewed as a continued fraction algorithm) is an example of an elementary result whose applications cover a large area having deep implications in interpolation and dilation theory, orthogonal polynomials and electrical engineering.

A comprehensive description of the geometric approach is based on commutant lifting theorem (Sz. Nagy and C. Foias) and this was applied to interpolation for rational matrix functions by Joseph Ball, Israel Gohberg, Leibe Rodman. Many other developments are based on the shift-invariant subspace approach (J. Ball and J. Helton); their key tool is the celebrated Beurling-Lax theorem. An algebraic approach permits applications to entropy.

The authors point out the genericity of the transmission line model, a physical model providing a unified view on a variety of problems in such fields as spectral factorizations and prediction of stationary second order stochastic processes, realization theory for linear time invariant systems, interpolation problems.

The graduate students, researchers in mathematics, engineers can find the book useful for their investigations.