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Orthogonal Rational Functions by Adhemar Bultheel
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This book generalises the classical theory of orthogonal polynomials on the complex unit circle, or on the real line to orthogonal rational functions whose poles are among a prescribed set of complex numbers. The first part treats the case where these poles are all outside the unit disk or in the lower half plane. Classical topics such as recurrence relations, numerical quadrature, interpolation properties, Favard theorems, convergence, asymptotics, and moment problems are generalised and treated in detail. The same topics are discussed for the different situation where the poles are located on the unit circle or on the extended real line. In the last chapter, several applications are mentioned including linear prediction, Pisarenko modelling, lossless inverse scattering, and network synthesis. This theory has many applications in theoretical real and complex analysis, approximation theory, numerical analysis, system theory, and in electrical engineering.
Cambridge University Press; February 1999
423 pages; ISBN 9780511836534
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Title: Orthogonal Rational Functions
Author: Adhemar Bultheel; Pablo Gonzalez-Vera; Erik Hendriksen; Olav Njastad