Orthogonal Rational Functions

by Adhemar Bultheel, Pablo Gonzalez-Vera, Erik Hendriksen, Olav Njastad

Overview

This volume generalizes 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 generalized and treated in detail. The same topics are discussed for a 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 modeling, lossless inverse scattering, and network synthesis. This theory has many applications both in theoretical real and complex analysis, approximation theory, numerical analysis, system theory, and electrical engineering.

Product Details

ISBN-13:	9780521115919
Publisher:	Cambridge University Press
Publication date:	07/23/2009
Series:	Cambridge Monographs on Applied and Computational Mathematics , #5
Pages:	424
Product dimensions:	6.00(w) x 8.90(h) x 1.20(d)

Table of Contents

List of symbols; Introduction; 1. Preliminaries; 2. The fundamental spaces; 3. The kernel functions; 4. Recurrence and second kind functions; 5. Para-orthogonality and quadrature; 6. Interpolation; 7. Density of the rational functions; 8. Favard theorems; 9. Convergence; 10. Moment problems; 11. The boundary case; 12. Some applications; Conclusion; Bibliography; Index.

Editorial Reviews

"...emphasizes new developments in contemporary mathematical and computational sciences, and it is intended that books in it should serve to inform a new generation of research students and professionals alike." Mathematical Reviews

"The text is written with great clarity and the order in which the material is presented is well designed...combined very successfully in this joint venture." Proceedings of the Edinburgh Mathematical Society

"This is the first book to discuss the rational orthogonal functions in such a great detail...The well-written, cohesive presentation should be of great assistance to new and experienced researchers, both mathematicians and physicists, in the area of rational orthogonal functions." Siam Review

"...an excellent book which is indispensable for anyone who wants to loosen the shackles of the fixed pole situation present in the ordinary theory on the real line and the unit circle. To top that, the price makes it possible for everyone working in the field to buy a personal copy." Mathematics of Computation

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