Orthogonal Rational Functions

The book generalises the classical theory of orthogonal polynomials to rational functions.

Adhemar Bultheel (Author), Pablo Gonzalez-Vera (Author), Erik Hendriksen (Author), Olav Njastad (Author)

9780521115919, Cambridge University Press

Paperback, published 23 July 2009

424 pages, 18 b/w illus.
22.9 x 15.2 x 2.4 cm, 0.62 kg

'The text is written with great clarity … A book with four authors is not common, but these four … are to be applauded for their achievement.' J. H. McCabe

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.

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.

Subject Areas: Applied mathematics [PBW]