Orthogonal Rational Functions (Cambridge Monographs on Applied and Computational Mathematics, Series Number 5)

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

• Publisher: Cambridge University Press
• Year: 1999
• Tongue: English
• Leaves: 422
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover
About
CAMBRIDGE MONOGRAPHS ON APPLIED AND COMPUTATIONAL MATHEMATICS 5
Orthogonal Rational Functions
Copyright - ISBN: 9780521650069
Contents
List of symbols
Introduction
1 Preliminaries
1.1 Hardy classes
1.2 The classes C and B
1.3 Factorizations
1.4 Reproducing kernel spaces
1.5 J-unitary and J-contractive matrices
2 The fundamental spaces
2.1 The spaces L_n
2.2 Calculus in L_n
2.3 Extremal problems in L_n
3 The kernel functions
3.1 Christoffel-Darboux relations
3.2 Recurrence relations for the kernels
3.3 Normalized recursions for the kernels
4 Recurrence and second kind functions
4.1 Recurrence for the orthogonal functions
4.2 Functions of the second kind
4.3 General solutions
4.4 Continued fractions and three-term recurrence
4.5 Points not on the boundary
5 Para-orthogonality and quadrature
5.1 Interpolatory quadrature
5.2 Para-orthogonal functions
5.3 Quadrature
5.4 The weights
5.5 An alternative approach
6 Interpolation
6.1 Interpolation properties for orthogonal functions
6.2 Measures and interpolation
6.3 Interpolation properties for the kernels
6.4 The interpolation algorithm of Nevanlinna-Pick
6.5 Interpolation algorithm for the orthonormal functions
7 Density of the rational functions
7.1 Density in L_p and H_p

8 Favard theorems
8.1 Orthogonal functions
8.2 Kernels
9 Convergence
9.1 Generalization of the Szego problem
9.2 Further convergence results and asymptotic behavior

9.4 Equivalence of conditions
9.5 Varying measures
9.6 Stronger results
9.7 Weak convergence
9.8 Erdos-Turan class and ratio asymptotics
9.9 Root asymptotics
9.10 Rates of convergence
10 Moment problems
10.1 Motivation and formulation of the problem
10.2 Nested disks
10.3 The moment problem
11 The boundary case
11.1Recurrence for points on the boundary
11.2 Functions of the second kind
11.3 Christoffel-Darboux relation
11.4 Green's formula
11.5 Quasi-orthogonal functions
11.6 Quadrature formulas
11.7 Nested disks
11.8 Moment problem
11.9 Favarcl type theorem
11.10 Interpolation
11.11 Convergence
12 Some applications
12.1 Linear prediction
12.2 Pisarenko modeling problem
12.3 Lossless inverse scattering
12.4 Network synthesis
12.5 H_infty problems
12.5.1 The standard H^ control problem
12.5.2 Hankel operators
12.5.3 Hankel norm approximation
Conclusion
Bibliography
Index