Orthogonal Rational Functions ( Cambridge Monographs on Applied and Computational Mathematics )

Publication series ：Cambridge Monographs on Applied and Computational Mathematics

Author： Adhemar Bultheel; Pablo Gonzalez-Vera; Erik Hendriksen

Publisher： Cambridge University Press‎

Publication year： 1999

E-ISBN: 9780511836534

P-ISBN(Paperback): 9780521650069

Subject： O174.21 orthogonal series (Fourier series)

Keyword：数值分析

Language： ENG

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Orthogonal Rational Functions

Description

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.

Chapter

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 Ln

2.2 Calculus in Ln

2.3 Extremal problems in Ln

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 Lp and Hp

7.2 Density in L2(µ) and H2(µ)

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

9.3 Convergence of φn*

9.4 Equivalence of conditions

9.5 Varying measures

9.6 Stronger results

9.7 Weak convergence

9.8 Erdos-Tuárn 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

11.1 Recurrence 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 Favard 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∞ problems

12.5.1 The standard H∞ control problem

12.5.2 Hankel operators

12.5.3 Hankel norm approximation

Conclusion

Bibliography

Index

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