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.
Author(s): Adhemar Bultheel, Pablo Gonzalez-Vera, Erik Hendriksen, Olav Njastad
Series: Cambridge Monographs on Applied and Computational Mathematics
Edition: 1
Publisher: Cambridge University Press
Year: 2009
Language: English
Pages: 422
Tags: Математика;Комплексное исчисление;
Cover......Page 1
About......Page 2
CAMBRIDGE MONOGRAPHS ON APPLIED AND COMPUTATIONAL MATHEMATICS 5......Page 4
Orthogonal Rational Functions......Page 6
Copyright - ISBN: 9780521650069......Page 7
Contents......Page 8
List of symbols......Page 12
Introduction......Page 16
1.1 Hardy classes......Page 30
1.2 The classes C and B......Page 38
1.3 Factorizations......Page 46
1.4 Reproducing kernel spaces......Page 49
1.5 J-unitary and J-contractive matrices......Page 51
2.1 The spaces L_n......Page 57
2.2 Calculus in L_n......Page 68
2.3 Extremal problems in L_n......Page 73
3.1 Christoffel-Darboux relations......Page 79
3.2 Recurrence relations for the kernels......Page 82
3.3 Normalized recursions for the kernels......Page 85
4.1 Recurrence for the orthogonal functions......Page 89
4.2 Functions of the second kind......Page 97
4.3 General solutions......Page 105
4.4 Continued fractions and three-term recurrence......Page 110
4.5 Points not on the boundary......Page 116
5.1 Interpolatory quadrature......Page 121
5.2 Para-orthogonal functions......Page 123
5.3 Quadrature......Page 127
5.4 The weights......Page 132
5.5 An alternative approach......Page 134
6.1 Interpolation properties for orthogonal functions......Page 136
6.2 Measures and interpolation......Page 144
6.3 Interpolation properties for the kernels......Page 150
6.4 The interpolation algorithm of Nevanlinna-Pick......Page 155
6.5 Interpolation algorithm for the orthonormal functions......Page 160
7.1 Density in L_p and H_p......Page 164
7.2 Density in L_2(μ) and H_2(μ)......Page 170
8.1 Orthogonal functions......Page 176
8.2 Kernels......Page 180
9 Convergence......Page 188
9.1 Generalization of the Szego problem......Page 189
9.2 Further convergence results and asymptotic behavior......Page 196
9.3 Convergence of φ_n^*......Page 198
9.4 Equivalence of conditions......Page 206
9.5 Varying measures......Page 207
9.6 Stronger results......Page 211
9.7 Weak convergence......Page 221
9.8 Erdos-Turan class and ratio asymptotics......Page 223
9.9 Root asymptotics......Page 241
9.10 Rates of convergence......Page 248
10.1 Motivation and formulation of the problem......Page 254
10.2 Nested disks......Page 256
10.3 The moment problem......Page 266
11.1Recurrence for points on the boundary......Page 272
11.2 Functions of the second kind......Page 282
11.3 Christoffel-Darboux relation......Page 287
11.4 Green's formula......Page 292
11.5 Quasi-orthogonal functions......Page 295
11.6 Quadrature formulas......Page 301
11.7 Nested disks......Page 305
11.8 Moment problem......Page 315
11.9 Favarcl type theorem......Page 322
11.10 Interpolation......Page 334
11.11 Convergence......Page 353
12 Some applications......Page 357
12.1 Linear prediction......Page 358
12.2 Pisarenko modeling problem......Page 371
12.3 Lossless inverse scattering......Page 374
12.4 Network synthesis......Page 384
12.5.1 The standard H^ control problem......Page 388
12.5.2 Hankel operators......Page 394
12.5.3 Hankel norm approximation......Page 400
Conclusion......Page 404
Bibliography......Page 408
Index......Page 420
