Interpolation, Schur Functions and Moment Problems (eBook)

Daniel Alpay, Israel Gohberg (Herausgeber)

eBook Download: PDF

2006
XI, 304 Seiten
Springer Basel (Verlag)
978-3-7643-7547-8 (ISBN)

Infos

Schur analysis originated with an 1917 article which associated to a function, which is analytic and contractive in the open unit disk, a sequence, finite or infinite, of numbers in the open unit disk, called Schur coefficients, often named reflection coefficients in signal processing. This volume comprises seven essays dedicated to the analysis of Schur and Carathéodory functions and to the solutions of problems for these classes.

Inhalt

Contents 5
Editorial Introduction 9
References 11
Basic Boundary Interpolation for Generalized Schur Functions and Factorization of Rational J-unitary Matrix Functions 12
1. Introduction 12
2. Auxiliary statements 17
3. The basic interpolation problem at one boundary point 22
4. Multipoint boundary interpolation 28
5. J-unitary factorization 31
6. A factorization algorithm 34
References 38
Discrete Analogs of Canonical Systems with Pseudo-exponential Potential. Inverse Problems 41
1. Introduction 41
2. Preliminaries 44
3. Inverse scattering problem 54
4. Other inverse problems 57
5. Inverse problem associated to the asymptotic equivalence matrix function 63
6. The case of two-sided .rst-order systems 64
7. A numerical example 66
8. An example of a non-strictly pseudo-exponential sequence 68
9. Jacobi matrices 69
References 73
Boundary Nevanlinna–Pick Interpolation Problems for Generalized Schur Functions 76
1. Introduction 76
2. Main results 81
3. Some preliminaries 86
4. Fundamental Matrix Inequality 93
5. Parameters and interpolation conditions 99
6. Negative squares of the function 113
7. The degenerate case 118
8. An example 124
References 127
A Truncated Matricial Moment Problem on a Finite Interval 129
0. Introduction and preliminaries 129
1. The moment problem 131
2. Main algebraic identities 136
3. From the moment problem to the system of fundamental matrix inequalities of Potapov-type 137
4. From the system of fundamental matrix inequalities to the moment problem 144
5. Nonnegative column pairs 151
6. Description of the solution set in the positive de.nite case 155
7. A necessary and su.cient condition for the existence of a solution of the moment problem 168
8. Appendix: Certain subclasses of holomorphic matrix-valued functions and a generalization of Stieltjes’ inversion formula 169
References 178
Shift Operators Contained in Contractions, Schur Parameters and Pseudocontinuable Schur Functions 182
0. Introduction 182
1. Shifts contained in contractions, unitary colligations and characteristic operator functions 185
2. Construction of a model of a unitary colligation via the Schur parameters of its c.o.f. in the scalar case 192
3. A model representation of the maximal shift VT contained in a contraction T 214
4. The connection of the maximal shifts VT and VT. with thepseudocontinuability of the corresponding c.o.f. . 227
5. Some criteria for the pseudocontinuability of a Schur function in terms of its Schur parameters 232
References 255
The Matricial Carath´ eodory Problem in Both Nondegenerate and Degenerate Cases 258
0. Introduction 258
1. Preliminaries 260
2. On particular matrix polynomials 264
3. Description of the set 270
4. Resolvent matrices which are constructed recursively 279
5. The nondegenerate case 286
6. The case of a unique solution 290
References 295
A Gohberg-Heinig Type Inversion Formula Involving Hankel Operators 298
0. Introduction 298
1. The indicator 300
2. The main theorem for kernel functions of stable exponential type 302
3. Proof of the main theorem (general case) 306
References 308
Linear Operators and Linear Systems 310

Produkt-Details

• Erscheint lt. Verlag: 22.6.2006
• Reihe/Serie: Linear Operators and Linear Systems
• Reihe/Serie: Linear Operators and Linear Systems
• Reihe/Serie: Operator Theory: Advances and Applications
• Reihe/Serie: Operator Theory: Advances and Applications
• Zusatzinfo: XI, 304 p.
• Verlagsort: Basel
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• Schlagworte: Factorization • Interpolation • Inverse Problems • Matrix • Moment problem • Shift operator
• ISBN-10: 3-7643-7547-7 / 3764375477
• ISBN-13: 978-3-7643-7547-8 / 9783764375478