Alpay / Gohberg Interpolation, Schur Functions and Moment Problems

2006
ISBN: 978-3-7643-7547-8
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 304 Seiten

Reihe: Linear Operators and Linear Systems

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.

Autoren/Hrsg.

Alpay, Daniel

Gohberg, Israel

Inhaltsverzeichnis

1;Contents;5
2;Editorial Introduction;9
2.1;References;11
3;Basic Boundary Interpolation for Generalized Schur Functions and Factorization of Rational J-unitary Matrix Functions;12
3.1;1. Introduction;12
3.2;2. Auxiliary statements;17
3.3;3. The basic interpolation problem at one boundary point;22
3.4;4. Multipoint boundary interpolation;28
3.5;5. J-unitary factorization;31
3.6;6. A factorization algorithm;34
3.7;References;38
4;Discrete Analogs of Canonical Systems with Pseudo-exponential Potential. Inverse Problems;41
4.1;1. Introduction;41
4.2;2. Preliminaries;44
4.3;3. Inverse scattering problem;54
4.4;4. Other inverse problems;57
4.5;5. Inverse problem associated to the asymptotic equivalence matrix function;63
4.6;6. The case of two-sided .rst-order systems;64
4.7;7. A numerical example;66
4.8;8. An example of a non-strictly pseudo-exponential sequence;68
4.9;9. Jacobi matrices;69
4.10;References;73
5;Boundary Nevanlinna–Pick Interpolation Problems for Generalized Schur Functions;76
5.1;1. Introduction;76
5.2;2. Main results;81
5.3;3. Some preliminaries;86
5.4;4. Fundamental Matrix Inequality;93
5.5;5. Parameters and interpolation conditions;99
5.6;6. Negative squares of the function;113
5.7;7. The degenerate case;118
5.8;8. An example;124
5.9;References;127
6;A Truncated Matricial Moment Problem on a Finite Interval;129
6.1;0. Introduction and preliminaries;129
6.2;1. The moment problem;131
6.3;2. Main algebraic identities;136
6.4;3. From the moment problem to the system of fundamental matrix inequalities of Potapov-type;137
6.5;4. From the system of fundamental matrix inequalities to the moment problem;144
6.6;5. Nonnegative column pairs;151
6.7;6. Description of the solution set in the positive de.nite case;155
6.8;7. A necessary and su.cient condition for the existence of a solution of the moment problem;168
6.9;8. Appendix: Certain subclasses of holomorphic matrix-valued functions and a generalization of Stieltjes’ inversion formula;169
6.10;References;178
7;Shift Operators Contained in Contractions, Schur Parameters and Pseudocontinuable Schur Functions;182
7.1;0. Introduction;182
7.2;1. Shifts contained in contractions, unitary colligations and characteristic operator functions;185
7.3;2. Construction of a model of a unitary colligation via the Schur parameters of its c.o.f. in the scalar case;192
7.4;3. A model representation of the maximal shift VT contained in a contraction T;214
7.5;4. The connection of the maximal shifts VT and VT. with thepseudocontinuability of the corresponding c.o.f. .;227
7.6;5. Some criteria for the pseudocontinuability of a Schur function in terms of its Schur parameters;232
7.7;References;255
8;The Matricial Carath´ eodory Problem in Both Nondegenerate and Degenerate Cases;258
8.1;0. Introduction;258
8.2;1. Preliminaries;260
8.3;2. On particular matrix polynomials;264
8.4;3. Description of the set;270
8.5;4. Resolvent matrices which are constructed recursively;279
8.6;5. The nondegenerate case;286
8.7;6. The case of a unique solution;290
8.8;References;295
9;A Gohberg-Heinig Type Inversion Formula Involving Hankel Operators;298
9.1;0. Introduction;298
9.2;1. The indicator;300
9.3;2. The main theorem for kernel functions of stable exponential type;302
9.4;3. Proof of the main theorem (general case);306
9.5;References;308
10;Linear Operators and Linear Systems;310