eBook ISBN:  9781470404604 
Product Code:  MEMO/181/856.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $39.00 
eBook ISBN:  9781470404604 
Product Code:  MEMO/181/856.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $39.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 181; 2006; 107 ppMSC: Primary 30; 47;
A number of interpolation problems are considered in the Schur class of \(p\times q\) matrix valued functions \(S\) that are analytic and contractive in the open unit disk. The interpolation constraints are specified in terms of nontangential limits and angular derivatives at one or more (of a finite number of) boundary points. Necessary and sufficient conditions for existence of solutions to these problems and a description of all the solutions when these conditions are met is given. The analysis makes extensive use of a class of reproducing kernel Hilbert spaces \({\mathcal{H}}(S)\) that was introduced by de Branges and Rovnyak. The Stein equation that is associated with the interpolation problems under consideration is analyzed in detail. A lossless inverse scattering problem is also considered.

Table of Contents

Chapters

1. Introduction

2. Preliminaries

3. Fundamental matrix inequalities

4. On $\mathcal {H}(\Theta )$ spaces

5. Parametrizations of all solutions

6. The equality case

7. Nontangential limits

8. The Nevanlinna–Pick boundary problem

9. A multiple analogue of the Carathéodory–Julia theorem

10. On the solvability of a Stein equation

11. Positive definite solutions of the Stein equation

12. A CarathéodoryFejér boundary problem

13. The full matrix CarathéodoryFejér boundary problem

14. The lossless inverse scattering problem


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A number of interpolation problems are considered in the Schur class of \(p\times q\) matrix valued functions \(S\) that are analytic and contractive in the open unit disk. The interpolation constraints are specified in terms of nontangential limits and angular derivatives at one or more (of a finite number of) boundary points. Necessary and sufficient conditions for existence of solutions to these problems and a description of all the solutions when these conditions are met is given. The analysis makes extensive use of a class of reproducing kernel Hilbert spaces \({\mathcal{H}}(S)\) that was introduced by de Branges and Rovnyak. The Stein equation that is associated with the interpolation problems under consideration is analyzed in detail. A lossless inverse scattering problem is also considered.

Chapters

1. Introduction

2. Preliminaries

3. Fundamental matrix inequalities

4. On $\mathcal {H}(\Theta )$ spaces

5. Parametrizations of all solutions

6. The equality case

7. Nontangential limits

8. The Nevanlinna–Pick boundary problem

9. A multiple analogue of the Carathéodory–Julia theorem

10. On the solvability of a Stein equation

11. Positive definite solutions of the Stein equation

12. A CarathéodoryFejér boundary problem

13. The full matrix CarathéodoryFejér boundary problem

14. The lossless inverse scattering problem