# On Boundary Interpolation for Matrix Valued Schur Functions

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*Vladimir Bolotnikov; Harry Dym*

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

# Table of Contents

## On Boundary Interpolation for Matrix Valued Schur Functions

- Contents v6 free
- 1. Introduction 18 free
- 2. Preliminaries 714 free
- 3. Fundamental matrix inequalities 1219
- 4. On H(θ) spaces 1825
- 5. Parametrizations of all solutions 2330
- 6. The equality case 2936
- 7. Nontangential limits 3239
- 8. The Nevanlinna–Pick boundary problem 4148
- 9. A multiple analogue of the Carathéodory–Julia theorem 4855
- 10. On the solvability of a Stein equation 6572
- 11. Positive definite solutions of the Stein equation 7683
- 12. A Carathéodory-Fejér boundary problem 8087
- 13. The full matrix Carathéodory-Fejér boundary problem 8794
- 14. The lossless inverse scattering problem 94101
- Bibliography 105112