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On Boundary Interpolation for Matrix Valued Schur Functions
Vladimir Bolotnikov, The College of William and Mary, Williamsburg, VA, and Harry Dym, Weizmann Institute of Science, Rehovot, Israel

Memoirs of the American Mathematical Society
2006; 107 pp; softcover
Volume: 181
ISBN-10: 0-8218-4047-9
ISBN-13: 978-0-8218-4047-4
List Price: US$61
Individual Members: US$36.60
Institutional Members: US$48.80
Order Code: MEMO/181/856
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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.

Table of Contents

  • Introduction
  • Preliminaries
  • Fundamental matrix inequalities
  • On \(\mathcal{H}(\Theta)\) spaces
  • Parametrizations of all solutions
  • The equality case
  • Nontangential limits
  • The Nevanlinna-Pick boundary problem
  • A multiple analogue of the Carathéodory-Julia theorem
  • On the solvability of a Stein equation
  • Positive definite solutions of the Stein equation
  • A Carathéodory-Fejér boundary problem
  • The full matrix Carathéodory-Fejér boundary problem
  • The lossless inverse scattering problem
  • Bibliography
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