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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
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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$65 Individual Members: US$39
Institutional Members: US\$52
Order Code: MEMO/181/856

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.

• 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