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On a boundary analogue of the Carathéodory-Schur interpolation problem


Author: Vladimir Bolotnikov
Journal: Proc. Amer. Math. Soc. 136 (2008), 3121-3131
MSC (2000): Primary 47A57
Published electronically: April 22, 2008
MathSciNet review: 2407075
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Abstract: Characterization of Schur functions in terms of their Taylor coefficients is due to C. Carathéodory and I. Schur. We discuss the boundary analogue of this problem.


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  • 1. Joseph A. Ball, Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions, Integral Equations Operator Theory 6 (1983), no. 6, 804–840. MR 719106, 10.1007/BF01691925
  • 2. Joseph A. Ball, Israel Gohberg, and Leiba Rodman, Interpolation of rational matrix functions, Operator Theory: Advances and Applications, vol. 45, Birkhäuser Verlag, Basel, 1990. MR 1083145
  • 3. Joseph A. Ball and J. William Helton, Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: parametrization of the set of all solutions, Integral Equations Operator Theory 9 (1986), no. 2, 155–203. MR 837514, 10.1007/BF01195006
  • 4. Vladimir Bolotnikov and Harry Dym, On boundary interpolation for matrix valued Schur functions, Mem. Amer. Math. Soc. 181 (2006), no. 856, vi+107. MR 2214130, 10.1090/memo/0856
  • 5. V. Bolotnikov and A. Kheifets, The Carathéodory-Julia theorem and related boundary interpolation problems, Recent Advances in Matrix and Operator Theory (J. A. Ball, Y. Eidelman, J. W. Helton, V. Olshevsky, and J. Rovnyak, eds.), Operator Theory: Advances and Applications 179, pp. 63-102, Birkhäuser-Verlag, Basel, 2007.
  • 6. C. Carathéodory,
    Über die Winkelderivierten von beschränkten Funktionen, Sitzungber. Preuss. Akad. Wiss. (1929), 39-52.
  • 7. G. Julia, Extension d'un lemma de Schwarz, Acta Math. 42 (1920), 349-355.
  • 8. C. Foias, A. E. Frazho, I. Gohberg, and M. A. Kaashoek, Metric constrained interpolation, commutant lifting and systems, Operator Theory: Advances and Applications, vol. 100, Birkhäuser Verlag, Basel, 1998. MR 1635831
  • 9. Dušan R. Georgijević, Solvability condition for a boundary value interpolation problem of Loewner type, J. Anal. Math. 74 (1998), 213–234. MR 1631662, 10.1007/BF02819451
  • 10. I. Gohberg, M. A. Kaashoek, and A. L. Sakhnovich, Taylor coefficients of a pseudo-exponential potential and the reflection coefficient of the corresponding canonical system, Math. Nachr. 278 (2005), no. 12-13, 1579–1590. MR 2169701, 10.1002/mana.200310323
  • 11. Pramod P. Khargonekar and Allen Tannenbaum, Non-Euclidean metrics and the robust stabilization of systems with parameter uncertainty, IEEE Trans. Automat. Control 30 (1985), no. 10, 1005–1013. MR 804138, 10.1109/TAC.1985.1103805
  • 12. I. V. Kovalishina, A multiple boundary value interpolation problem for contracting matrix functions in the unit disk, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 51 (1989), 38–55 (Russian); English transl., J. Soviet Math. 52 (1990), no. 6, 3467–3481. MR 1009144, 10.1007/BF01095405
  • 13. Donald Sarason, Nevanlinna-Pick interpolation with boundary data, Integral Equations Operator Theory 30 (1998), no. 2, 231–250. Dedicated to the memory of Mark Grigorievich Krein (1907–1989). MR 1607902, 10.1007/BF01238220
  • 14. I. Schur, Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, J. Reine Angew. Math. 147 (1917), 205-232.

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Additional Information

Vladimir Bolotnikov
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09059-X
Keywords: Schur functions, boundary interpolation problem.
Received by editor(s): December 27, 2005
Received by editor(s) in revised form: December 29, 2006
Published electronically: April 22, 2008
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.