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A uniqueness result on boundary interpolation

Author: Vladimir Bolotnikov
Journal: Proc. Amer. Math. Soc. 136 (2008), 1705-1715
MSC (2000): Primary 47A57
Published electronically: November 28, 2007
MathSciNet review: 2373600
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Abstract: Let $ f$ be an analytic function mapping the unit disk $ \mathbb{D}$ into itself. We give necessary and sufficient conditions on the local behavior of $ f$ near a finite set of boundary points that require $ f$ to be a finite Blaschke product.

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  • 1. J. A. Ball,
    Interpolation problems of Pick-Nevanlinna and Loewner type for meromorphic matrix functions,
    Integral Equations Operator Theory 6 (1983), 804-840. MR 719106 (85k:30054)
  • 2. J.A. Ball, I. Gohberg and L. Rodman, Interpolation of Rational Matrix Functions, OT45, Birkhäuser-Verlag, Basel-Boston, 1990. MR 1083145 (92m:47027)
  • 3. J. A. Ball and J. W. 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), 155-203. MR 837514 (87j:30085)
  • 4. V. Bolotnikov and H. Dym, On boundary interpolation for matrix valued Schur functions, Mem. Amer. Math. Soc. 181 (2006), no. 856. MR 2214130 (2007g:47022)
  • 5. V. Bolotnikov and A. Kheifets, A higher order analogue of the Carathéodory-Julia theorem, J. Funct. Anal. 237 (2006), no. 1, 350-371. MR 2239269 (2007d:46023)
  • 6. V. Bolotnikov and A. Kheifets, The higher order Carathéodory-Julia theorem and related boundary interpolation problems, Oper. Theory: Adv. Appl. OT 179, pages 63-102, Birkhäuser-Verlag, Basel, 2007.
  • 7. V. Bolotnikov and A. Kheifets, Boundary Nevanlinna-Pick interpolation problems for generalized Schur functions, in Interpolation, Schur Functions and Moment Problems (D. Alpay and I. Gohberg, Eds.), Oper. Theory: Adv. Appl. OT 165, pages 67-119, Birkhäuser-Verlag, Basel, 2006. MR 2222520 (2007d:30027)
  • 8. V. Bolotnikov and L. Rodman, Krein-Langer factorizations via pole triples, Integral Equations Operator Theory 47 (2003), no. 2, 169-195. MR 2002664 (2004h:47021)
  • 9. L. de Branges and J. Rovnyak, Square summable power series, Holt, Rinehart and Winston, New York, 1966. MR 0215065 (35:5909)
  • 10. D. Burns and S. G. Krantz, Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary, J. Amer. Math. Soc. 7 (1994), no. 3, 661-676. MR 1242454 (94j:32016)
  • 11. D. Chelst, A generalized Schwarz lemma at the boundary, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3275-3278. MR 1845002 (2002c:30033)
  • 12. C. Carathéodory,
    Über die Winkelderivierten von beschränkten Funktionen, Sitzungber. Preuss. Akad. Wiss. (1929), 39-52.
  • 13. I. V. Kovalishina, Multiple boundary interpolation problem for contractive matrix-valued functions in the unit circle, J. Soviet Math. 52 (6), (1990), 3467-3481. MR 1009144 (90i:41007)
  • 14. G. Pick, Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1916), no. 1, 7-23. MR 1511844
  • 15. D. Sarason,
    Nevanlinna-Pick interpolation with boundary data,
    Integral Equations Operator Theory 30 (1998), no. 2, 231-250. MR 1607902 (98m:47017)
  • 16. 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

Received by editor(s): January 10, 2006
Received by editor(s) in revised form: February 20, 2007
Published electronically: November 28, 2007
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2007 American Mathematical Society

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