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The Schur algorithm and coefficient characterizations for generalized Schur functions


Authors: Tiberiu Constantinescu and Aurelian Gheondea
Journal: Proc. Amer. Math. Soc. 128 (2000), 2705-2713
MSC (1991): Primary 30C50, 47B50, 30E05
DOI: https://doi.org/10.1090/S0002-9939-00-05375-2
Published electronically: February 28, 2000
MathSciNet review: 1670430
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Abstract:

In this paper we analyze the existence of a Schur algorithm and obtain coefficient characterizations for the functions in a generalized Schur class. An application to an interpolation problem of Carathéodory type raised by M.G. Krein and H. Langer is indicated.


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  • 1. V.M. ADAMJAN, D.Z. AROV, M.G. KREIN: Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem, Mat. Sb. (N.S), 86(128)(1971), 34-75; English transl.: Math USSR-Sb., 15(1971), 31-73. MR 45:7505
  • 2. D. ALPAY, A. DIJKSMA, J. ROVNYAK, H.S.V. DE SNOO: Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces, Birkhäuser, Basel-Boston-Berlin, 1996. CMP 97:17
  • 3. T.YA. AZIZOV, I.S. IOKHVIDOV: Linear Operators in Spaces with an Indefinite Metric, Wiley&Sons, 1989. MR 90j:47042
  • 4. J.A. BALL, J.W. HELTON: A Beurling-Lax theorem for the Lie group $U(m,n)$ which contains most interpolation theory, J. Operator Theory, 9(1983), 107-142. MR 84m:47046
  • 5. G. CHRISTNER, J. ROVNYAK: Julia operators and the Schur algorithm, in Harmonic Analysis and Operator Theory, Contemporary Mathematics, Vol. 189, 1995, pp. 135-160. MR 96h:47022
  • 6. T. CONSTANTINESCU, A. GHEONDEA: Minimal signature in lifting of operators.I, J. Operator Theory, 22(1989), 345-367; II, J. Functional Analysis, 103(1992), 317-351. MR 93c:47041
  • 7. T. CONSTANTINESCU, A. GHEONDEA: Kolmogorov decompositions and the realization of time dependent systems, preprint 1997.
  • 8. I.S. IOHVIDOV, M.G. KREIN : Spectral theory of operators in indefinite metric.II, Trud. Mosk. Mat. Obshch., 9(1959), 413-496; English transl., Amer. Math. Soc. Transl.(2), 34(1963), 283-373. MR 21:6543
  • 9. M.G. KREIN, H. LANGER: Über die verallgemeinerten Resolventen und die characteristische Funktion eines isometrischen Operators im Raume $\Pi _{\kappa }$, in Hilbert Space Operators and Operator Algebras (Proc. Internat. Conf. Tihany, 1970), Colloq. Math. Soc. Janos Bolyai, Vol.5, Norht Holland, Amsterdam, 1972. MR 54:11103
  • 10. M.G. KREIN, H. LANGER: Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume $\Pi _{\kappa }$ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr., 77(1977), 187-236; II. Verallgemeinerte Resolventen, $u$-Resolventen und ganze Operatoren, J. Functional Analysis, 30(1978), 390-447. MR 57:1173
  • 11. M.G. KREIN, A.A. NUDELMAN: The Markov Moment Problem and Extremal Problems, Amer. Math. Soc. Transl. Math. Monographs, Providence, RI, 1977. MR 56:16284

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Additional Information

Tiberiu Constantinescu
Affiliation: Department of Mathematics, University of Texas at Dallas, Richardson, Texas 75083-0688
Email: tiberiu@utdallas.edu

Aurelian Gheondea
Affiliation: Institutul de Matematică al Academiei Române, CP 1-764, 70700 Bucureşti, România
Email: gheondea@imar.ro

DOI: https://doi.org/10.1090/S0002-9939-00-05375-2
Keywords: Schur functions, Schur algorithm, coefficient characterization, Carath\'eodory problem, Kre\u\i n space
Received by editor(s): March 30, 1998
Received by editor(s) in revised form: October 29, 1998
Published electronically: February 28, 2000
Additional Notes: The second author’s research was partially supported by the Ministry of Research and Technology of Romania grant 4022GR/1998.
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

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