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On the Loewner problem in the class $\mathbf{N}_{\kappa}$

Authors: D. Alpay, A. Dijksma and H. Langer
Journal: Proc. Amer. Math. Soc. 130 (2002), 2057-2066
MSC (2000): Primary 30E05, 47A57; Secondary 47B25, 47B50
Published electronically: December 31, 2001
MathSciNet review: 1896042
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Abstract: Loewner's theorem on boundary interpolation of $\mathbf{N}_{\kappa}$ functions is proved under rather general conditions. In particular, the hypothesis of Alpay and Rovnyak (1999) that the function $f$, which is to be extended to an $\mathbf{N}_{\kappa}$function, is defined and continuously differentiable on a nonempty open subset of the real line, is replaced by the hypothesis that the set on which $f$ is defined contains an accumulation point at which $f$satisfies some kind of differentiability condition. The proof of the theorem in this note uses the representation of $\mathbf{N}_{\kappa}$ functions in terms of selfadjoint relations in Pontryagin spaces and the extension theory of symmetric relations in Pontryagin spaces.

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Additional Information

D. Alpay
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel

A. Dijksma
Affiliation: Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands

H. Langer
Affiliation: Department of Mathematics, Technical University Vienna, Wiedner Hauptstrasse 8–10, A–1040 Vienna, Austria

Received by editor(s): October 26, 2000
Received by editor(s) in revised form: February 9, 2001
Published electronically: December 31, 2001
Additional Notes: The second and third authors gratefully acknowledge support through Harry T. Dozor fellowships at the Ben-Gurion University of the Negev, Beer-Sheva, Israel, in the years 1999 and 2000 respectively, and support from NWO, the Netherlands Organization for Scientific Research (grant B 61-453).
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2001 American Mathematical Society