On the Loewner problem in the class |
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Authors: | D Alpay A Dijksma H Langer |
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Institution: | Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel ; Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands ; Department of Mathematics, Technical University Vienna, Wiedner Hauptstrasse 8--10, A--1040 Vienna, Austria |
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Abstract: | Loewner's theorem on boundary interpolation of functions is proved under rather general conditions. In particular, the hypothesis of Alpay and Rovnyak (1999) that the function , which is to be extended to an 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 is defined contains an accumulation point at which satisfies some kind of differentiability condition. The proof of the theorem in this note uses the representation of functions in terms of selfadjoint relations in Pontryagin spaces and the extension theory of symmetric relations in Pontryagin spaces. |
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