On the Loewner problem in the class 𝐍_{𝜅}

Alpay, D.; Dijksma, A.; Langer, H.

doi:10.1090/S0002-9939-01-06345-6

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
DOI: https://doi.org/10.1090/S0002-9939-01-06345-6
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 , 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 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 $\mathbf{N}_{\kappa}$ functions in terms of selfadjoint relations in Pontryagin spaces and the extension theory of symmetric relations in Pontryagin spaces.

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