On the Loewner problem in the class N_κ

Loewner's theorem on boundary interpolation of N_κ 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 N_κ 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 N_κ functions in terms of selfadjoint relations in Pontryagin spaces and the extension theory of symmetric relations in Pontryagin spaces.