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

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.