On the Loewner problem in the class \(\mathbf{N}_{\kappa}\) (Q2782637)

The Nevanlinna class \(N_k\) is the class of functions \(f\) meromorphic in the upper half plane and for which the kernel \([f(z)-f(w)^*]/[z-w^*]\) has \(k\) negative squares. The Loewner problem in the class \(N_k\) is the following: given a real set \(A\), values \(f(s)\) and some other interpolation conditions characterized by the numbers \(\widehat{f}(s)\) for \(s\in A\), find conditions such that \(f\) can be extrapolated to a function in the class \(N_k\). The solvability conditions are basically that the kernel function has \(k\) negative squares. In [Proc. Am. Math. Soc. 127, No. 4, 1109-1117 (1999, Zbl 0921.30027)], this is proved when it is assumed that \(f\) is continuously differentiable in the open set \(A\) on which \(f\) is defined and that \(\widehat{f}\) is the derivative \(f'\). In this paper, another assumption is made, namely the set on which \(f\) is defined has an accumulation point \(t\) at which \(\lim [f(s)-f(t)]/[s-t]=\widehat{f}(t)\) where the limit is a limit for \(s\to t\), \(s\in A\). If \(A\) has no interior points, then the solution is unique. The method used in this paper is by reducing the problem to an extension problem of a symmetric linear relation in a Pontryagin space.