Distributional representations of Nκ(∞)‐functions. Issue 10 (12th February 2015) - Langer, Matthias; Woracek, Harald
<x xml:space="preserve">Abstract</x> The subclasses <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj1d0wjdc5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Nκ(∞)</alternatives> of the classes <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj1d0wjd49" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Nκ</alternatives> of generalized Nevanlinna functions appear in the context of Pontryagin space models, where they correspond to model relations having a particular spectral behaviour. Applications are found, for instance, in the investigation of differential expressions with singular coefficients. We study representations of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj1d0wjd3r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Nκ(∞)</alternatives>‐functions as Cauchy‐type integrals in a distributional sense and characterize the class of distributions occurring in such representations. We make explicit how the Pontryagin space model of an <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj1d0wjd26" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Nκ(∞)</alternatives>‐function is related to the multiplication operator in the L2‐space of the measure which describes the action of the representing distribution away from infinity. Moreover, we determine the distributional representations of a pair of functions associated with a symmetric generalized Nevanlinna function.