Ludvig Fagrell: Rational Herglotz–Nevanlinna functions of several variables

Abstract

A Herglotz–Nevanlinna function of several variables is a holomorphic function from the poly-upper half-plane to the closed upper half-plane. A recent development shows that a function of this type admits an integral representation determined by a set of constants and a particular positive Borel measure, known as a Nevanlinna measure. The appearance of the representation has sparked further research on Herglotz–Nevanlinna functions, often with emphasis on the class of Nevanlinna measures.

This thesis is concerned with the class of real-rational functions, which are the rational Herglotz–Nevanlinna functions whose boundary values on R^n are real. The theory on real-rational functions is rather sparse, and many of the results seem to lack formal proofs. This leads to the first of two aims of this thesis, which is to provide a more complete overview of real-rational functions of several variables than what currently exists in the literature. The results include, but are not limited to, a characterization of the real-rational functions in terms of the structure of the functions themselves, a support theorem for Nevanlinna measures representing real-rational functions, an invariance property of real-rational functions and their representing Nevanlinna measures, and an exact description of the relation between any real-rational function of two variables with an affine denominator (possibly after a biholomorphic change of variables) and its representing Nevanlinna measure.

The second aim of this thesis is to present a result on real-rational functions. The result and its context was communicated to the author by the thesis supervisor and, to the best of our knowledge, has not previously appeared in the literature. Specifically, it is shown that a real-rational function of two variables with a denominator that is a product of two affine factors has a nontrivial decomposition into a sum of two real-rational functions.