On Herglotz-Nevanlinna functions in several variables

In this thesis, we investigate different aspects of the class of Herglotz-Nevanlinna functions in several variables. These are holomorphic functions on the poly-upper half-plane having non-negative imaginary part. Our results are presented in the four research articles A1 - A4, which are included in this thesis. Articles A1 and A2 establish a characterization of Herglotz-Nevanlinna functions in terms of an integral representation formula. The case of functions of two complex variables is presented in article A1, while the general case is treated in article A2, where different symmetry properties of Herglotz-Nevanlinna functions are also discussed. Article A3 discusses, in detail, the convex combination problem for Herglotz-Nevanlinna functions. This problem asks us to relate the representing parameters of different Herglotz-Nevanlinna functions under the assumption that these functions are related in a very particular way involving the convex combination of several independent variables. A related class of boundary measures is also discussed. Article A4 investigates the properties of Nevanlinna measures with respect to restrictions to coordinate orthogonal hyperplanes and the geometry of the support. A related class of measures on the unit poly-torus is also considered. Furthermore, this thesis is supplemented by three additional publications concerning Herglotz-Nevanlinna functions in one variable, related topics and applications. Article B1 concerns a particular class of convolution operators on the space of distributions that generalizes the well-studied class of passive operators. Article B2 introduces the class of quasi-Herglotz functions and discusses their integral representations, boundary values and sum-rules, as well as their applications in connection with convex optimization. Finally, the summary book-chapter C1 provides a general overview of the applications of Herglotz-Nevanlinna functions in electromagnetics. I denna avhandling undersöker vi de olika aspekterna av klassen av Herglotz-Nevanlinnafunktioner i flera variabler. Dessa är holomorfa funktioner definierade i det polyövre halvplanet som har en icke-negativ imaginärdel. Våra resultat presenteras i de fyra vetenskapliga artiklarna A1 – A4 som inkluderas i avhandlingen. Artiklar A1 och A2 sätter upp en karakterisering av Herglotz-Nevnlinnafunktioner genom en integralframställning. Fallet där vi behandlar funktioner av två komplexa variabler presenteras i artikeln A1 medan det generella f