In this paper, the class of (complex) quasi-Herglotz functions is introduced as the complex vector space generated by the convex cone of ordinary Herglotz functions. We prove characterization theorems, in particular, an analytic characterization. The subclasses of quasi-Herglotz functions that are identically zero in one half-plane as well as rational quasi-Herglotz functions are investigated in detail. Moreover, we relate to other areas such as weighted Hardy spaces, definitizable functions, the Cauchy transform on the unit circle, sum-rule identities and matrix-valued Herglotz functions. 

Dans le présent article, la classe des fonctions (complexes) quasi-Herglotz est présentée comme l’espace vectoriel complexe engendré par le cône convexe des fonctions de Herglotz usuelles. Nous démontrons des théorèmes de caractérisation, en particulier une caractérisation analytique. Les sous-classes des fonctions quasi-Herglotz qui sont identiquement nulles dans un demi-plan ainsi que les fonctions quasi-Herglotz rationnelles sont étudiées en détail. De plus, nous faisons le lien avec d’autres domaines tels que les espaces de Hardy avec poids, les fonctions définissables, la transformée de Cauchy sur le cercle unité, les identités « somme-règle » et les fonctions de Herglotz à valeur matricielle. 

In this paper, we study the properties of a certain class of Borel measures on Rnthat arise in the integral representation of Herglotz-Nevanlinna functions. In particular, we find that restrictions to certain hyperplanes are of a surprisingly simple form and show that the supports of such measures cannot lie within particular geometric regions, e.g., strips with positive slope. Corresponding results are derived for measures on the unit poly-torus with vanishing mixed Fourier coefficients. These measures are closely related to functions mapping the unit polydisk analytically into the right half-plane.

In this paper, we study a class of Borel measures on Double-struck capital Rn that arises as the class of representing measures of Herglotz-Nevanlinna functions. In particular, we study product measures within this class where products with the Lebesgue measures play a special role. Hence, we give several characterizations of the n-dimensional Lebesgue measure among all such measures and characterize all product measures that appear in this class of measures. Furthermore, analogous results for the class of positive Borel measures on the unit poly-torus with vanishing mixed Fourier coefficients are also presented, and the relation between the two classes of measures with regard to the obtained results is discussed.

We introduce the set of quasi-Herglotz functions and demonstrate that it has properties useful in the modelling of non-passive systems. The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of Herglotz functions. It consists of differences of Herglotz functions and we show that several of the important properties and modelling perspectives are inherited by the new set of quasi-Herglotz functions. In particular, this applies to their integral representations, the associated integral identities or sum rules (with adequate additional assumptions), their boundary values on the real axis and the associated approximation theory. Numerical examples are included to demonstrate the modelling of a non-passive gain medium formulated as a convex optimization problem, where the generating measure is modelled by using a finite expansion of B-splines and point masses.

In this paper, we begin by investigating a particular subclass of boundary measures of Herglotz-Nevanlinna functions in two variables. Based on this, we then proceed to solve the convex combination problem for Herglotz-Nevanlinna functions in several variables.

In this article, a characterization of the class of Herglotz-Nevanlinna functions in several variables is given in terms of an integral representation. The conditions on the representing measure are discussed in detail, and, furthermore, the properties of the symmetric extension of a Herglotz-Nevanlinna function are also investigated.

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 fallet diskuteras i artikeln A2. Där diskuteras även egenskaper av den symmetriska utvidgningen av Herglotz-Nevnlinnafunktioner.

I artikeln A3 presenteras en detaljerad undersökning av det konvexa kombinationsproblemet för Herglotz-Nevanlinnafunktioner. Detta problem frågar efter ett samband mellan representationsparametrar av olika Herglotz-Nevanlinnafunktioner under antagandet att dessa funktioner kan förknippas med varandra genom en konvex kombination av flera oberoende variabler. En relaterad klass av randmått undersöks också.

Artikeln A4 undersöker egenskaper av Nevanlinnamått med avseende på deras restriktioner till koordinatortogonala hyperplan och geometrin av stödet. En relaterad klass av mått i enhets polytorusen diskuteras också.

Vidare kompletteras avhandlingen med ytterligare tre publikationer som handlar om Herglotz-Nevanlinnafunktioner i en variabler, relaterade ämnena och deras tillämpningar.

Artikeln B1 behandlar en viss klass av faltningsoperatorer definierade på rummet av distributioner som generalisera den välkända klassen av passiva operatorer. Artikeln B2 introducerar klassen av quasi-Herglotzfunktioner och presenterar deras integralframställningar, randvärdena och summaregler så som deras tillämpningar i sambandet med konvex optimering. Till slut förser sammanfattningsbokkapitel C1 oss en överblick över tillämpningar av Herglotz-Nevanlinnafunktioner inom elektromagnetisk teori.

We introduce the set of quasi-Herglotz functions and demonstrate that it has properties useful in the modeling of non-passive systems.The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of Herglotz functions. It consists of differences of Herglotz functions, and we show that several of the important properties and modeling perspectives of Herglotz functions are inherited by the new set of quasi-Herglotz functions.In particular, this applies to their integral representations, the associated integral identities or sum rules (with adequate additional assumptions), their boundary values on the real axis and the associated approximation theory.Numerical examples are included to demonstrate the modeling of a non-passive gain media formulated as a convex optimization problem,where the generating measure is modeled by using a finite expansion of B-splines and point masses.

We derive an integral representation for Herglotz-Nevanlinna functions in two variables, which provides a complete characterization of this class in terms of a real number, two non-negative numbers and a positive measure satisfying certain conditions. Further properties of the class of representing measures are also discussed.

In this article, a characterization of the class of Herglotz-Nevanlinna functions in n variables is given in terms of an integral representation. Furthermore, the properties of the class of representing measures are discussed in detail. Symmetry properties induced by the integral representation are also investigated.