In this article it is shown that every function meromorphic in the upper halfplane and of bounded type does have a realization with the resolvent of a self-adjoint relation in a Krein space.

This is the first part of the review article that focuses on theory and applications of Herglotz-Nevanlinna functions in material sciences. It starts with the definition of scalar-valued Herglotz-Nevanlinna functions and explains in detail the theorems that are pertinent to applications, followed by a short overview of the matrix-valued and operator-valued versions of these functions and the properties that carry over from scalar cases. The theory is complemented by some applications from electromagnetics that are related to the sum rules. More applications of Herglotz-Nevanlinna functions in material sciences can be found in Part II. 

Part II of the review article focuses on the applications of Herglotz–Nevanlinna functions in material sciences. It presents a diverse set of applications with details and the role of Herglotz–Nevanlinna functions clearly pointed out. This chapter is concluded by a collection of existent generalizations of the class of Herglotz–Nevanlinna functions that are motivated by potential applications. 

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.

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 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 paper self-adjoint realizations of the formal expression A(alpha ):= A + alpha <phi, .> phi are described, where alpha is an element of R boolean OR {infinity}, the operator A is self-adjoint in a Hilbert space H and phi is a supersingular element from the scale space H--(n) (-2) (A) \H--(n) (-1) (A) for n >= 1. The crucial point is that the spectrum of A may consist of the whole real line. We construct two models to describe the family (A(alpha)). It can be interpreted in a Hilbert space with a twisted version of Krein's formula, or with a more classical version of Krein's formula but in a Pontryagin space. Finally, we compare the two approaches in terms of the respective Q-functions.

A passive approximation problem is formulated where the target function is an arbitrary complex-valued continuous function defined on an approximation domain consisting of a finite union of closed and bounded intervals on the real axis. The norm used is a weighted L-p-norm where 1 <= p <= infinity. The approximating functions are Herglotz functions generated by a measure with Holder continuous density in an arbitrary neighborhood of the approximation domain. Hence, the imaginary and the real parts of the approximating functions are Holder continuous functions given by the density of the measure and its Hilbert transform, respectively. In practice, it is useful to employ finite B-spline expansions to represent the generating measure. The corresponding approximation problem can then be posed as a finite-dimensional convex optimization problem which is amenable for numerical solution. A constructive proof is given here showing that the convex cone of approximating functions generated by finite uniform B-spline expansions of fixed arbitrary order (linear, quadratic, cubic, etc.) is dense in the convex cone of Herglotz functions which are locally Holder continuous in a neighborhood of the approximation domain, as mentioned above. As an illustration, typical physical application examples are included regarding the passive approximation and optimization of a linear system having metamaterial characteristics, as well as passive realization of optimal absorption of a dielectric small sphere over a finite bandwidth.

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.