Related papers: A characterization of Herglotz-Nevanlinna function…
In this article, a characterization of the class of Herglotz-Nevanlinna functions in $n$ variables is given in terms of an integral representation. Furthermore, alternative conditions on the measure appearing in this representation are…
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 this paper, we derive an analytic characterization of the symmetric extension of a Herglotz-Nevanlinna function in several variables. Here, the main tools used are the so-called variable non-dependence property and the symmetry formula…
A multidimensional generalization of the Bernstein class of functions and the properties of functions of the introduced class are examined. In particular, a new proof of the integral representation of Bernstein functions of many variables…
In this paper, we give several characterizations of Herglotz-Nevanlinna functions in terms of a specific type of positive semi-definite functions called Poisson-type functions. This allows us to propose a multidimensional analogue of the…
The Herglotz representation theorem for holomorphic functions with non-negative real part is a fundamental result in the theory of holomorphic functions. In this paper, we reinterpret the Herglotz representation in the context of modern…
A Herglotz function is a holomorphic map from the open complex unit disk into the closed complex right halfplane. A classical Herglotz function has an integral representation against a positive measure on the unit circle. We prove a free…
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…
In this note I prove the following property of Herglotz functions, which to my knowledge is new: For a Herglotz function $h(z)$ and a real number $r \in \mathbb R$ define a Herglotz function $g_r(z) = (r - h(z))^{-1}.$ Let $\mu_r^{(s)}$ be…
We generalize two integral representation formulae of Nevanlinna to functions of several variables. We show that for a large class of analytic functions that have non-negative imaginary part on the upper polyhalfplane there are…
In connection with the Herglotz-Nevanlinna integral representation of so-called Pick functions, we introduce the notion of boundary measure of holomorphic functions on the imaginary domain and elucidate some of basic properties.
In this paper, we study the structural properties of Nevanlinna measures, i.e. Borel measures that arise in the integral representation of Herglotz-Nevanlinna functions. In particular, we give a characterization of these measures in terms…
Scalar-valued meromorphic Herglotz-Nevanlinna functions are characterized by the interlacing property of their poles and zeros together with some growth properties. We give a characterization of matrix-valued Herglotz-Nevanlinna functions…
Integral representations play a prominent role in the analysis of entire functions. The representations of generalized Mittag-Leffler type functions and their asymptotics have been (and still are) investigated by plenty of authors in…
We supplement the Herglotz-Nevanlinna integral representation of so-called Pick functions by adding the formula for M\"obius transforms and the positivity characterization near boundary supports.
A complex function $f(z)$ is called a Herglotz-Nevanlinna function if it is holomorphic in the upper half-plane ${\mathbb C}_+$ and maps ${\mathbb C}_+$ into itself. By a maximum principle a Herglotz-Nevanlinna function which takes a real…
This is the first part of the review article which 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 paper explores various special functions which generalize the two-parametric Mittag-Leffler type function of two variables. Integral representations for these functions in different domains of variation of arguments for certain values…
In this paper, we study a class of Borel measures on $\mathbb{R}^n$ 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…
We consider a two-spectra inverse problem for the one-dimensional Schr\"{o}dinger equation with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter and provide a complete solution of this…