[Paper Review] An integral representation for Herglotz-Nevanlinna functions in several variables

Annemarie Luger, Mitja Nedic|KTH Publication Database DiVA (KTH Royal Institute of Technology)|May 30, 2017

Social and Educational Sciences2 references4 citations

TL;DR

This paper establishes a complete integral representation for Herglotz-Nevanlinna functions in several complex variables, generalizing the classical one-variable Nevanlinna representation. It proves that such functions admit a representation involving a real constant, non-negative linear terms, and a positive Borel measure satisfying a growth and Nevanlinna condition, enabling characterization and extension analysis beyond the poly-upper half-plane.

ABSTRACT

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 discussed in detail. Symmetry properties induced by the integral representation are also investigated.

Motivation & Objective

• To provide a complete integral representation for Herglotz-Nevanlinna functions in $n$ variables, generalizing the classical one-variable case.
• To eliminate the need for distribution theory and boundary measure assumptions present in prior works, such as Vladimirov's approach.
• To characterize the class of representing measures via natural growth and Nevanlinna conditions.
• To analyze the analytic continuation of these functions to $(\mathbb{C} \setminus \mathbb{R})^n$ and their symmetry properties.
• To extend and simplify earlier results for $n=2$ to the general $n$-variable case.

Proposed method

• Derives the integral representation using a direct approach based on Cauchy’s integral formula and Helly’s selection principle.
• Adapts an integral representation for functions with non-negative real part on the unit polydisk to the upper half-plane via conformal mapping.
• Defines the kernel function $K_n(\vec{z}, \vec{t})$ explicitly in terms of the imaginary part of the Cauchy kernel in $n$ variables.
• Imposes two key conditions on the measure $\mu$: a growth condition $\int_{\mathbb{R}^n} \frac{1}{1+|\vec{t}|^2} d\mu(\vec{t}) < \infty$ and the Nevanlinna condition (4.2) ensuring non-negative imaginary part.
• Uses the structure of the kernel and measure to prove that any function with such a representation is a Herglotz-Nevanlinna function.
• Analyzes the extension of the function to $(\mathbb{C} \setminus \mathbb{R})^n$ by studying the behavior of the integral and deriving symmetry formulas involving conjugate variables.

Experimental results

Research questions

• RQ1 Can a complete integral representation be established for Herglotz-Nevanlinna functions in $n$ complex variables that generalizes the classical one-variable case?
• RQ2 What are the necessary and sufficient conditions on the representing measure $\mu$ for the integral representation to yield a Herglotz-Nevanlinna function?
• RQ3 How do the symmetry properties of the function manifest in the extended domain $(\mathbb{C} \setminus \mathbb{R})^n$?
• RQ4 Under what conditions can the function be constant in the extended domain even if it is non-constant in the poly-upper half-plane?
• RQ5 How do the Nevanlinna condition and growth condition relate to each other and to the function's analytic and boundary behavior?

Key findings

• A Herglotz-Nevanlinna function $q$ in $n$ variables admits an integral representation of the form $q(\vec{z}) = a + \sum_{\ell=1}^n b_\ell z_\ell + \frac{1}{\pi^n} \int_{\mathbb{R}^n} K_n(\vec{z}, \vec{t}) d\mu(\vec{t})$, where $a \in \mathbb{R}$, $\vec{b} \in [0,\infty)^n$, and $\mu$ is a positive Borel measure satisfying the growth and Nevanlinna conditions.
• The Nevanlinna condition (4.2) is equivalent to the requirement that the imaginary part of $q$ is non-negative in the poly-upper half-plane, and it is shown to be necessary and sufficient for the representation.
• The integral representation allows for analytic continuation of $q$ to the set $(\mathbb{C} \setminus \mathbb{R})^n$, where the function may become constant even if it is non-constant in $\mathbb{C}^{+n}$.
• A symmetry formula (6.4) is derived for the extension, expressing $q(\vec{z})$ in terms of conjugate variables and the function $q_0$ associated with the measure, revealing dependence on the sign of imaginary parts.
• The function $q_0$, corresponding to the measure $\mu$ with zero constant and linear terms, satisfies $q_0(\vec{z}) = \frac{1}{\pi^n} \int_{\mathbb{R}^n} K_n(\vec{z}, \vec{t}) d\mu(\vec{t})$, and its behavior under conjugation is central to the symmetry analysis.
• An explicit example shows that the symmetry formula (6.4) correctly reproduces the extension of $q(z_1,z_2) = 2z_2 - \frac{1}{z_1 + z_2}$ to $\mathbb{C}^2 \setminus \mathbb{R}^2$, verifying the theoretical framework.