# An analytic characterization of the symmetric extension of a Herglotz-
Nevanlinna function in several variables

Mitja Nedic  Mitja Nedic, Department of Mathematics and Statistics, University
of Helsinki, PO Box 68, FI-00014 Helsinki, Finland, orc-id:
0000-0001-7867-5874<EMAIL_ADDRESS>
(Date: 2019-11-27)

###### Abstract.

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 satisfied by Herglotz-Nevanlinna and Cauchy-type functions.
We also provide an extension of the Stieltjes inversion formula for Cauchy-
type functions.

###### 2010 Mathematics Subject Classification:

32A36, 32A99.

Key words. Herglotz-Nevanlinna functions, Cauchy-type functions, symmetric
extension.

##  1\. Introduction

On the upper half-plane $\mathbb{C}^{+}:=\\{z\in\mathbb{C}\leavevmode\nobreak\
|\leavevmode\nobreak\ \mathrm{Im}[z]>0\\}$, the class of holomorphic functions
with non-negative imaginary part plays an important role in many areas of
analysis and applications. These functions, called _Herglotz-Nevanlinna
functions_ , appear, to name but a few examples, in the theory of Sturm-
Liouville operators and their perturbations [3, 4, 7, 10], when studying the
classical moment problem [2, 16, 17], when deriving physical bounds for
passive systems [5] or as approximating functions in certain convex
optimization problems [8, 9].

A classical integral representation theorem [6, 16] states that any Herglotz-
Nevanlinna function $h$ can be written, for $z\in\mathbb{C}^{+}$, as

(1.1)
$h(z)=a+bz+\frac{1}{\pi}\int_{\mathbb{R}}\left(\frac{1}{t-z}-\frac{t}{1+t^{2}}\right)\mathrm{d}\mu(t),$

where $a\in\mathbb{R}$, $b\geq 0$ and $\mu$ is a positive Borel measure on
$\mathbb{R}$ for which
$\int_{\mathbb{R}}(1+t^{2})^{-1}\mathrm{d}\mu(t)<\infty$. Although this
representation is _a prioir_ established for $z\in\mathbb{C}^{+}$, it is well-
defined, as an algebraic expression, for any
$z\in\mathbb{C}\setminus\mathbb{R}$. Hence, for a Herglotz-Nevanlinna function
$h$, we define its _symmetric extension_ $h_{\mathrm{sym}}$ as the right-hand
side of representation (1.1) where we now take
$z\in\mathbb{C}\setminus\mathbb{R}$. It is now an easy consequence of the
definitions that a holomorphic function
$f\colon\mathbb{C}\setminus\mathbb{R}\to\mathbb{C}$ equals the symmetric
extension of some Herglotz-Nevanlinna function if and only if it holds that
$\mathrm{Im}[f(z)]\geq 0$ for $z\in\mathbb{C}^{+}$ and
$f(z)=\overline{f(\overline{z})}$ for all $z\in\mathbb{C}\setminus\mathbb{R}$.
In this way, we obtain an analytic characterzation of the symmetric extension.

When considering, instead, functions in the poly-upper half-plane

$\mathbb{C}^{+n}:=(\mathbb{C}^{+})^{n}=\big{\\{}\bm{z}\in\mathbb{C}^{n}\,\big{|}\,\forall
j=1,2,\ldots,n:\mathrm{Im}[z_{j}]>0\big{\\}},$

the analogous situation becomes more involved. Herglotz-Nevanlinna functions
in several variables, cf. Definition 2.2, appear e.g. when considering
operator monotone functions [1] or with representations of multidimensional
passive systems [19]. Their corresponding integral representation is recalled
in detail in Theorem 2.3 later on and leads, in an analogous way as in the
one-variable case, to the definition of a _symmetric extension_ , which is now
a holomorphic function on $(\mathbb{C}\setminus\mathbb{R})^{n}$. As such, the
main goal of this paper is to give an analytic characterization of symmetric
extensions of a Herglotz-Nevanlinna function in several variables, i.e. we
wish to be able to determine when a function
$f\colon(\mathbb{C}\setminus\mathbb{R})^{n}\to\mathbb{C}$ is, in fact, equal
to the symmetric extension of a Herglotz-Nevanlinna function. This is answered
by Theorem 3.3 and Corollary 3.4.

The structure of the paper is as follows. After the introduction in Section 1
we review the different classes of functions that will appear throughout the
paper in Section 2. Section 3 is then devoted to presenting the main result of
the paper as well as some important examples. Finally, Section 4 discusses how
the Stieltjes inversion formula can be extended to certain functions on
$(\mathbb{C}\setminus\mathbb{R})^{n}$.

##  2\. Classes of functions in the poly cut-plane

Throughout this paper, we will primarily consider two classes of holomorphic
functions on the poly cut-plane $(\mathbb{C}\setminus\mathbb{R})^{n}$, both of
which are intricately connected to a certain kernel function. These objects
are defined as follows.

###  2.1. The kernel $K_{n}$ and Cauchy-type functions

We begin by introducing the kernel
$K_{n}\colon(\mathbb{C}\setminus\mathbb{R})^{n}\times\mathbb{R}^{n}\to\mathbb{C}$
as

(2.1)
$K_{n}(\bm{z},\bm{t}):=\mathfrak{i}\left(\frac{2}{(2\mathfrak{i})^{n}}\prod_{\ell=1}^{n}\left(\frac{1}{t_{\ell}-z_{\ell}}-\frac{1}{t_{\ell}+\mathfrak{i}}\right)-\frac{1}{(2\mathfrak{i})^{n}}\prod_{\ell=1}^{n}\left(\frac{1}{t_{\ell}-\mathfrak{i}}-\frac{1}{t_{\ell}+\mathfrak{i}}\right)\right).$

If the vector $\bm{z}$ is restricted to $\mathbb{C}^{+n}$, then the kernel
$K_{n}$ is a complex-constant multiple of the Schwartz kernel of
$\mathbb{C}^{+n}$ viewed as a tubular domain over the cone $[0,\infty)^{n}$
[19, Sec. 12.5].

When $n=1$, it holds that

$K_{1}(z,t)=\frac{1}{t-z}-\frac{t}{1+t^{2}}.$

As such, the kernel $K_{1}$ satisfies, for all
$z\in\mathbb{C}\setminus\mathbb{R}$ and all $t\in\mathbb{R}$, the symmetry
property

$K_{1}(z,t)=\overline{K_{1}(\overline{z},t)}.$

When $n\geq 2$, the symmetry satisfied by the kernel becomes more involved and
requires the introduction of some additional notation. First, given two
numbers $z,w\in\mathbb{C}$, an indexing set $B\subseteq\\{1,2,\ldots,n\\}$ and
an index $j\in\\{1,2,\ldots,n\\}$, define

$\psi_{B}^{j}(z,w):=\left\\{\begin{array}[]{rcl}z&;&j\not\in B,\\\
\overline{w}&;&j\in B.\end{array}\right.$

Second, given an indexing set $B\subseteq\\{1,2,\ldots,n\\}$, define the map
$\Psi_{B}\colon\mathbb{C}^{n}\times\mathbb{C}^{n}\to\mathbb{C}^{n}$ as
$\Psi_{B}(\bm{z},\bm{w}):=\bm{\zeta}$ with
$\zeta_{j}:=\psi_{B}^{j}(z_{j},w_{j})$. In other words, the map $\Psi_{B}$
functions as a way of selectively combining two vectors into one where the set
$B$ determines which components of $\bm{z}$ should be replaced by the
conjugates of the components of $\bm{w}$. It now holds that

(2.2)
$K_{n}(\bm{z},\bm{t})=\sum_{\begin{subarray}{c}B\subseteq\\{1,\ldots,n\\}\\\
B\neq\emptyset\end{subarray}}(-1)^{|B|+1}\overline{K_{n}(\Psi_{B}(\mathfrak{i}\,\bm{1},\bm{z}),\bm{t})}$

for every $\bm{z}\in(\mathbb{C}\setminus\mathbb{R})^{n}$ and every
$\bm{t}\in\mathbb{R}^{n}$ [13, Prop. 6.1].

Using the kernel $K_{n}$, the largest class of functions that will be
considered is the following.

######  Definition 2.1.

A function $g\colon(\mathbb{C}\setminus\mathbb{R})^{n}\to\mathbb{C}$ is called
a _Cauchy-type function_ if there exists a positive Borel measure $\mu$ on
$\mathbb{R}^{n}$ satisfying the growth condition

(2.3)
$\int_{\mathbb{R}^{n}}\prod_{\ell=1}^{n}\frac{1}{1+t_{\ell}^{2}}\mathrm{d}\mu(\bm{t})<\infty$

such that

$g(\bm{z})=\frac{1}{\pi^{n}}\int_{\mathbb{R}^{n}}K_{n}(\bm{z},\bm{t})\mathrm{d}\mu(\bm{t})$

for every $\bm{z}\in(\mathbb{C}\setminus\mathbb{R})^{n}$.

Note that this definition is different from [13, Def. 3.1] in that it assumes
from the beginning that a Cauchy-type function is defined on
$(\mathbb{C}\setminus\mathbb{R})^{n}$ and not only on $\mathbb{C}^{+n}$.
Furthermore, it would be possible to define an even larger class of functions
using the same kernel, but general distributions instead of measures, see [14,
Ex. 7.7] for an example. However, this extension will not be considered here.
Moreover, Definition 2.1 allows, in principle, for two (or more) different
measure to yield the same function $g$, though we will show that this is not
the case later in Section 4.

An immediate consequence of the symmetry formula (2.2) is an analogous
symmetry formula for Cauchy-type functions. In particular, it holds, for any
Cauchy-type function $g$, that

(2.4) $g(\bm{z})=\sum_{\begin{subarray}{c}B\subseteq\\{1,\ldots,n\\}\\\
B\neq\emptyset\end{subarray}}(-1)^{|B|+1}\overline{g(\Psi_{B}(\mathfrak{i}\,\bm{1},\bm{z}))}$

for every $\bm{z}\in(\mathbb{C}\setminus\mathbb{R})^{n}$ and every
$\bm{t}\in\mathbb{R}^{n}$ [13, Prop. 6.5].

The growth of a Cauchy-type function along a coordinate parallel complex line
can be described using non-tangential limits. These are taken in so-called
_Stoltz domains_ and are defined as follows. An _upper Stoltz domain_ with
centre $0\in\mathbb{R}$ and angle $\theta\in(0,\frac{\pi}{2}]$ is the set
$\\{z\in\mathbb{C}^{+}\leavevmode\nobreak\ |\leavevmode\nobreak\
\theta\leq\arg(z)\leq\pi-\theta\\}$ and the symbol
$z\>\scriptsize{\xrightarrow{\vee\>}}\>\infty$ then denotes the limit
$|z|\to\infty$ in any upper Stoltz domain with centre $0$. A _lower Stoltz
domain_ and the symbol $z\>\scriptsize{\xrightarrow{\wedge\>}}\>\infty$ are
defined analogously. Furthermore, we note that in the literature, slightly
different notations are sometimes used to describe these limits. Two examples
of Stlotz domains are visualized in Figure 1 below.

$x$$\mathfrak{i}\,y$$\theta_{1}$

$x$$\mathfrak{i}\,y$$\theta_{2}$

Figure 1. An upper Stoltz domain with centre $0$ and angle $\theta_{1}$ (left)
and a lower Stoltz domain with centre $0$ and angle $\theta_{2}$ (right).

For any Cauchy-type function $g$ it now holds, for any
$\bm{z}\in(\mathbb{C}\setminus\mathbb{R})^{n}$ and any $j\in\\{1,\ldots,n\\}$,
that

$\lim\limits_{z_{j}\>\scriptsize{\xrightarrow{\vee\>}}\>\infty}\frac{g(\bm{z})}{z_{j}}=\lim\limits_{z_{j}\>\scriptsize{\xrightarrow{\wedge\>}}\>\infty}\frac{g(\bm{z})}{z_{j}}=0,$

see [13, Lem. 3.2].

###  2.2. Herglotz-Nevanlina functions

These functions are defined as follows, cf. [12, 13, 18, 19].

######  Definition 2.2.

A holomorphic function $h\colon\mathbb{C}^{+n}\to\mathbb{C}$ is called a
_Herglotz-Nevanlinna function_ if it is holomorphic with non-negative
imaginary part.

In contrast to the definition of Cauchy-type function, the above definition is
analytic in nature, i.e. it describes the function class in terms of
conditions on the function itself. In order to be able to relate it to the
kernel $K_{n}$, we introduce, given ambient numbers
$z\in\mathbb{C}\setminus\mathbb{R}$ and $t\in\mathbb{R}$, the expressions

$\begin{array}[]{RCL}N_{-1}(z,t)&:=&\frac{1}{2\,\mathfrak{i}}\left(\frac{1}{t-z}-\frac{1}{t-\mathfrak{i}}\right),\\\\[9.95863pt]
N_{0}(z,t)&:=&\frac{1}{2\,\mathfrak{i}}\left(\frac{1}{t-\mathfrak{i}}-\frac{1}{t_{j}+\mathfrak{i}}\right),\\\\[9.95863pt]
N_{1}(z,t)&:=&\frac{1}{2\,\mathfrak{i}}\left(\frac{1}{t+\mathfrak{i}}-\frac{1}{t-\overline{z}}\right).\end{array}$

Note that $N_{0}$ is independent of $z\in\mathbb{C}\setminus\mathbb{R}$ and
$N_{0}(z,t)\in\mathbb{R}$ while

$\overline{N_{-1}(z,t)}=N_{1}(z,t)$

for all $z\in\mathbb{C}\setminus\mathbb{R}$ and $t\in\mathbb{R}$. Using these
expression, one may give an integral representation formula for Herglotz-
Nevanlinna functions involving the kernel $K_{n}$ [13, Thm. 4.1].

######  Theorem 2.3.

A function $h\colon\mathbb{C}^{+n}\to\mathbb{C}$ is a Herglotz-Nevanlinna
function if and only if $h$ can be written as

(2.5)
$h(\bm{z})=a+\sum_{j=1}^{n}b_{j}\,z_{j}+\frac{1}{\pi^{n}}\int_{\mathbb{R}^{n}}K_{n}(\bm{z},\bm{t})\mathrm{d}\mu(\bm{t}),$

where $a\in\mathbb{R}$, $\bm{b}\in[0,\infty)^{n}$, the kernel $K_{n}$ is as
before and $\mu$ is a positive Borel measure on $\mathbb{R}^{n}$ satisfying
the growth condition (2.3) and the Nevanlinna condition

(2.6) $\sum_{\begin{subarray}{c}\bm{\rho}\in\\{-1,0,1\\}^{n}\\\
-1\in\bm{\rho}\wedge
1\in\bm{\rho}\end{subarray}}\int_{\mathbb{R}^{n}}\prod_{j=1}^{n}N_{\rho_{j}}(z_{j},t_{j})\mathrm{d}\mu(\bm{t})=0$

for all $\bm{z}\in\mathbb{C}^{+n}$. Furthermore, for a given function $h$, the
triple of representing parameters $(a,\bm{b},\mu)$ is unique.

The integral representation in formula (2.5) is well-defined for any
$\bm{z}\in(\mathbb{C}\setminus\mathbb{R})^{n}$, which may be used to extend
any Herglotz-Nevanlinna function $h$ from $\mathbb{C}^{+n}$ to
$(\mathbb{C}\setminus\mathbb{R})^{n}$. This extension is called the _symmetric
extension_ of the function $h$ and is denoted as $h_{\mathrm{sym}}$. The
symmetric extension of a Herglotz-Nevanlinna function $h$ is different from
its possible analytic extension as soon as $\mu\neq 0$ [13, Prop. 6.10] and
satisfies the following variable-dependence property [13, Prop. 6.9].

######  Proposition 2.4.

Let $n\geq 2$ and let $h_{\mathrm{sym}}$ be the symmetric extension of a
Herglotz-Nevanlinna function $h$ in $n$ variables for which $\bm{b}=\bm{0}$.
Let $\bm{z}\in(\mathbb{C}\setminus\mathbb{R})^{n}$ be such that
$z_{j}\in\mathbb{C}^{-}$ for some index $j\in\\{1,2,\ldots,n\\}$. Then, the
value $h_{\mathrm{sym}}(\bm{z})$ does not depend on the components of $\bm{z}$
that lie in $\mathbb{C}^{+}$.

Furthermore, if $h$ is a Herglotz-Nevanlinna function for which
$\bm{b}=\bm{0}$, then its symmetric extension $h_{\mathrm{sym}}$ will satisfy
the symmetry formula

(2.7)
$h_{\mathrm{sym}}(\bm{z})=\sum_{\begin{subarray}{c}B\subseteq\\{1,\ldots,n\\}\\\
B\neq\emptyset\end{subarray}}(-1)^{|B|+1}\overline{h_{\mathrm{sym}}(\Psi_{B}(\mathfrak{i}\,\bm{1},\bm{z}))},$

where $\bm{z}\in(\mathbb{C}\setminus\mathbb{R})^{n}$ [13, Prop. 6.7]. When
$n=1$, it is not necessary to assume that $b=0$ for formula (2.7) to hold.
However, when $n>1$, this is required.

The representing vector $\bm{b}$ describes the growth of the function $h$
along coordinate parallel complex lines in $\mathbb{C}^{+n}$. More precisely,
we recall from [13, Cor. 4.6(iv)] that, for any $j\in\\{1,\ldots,n\\}$, we
have

(2.8)
$b_{j}=\lim\limits_{z_{j}\>\scriptsize{\xrightarrow{\vee\>}}\>\infty}\frac{h(\bm{z})}{z_{j}}.$

In particular, the above limit is independent of the entries of the vector
$\bm{z}$ at the non-$j$-th positions. This result carries over to the
symmetric extension, for which it holds, for any $j\in\\{1,\ldots,n\\}$, that

$b_{j}=\lim\limits_{z_{j}\>\scriptsize{\xrightarrow{\vee\>}}\>\infty}\frac{h_{\mathrm{sym}}(\bm{z})}{z_{j}}=\lim\limits_{z_{j}\>\scriptsize{\xrightarrow{\wedge\>}}\>\infty}\frac{h_{\mathrm{sym}}(\bm{z})}{z_{j}}.$

Every Herglotz-Nevanlinna function that is represented by a data-triple of the
form $(0,\bm{0},\mu)$ in the sense of Theorem 2.3 is also a Cauchy-type
function. The converse, i.e. that every Cauchy-type function equals a
Herglotz-Nevanlinna function represented by a data-triple of the form
$(0,\bm{0},\mu)$, is true only when $n=1$. This is due to the fact that when
$n=1$, the Nevanlinna condition (2.6) becomes emptily fulfilled by every
positive Borel measures $\mu$ satisfying the growth condition (2.3).

##  3\. Symmetry and variable non-dependence

We begin by recalling that the symmetric extension of a Herglotz-Nevanlinna
function $h$ in one variable is uniquely determined by its values in
$\mathbb{C}^{+}$. Indeed, when $n=1$, the symmetry formula (2.7) takes the
form $h_{\mathrm{sym}}(z)=\overline{h_{\mathrm{sym}}(\overline{z})}$,
providing a way to recover the values of the function in $\mathbb{C}^{-}$
using only the values of the function in $\mathbb{C}^{+}$.

For functions of several variables, the appropriate analogue involves the
following definition.

######  Definition 3.1.

A function $f\colon(\mathbb{C}\setminus\mathbb{R})^{n}\to\mathbb{C}$ is said
to satisfy the _variable non-dependence property_ if for every vector
$\bm{z}\in(\mathbb{C}\setminus\mathbb{R})^{n}$ such that
$z_{j}\in\mathbb{C}^{-}$ for some index $j\in\\{1,2,\ldots,n\\}$ the value
$f(\bm{z})$ does not depend on the components of $\bm{z}$ that lie in
$\mathbb{C}^{+}$.

By Proposition 2.4, the symmetric extension of a Herglotz-Nevanlinna function
satisfies the variable non-dependence property 3.1 if $\bm{b}=\bm{0}$. In
particular, the symmetric extension of any Herglotz-Nevanlinna function that
is also a Cauchy-type function will always satisfy the variable non-dependence
property 3.1. However, a general Cauchy-type function need not satisfy it, as
shown by the function $f_{2}$ in Example 3.5 later on.

We may now describe the precise circumstances under which we can recover the
values of a function defined on $(\mathbb{C}\setminus\mathbb{R})^{n}$ purely
in terms of its values in $\mathbb{C}^{+n}$.

######  Proposition 3.2.

Let $f\colon(\mathbb{C}\setminus\mathbb{R})^{n}\to\mathbb{C}$ be a holomorphic
function satisfying the symmetry formula (2.7) and the variable non-dependence
property 3.1. Then, the values of the function $f$ on
$(\mathbb{C}\setminus\mathbb{R})^{n}$ are uniquely determined by its values in
$\mathbb{C}^{+n}$.

###### Proof.

Using the symmetry formula (2.7), let us investigate the values of the
function $f$ in a connected component of $(\mathbb{C}\setminus\mathbb{R})^{n}$
where at least one of the coordinates has a negative sign of the imaginary
part, i.e. we are investigating a connected component
$X\subseteq(\mathbb{C}\setminus\mathbb{R})^{n}$ where exist at lest one index
$j\in\\{1,\ldots,n\\}$ such that the $j$-th coordinate lies in
$\mathbb{C}^{-}$. For any such chosen connected component $X$, let
$B^{\prime}\subseteq\\{1,\ldots,n\\}$ be the set of those indices for which
the corresponding variables lie in $\mathbb{C}^{-}$. In particular,
$1\leq|B^{\prime}|\leq n$. For $\bm{z}\in X$, it holds, by the symmetry
formula (2.7), that

$f(\bm{z})=\sum_{\begin{subarray}{c}B\subseteq\\{1,\ldots,n\\}\\\
B\neq\emptyset\end{subarray}}(-1)^{|B|+1}\overline{f(\Psi_{B}(\mathfrak{i}\,\bm{1},\bm{z}))}\\\
=\sum_{\begin{subarray}{c}B\subseteq\\{1,\ldots,n\\}\\\ B\neq\emptyset\wedge
B\subseteq
B^{\prime}\end{subarray}}(-1)^{|B|+1}\overline{f(\Psi_{B}(\mathfrak{i}\,\bm{1},\bm{z}))}+\sum_{\begin{subarray}{c}B\subseteq\\{1,\ldots,n\\}\\\
B\not\subseteq
B^{\prime}\end{subarray}}(-1)^{|B|+1}\overline{f(\Psi_{B}(\mathfrak{i}\,\bm{1},\bm{z}))}.$

Due to the definition of the set $B^{\prime}$, it holds that
$\Psi_{B}(\mathfrak{i}\,\bm{1},\bm{z})\in\mathbb{C}^{+n}$ for any $\bm{z}\in
X$ and any indexing set $B\subseteq B^{\prime}$. Furthermore, by the variable
non-dependence property 3.1, it holds that

$\Psi_{B}(\mathfrak{i}\,\bm{1},\bm{z})=\Psi_{B\setminus
B^{\prime}}(\mathfrak{i}\,\bm{1},\bm{z})$

for any $\bm{z}\in X$ and any indexing set $B$ where $B\not\subseteq
B^{\prime}$. Hence,

$f(\bm{z})=\sum_{\begin{subarray}{c}B\subseteq\\{1,\ldots,n\\}\\\
B\neq\emptyset\wedge B\subseteq
B^{\prime}\end{subarray}}(-1)^{|B|+1}\overline{f(\Psi_{B}(\mathfrak{i}\,\bm{1},\bm{z}))}+\sum_{\begin{subarray}{c}B\subseteq\\{1,\ldots,n\\}\\\
B\not\subseteq
B^{\prime}\end{subarray}}(-1)^{|B|+1}\overline{f(\Psi_{B\setminus
B^{\prime}}(\mathfrak{i}\,\bm{1},\bm{z}))}.$

We now claim that the second sum is always equal to zero. Indeed, if
$|B^{\prime}|=n$, there is nothing left to prove. Otherwise, we may assume
that $|B^{\prime}|<n$, where we claim that there is a way to "pair up" the
indexing sets in the second sum in such a way that the two sets in each pair
only differ by one element in $B^{\prime}$. We construct this pairing in the
following way. Let $j_{1}$ be the smallest index in $B^{\prime}$. Then,
exactly half of the sets $B\subseteq\\{1,\ldots,n\\}$ that are not subsets of
$B^{\prime}$ contain the index $j_{1}$ and exactly half of them do not contain
the index $j_{1}$. This follows from the general observation that exactly half
of the subsets of a given set contain a specific element of the set. An
indexing set $B_{1}$ is then paired with the indexing set
$B_{1}\cup\\{j_{1}\\}$. In this case, $(B_{1}\cup\\{j_{1}\\})\setminus
B^{\prime}=B_{1}\setminus B^{\prime}$ and

$(-1)^{|B_{1}|+1}\overline{f(\Psi_{B_{1}\setminus
B^{\prime}}(\mathfrak{i}\,\bm{1},\bm{z}))}+(-1)^{|B_{1}\cup\\{j_{1}\\}|+1}\overline{f(\Psi_{(B_{1}\cup\\{j_{1}\\})\setminus
B^{\prime}}(\mathfrak{i}\,\bm{1},\bm{z}))}\\\
=(-1)^{|B_{1}|+1}\overline{f(\Psi_{B_{1}\setminus
B^{\prime}}(\mathfrak{i}\,\bm{1},\bm{z}))}-(-1)^{|B_{1}|+1}\overline{f(\Psi_{B_{1}\setminus
B^{\prime}}(\mathfrak{i}\,\bm{1},\bm{z}))}=0,$

yielding the desired result. ∎

Using Proposition 3.2, we may now given an analytic characterization of the
symmetric extension of a Herglotz-Nevanlinna function.

######  Theorem 3.3.

Let $f\colon(\mathbb{C}\setminus\mathbb{R})^{n}\to\mathbb{C}$ be a holomorphic
function such that

$\lim\limits_{z_{j}\>\scriptsize{\xrightarrow{\vee\>}}\>\infty}\frac{f(\bm{z})}{z_{j}}=\lim\limits_{z_{j}\>\scriptsize{\xrightarrow{\wedge\>}}\>\infty}\frac{f(\bm{z})}{z_{j}}=0$

for all indices $j\in\\{1,\ldots,n\\}$. Then $f=h_{\mathrm{sym}}$ for some
Herglotz-Nevanlinna function $h$ if and only if

  * (i)

it holds that $\mathrm{Im}[f(\bm{z})]\geq 0$ for all
$\bm{z}\in\mathbb{C}^{+n}$,

  * (ii)

the function $f$ satisfies the symmetry formula (2.7),

  * (iii)

the function $f$ satisfies the variable non-dependence property 3.1.

###### Proof.

If $f=h_{\mathrm{sym}}$ for some Herglotz-Nevanlinna function $h$, then this
function must have $\bm{b}=\bm{0}$ due to the assumption on the growth of $f$.
Then, properties (i) – (iii) are satisfied by the previously known results
discussed in Section 2.2. Conversely, if we are given the function $f$
satisfies the properties (i) – (iii), we construct a Herglotz-Nevanlinna
function out the function $f$ by setting

$h:=f|_{\mathbb{C}^{+n}}.$

This function $h$ may then be symmetrically extended to
$(\mathbb{C}\setminus\mathbb{R})^{n}$. However, $f$ and $h_{\mathrm{sym}}$ are
now two holomorphic functions on $(\mathbb{C}\setminus\mathbb{R})^{n}$
satisfying the symmetry formula (2.7) and the variable non-dependence property
3.1 which, furthermore, agree on $\mathbb{C}^{+n}$. Therefore, by Proposition
3.2, they agree everywhere on $(\mathbb{C}\setminus\mathbb{R})^{n}$, as
desired. ∎

The assumption on the growth of the function $f$ may be slightly weakened,
but, to compensate, conditions (ii) and (iii) need to be slightly modified.

######  Corollary 3.4.

Let $f\colon(\mathbb{C}\setminus\mathbb{R})^{n}\to\mathbb{C}$ be a holomorphic
function such that

$\lim\limits_{z_{j}\>\scriptsize{\xrightarrow{\vee\>}}\>\infty}\frac{f(\bm{z})}{z_{j}}=\lim\limits_{z_{j}\>\scriptsize{\xrightarrow{\wedge\>}}\>\infty}\frac{f(\bm{z})}{z_{j}}=d_{j}\geq
0$

for all indices $j\in\\{1,\ldots,n\\}$. In particular, for a fixed
$j\in\\{1,\ldots,n\\}$, the above limits are assumed to be independent of the
values of the vector $\bm{z}\in(\mathbb{C}\setminus\mathbb{R})^{n}$ at the
non-$j$-th positions. Then $f=h_{\mathrm{sym}}$ for some Herglotz-Nevanlinna
function $h$ if and only if

  * (i)

it holds that $\mathrm{Im}[f(\bm{z})]\geq 0$ for all
$\bm{z}\in\mathbb{C}^{+n}$,

  * (ii’)

the function $\bm{z}\mapsto f(\bm{z})-\sum_{j=1}^{n}d_{j}z_{j}$ satisfies the
symmetry formula (2.7),

  * (iii’)

the function $\bm{z}\mapsto f(\bm{z})-\sum_{j=1}^{n}d_{j}z_{j}$ satisfies the
variable non-dependence property 3.1.

The three conditions on the function $f$ in Theorem 3.3 are independent of
each-other. To verify this, consider the following functions on
$(\mathbb{C}\setminus\mathbb{R})^{n}$.

######  Example 3.5.

Table 1 presents eight explicit functions defined on
$(\mathbb{C}\setminus\mathbb{R})^{n}$ and Table 2 summarizes which conditions
of Theorem 3.3 are fulfilled by which function. Note also that all eight
functions satisfy the assumption on the growth of the function from Theorem
3.3. The functions are constructed as follows.

The functions $f_{0}$ is defined to equal: a negative imaginary constant on
$\mathbb{C}^{+}\times\mathbb{C}^{+}$, breaking condition (i); a function
depending only on the second variable on $\mathbb{C}^{-}\times\mathbb{C}^{+}$,
breaking condition (iii); and identically zero in the remaining connected
components of $(\mathbb{C}\setminus\mathbb{R})^{n}$, ensuring that condition
(ii) is not satisfied. The function $f_{1}$ is obtained form $f_{0}$ by
changing the definition on $\mathbb{C}^{+}\times\mathbb{C}^{+}$ to a positive
imaginary constant, thereby satisfying condition (i), but still neither (ii)
nor (iii).

The function $f_{2}$ is the Cauchy-type function given by a measure $\mu_{2}$
on $\mathbb{R}^{2}$ defined on Borel subsets $U\subseteq\mathbb{R}^{2}$ as

$\mu_{2}(U):=\pi\int_{\mathbb{R}}\chi_{U}(t,t)\mathrm{d}t,$

where $\chi$ denotes the characteristic function of a set. This measure
obviously satisfies the growth condition (2.3) and it does not satisfy the
Nevanlinna condition (2.6) as it supported on the diagonal in $\mathbb{R}^{2}$
\- an impossibility for Nevanlinna measures as shown in [15, Ex. 3.14]. This
function does not satisfy condition (i) as, for example,
$f_{2}(4\,\mathfrak{i},4\,\mathfrak{i})=-\frac{\mathfrak{i}}{10}$. As a
Cauchy-type function, it is guaranteed to satisfy condition (ii). It also
clearly does not satisfy condition (iii) as the values in e.g.
$\mathbb{C}^{+}\times\mathbb{C}^{-}$ depend explicitly on both variables. Note
now that while the function $f_{2}$ takes values with negative imaginary part
in $\mathbb{C}^{+}\times\mathbb{C}^{+}$, its imaginary part is bounded form
below. Indeed, the functions $z_{1}\mapsto-\frac{1}{\mathfrak{i}+z_{1}}$ and
$z_{2}\mapsto-\frac{1}{\mathfrak{i}+z_{2}}$ are Herglotz-Nevanlinna functions
of one variable, implying that
$\mathrm{Im}[f_{2}(z_{1},z_{2})]\geq-\frac{1}{2}$ for all
$(z_{1},z_{2})\in\mathbb{C}^{+}\times\mathbb{C}^{+}$. Hence, the function
$f_{4}$ is obtained by adding to the function $f_{2}$ the symmetric extension
of the Herglotz-Nevanlinna function $(z_{1},z_{2})\mapsto 5\,\mathfrak{i}$
(represented by the measure $5\lambda_{\mathbb{R}^{2}}$). This new function
now satisfies condition (i) in addition to (ii), while clearly still not
satisfying condition (iii). Note that the function

$f_{4}|_{\mathbb{C}^{+}\times\mathbb{C}^{+}}(z_{1},z_{2})=\frac{9\,\mathfrak{i}}{2}-\frac{1}{\mathfrak{i}+z_{1}}-\frac{1}{\mathfrak{i}+z_{2}}$

as a Herglotz-Nevanlinna function is not represented by the measure
$\mu_{2}+5\lambda_{\mathbb{R}^{2}}$ in the sense of Theorem 2.3, but rather by
the measure

$\genfrac{}{}{}{1}{9}{2}\lambda_{\mathbb{R}^{2}}+(\tau\mapsto(1+\tau^{2})^{-1})\lambda_{\mathbb{R}}\otimes\lambda_{\mathbb{R}}+\lambda_{\mathbb{R}}\otimes(\tau\mapsto(1+\tau^{2})^{-1})\lambda_{\mathbb{R}}.$

The function $f_{3}$ is defined as zero on all the connected components of
$(\mathbb{C}\setminus\mathbb{R})^{2}$ other than
$\mathbb{C}^{+}\times\mathbb{C}^{+}$ to ensure that it satisfies condition
(iii), while setting the function equal to a negative imaginary constant in
$\mathbb{C}^{+}\times\mathbb{C}^{+}$ ensures that it satisfies neither
condition (i) nor (ii). Changing this definition to a positive imaginary
constant in $\mathbb{C}^{+}\times\mathbb{C}^{+}$ gives the function $f_{5}$
which satisfies condition (i) and (iii), but not (ii).

The function $f_{7}$ is simply taken as the symmetric extension of a Herglotz-
Nevanlinna function, thereby satisfying all three properties automatically.
Finally, the function $f_{6}$ is chosen as $f_{6}:=-f_{7}$, satisfying
conditions (ii) and (iii), but not (i). $\lozenge$

| $\mathbb{C}^{+}\times\mathbb{C}^{+}$ | $\mathbb{C}^{-}\times\mathbb{C}^{+}$ | $\mathbb{C}^{+}\times\mathbb{C}^{-}$ | $\mathbb{C}^{-}\times\mathbb{C}^{-}$
---|---|---|---|---
$f_{0}$ | $-\mathfrak{i}$ | $\frac{1}{z_{2}}$ | $0$ | $0$
$f_{1}$ | $\mathfrak{i}$ | $\frac{1}{z_{2}}$ | $0$ | $0$
$f_{2}$ | $-\frac{\mathfrak{i}}{2}-\frac{1}{\mathfrak{i}+z_{1}}-\frac{1}{\mathfrak{i}+z_{2}}$ | $-\frac{\mathfrak{i}}{2}+\frac{1}{z_{2}-z_{1}}-\frac{1}{\mathfrak{i}+z_{2}}$ | $-\frac{\mathfrak{i}}{2}-\frac{1}{\mathfrak{i}+z_{1}}+\frac{1}{z_{1}-z_{2}}$ | $-\frac{\mathfrak{i}}{2}$
$f_{3}$ | $-\mathfrak{i}$ | $0$ | $0$ | $0$
$f_{4}$ | $\frac{9\,\mathfrak{i}}{2}-\frac{1}{\mathfrak{i}+z_{1}}-\frac{1}{\mathfrak{i}+z_{2}}$ | $-\frac{11\,\mathfrak{i}}{2}+\frac{1}{z_{2}-z_{1}}-\frac{1}{\mathfrak{i}+z_{2}}$ | $-\frac{11\,\mathfrak{i}}{2}-\frac{1}{\mathfrak{i}+z_{1}}+\frac{1}{z_{1}-z_{2}}$ | $-\frac{11\,\mathfrak{i}}{2}$
$f_{5}$ | $\mathfrak{i}$ | $0$ | $0$ | $0$
$f_{6}$ | $-\mathfrak{i}$ | $\mathfrak{i}$ | $\mathfrak{i}$ | $\mathfrak{i}$
$f_{7}$ | $\mathfrak{i}$ | $-\mathfrak{i}$ | $-\mathfrak{i}$ | $-\mathfrak{i}$
Table 1. Examples of eight functions defined on $(\mathbb{C}\setminus\mathbb{R})^{2}$. | (i) | (ii) | (iii)
---|---|---|---
$f_{0}$ | $\times$ | $\times$ | $\times$
$f_{1}$ | ✓ | $\times$ | $\times$
$f_{2}$ | $\times$ | ✓ | $\times$
$f_{3}$ | $\times$ | $\times$ | ✓
$f_{4}$ | ✓ | ✓ | $\times$
$f_{5}$ | ✓ | $\times$ | ✓
$f_{6}$ | $\times$ | ✓ | ✓
$f_{7}$ | ✓ | ✓ | ✓
Table 2. The relation of the eight functions from Table 1 to the three
conditions from Theorem 3.3.

##  4\. The Stieltjes inversion formula for Cauchy-type functions

For Herglotz-Nevanlinna functions, the Stieltjes inversion formula describes
how to reconstruct the representing measure $\mu$ of a Herglotz-Nevanlinna
function $h$ from the values of the imaginary part of the function in
$\mathbb{C}^{+n}$. More precisely, it holds that

$\int_{\mathbb{R}^{n}}\varphi(\bm{t})\mathrm{d}\mu(\bm{t})=\lim\limits_{\bm{y}\to\bm{0}^{+}}\int_{\mathbb{R}^{n}}\varphi(\bm{x})\mathrm{Im}[h(\bm{x}+\mathfrak{i}\,\bm{y})]\mathrm{d}\bm{x}$

for all $\mathcal{C}^{1}$-functions $\varphi\colon\mathbb{R}^{n}\to\mathbb{R}$
for which there exists a constant $D\geq 0$ such that $|\varphi(\bm{x})|\leq
D\prod_{j=1}^{n}(1+x_{j}^{2})^{-1}$ for all $\bm{x}\in\mathbb{R}^{n}$, see
e.g. [10] or [5, Lem. 4.1] for the case $n=1$ and [13, Cor. 4.6(viii)] for the
general case.

As noted in Section 2.2, Cauchy-type functions are a subclass of Herglotz-
Nevanlinna functions when $n=1$ and, hence, one only need the values of (the
imaginary part of) a Cauchy-type function in $\mathbb{C}^{+}$ to reconstruct
its measure. However, in Example 3.5, we have seen two different positive
Borel measures on $\mathbb{R}^{2}$ for which the corresponding Cauchy-type
functions agree on $\mathbb{C}^{+2}$, but not on the remaining connected
components of $(\mathbb{C}\setminus\mathbb{R})^{2}$.

The crucial role in the proof of the Stieltjes inversion formula is held by
the Poisson kernel of $\mathbb{C}^{+n}$, which, we recall, is defined for
$\bm{z}\in\mathbb{C}^{+n}$ and $\bm{t}\in\mathbb{R}^{n}$ as

$\mathcal{P}_{n}(\bm{z},\bm{t}):=\prod_{j=1}^{n}\frac{\mathrm{Im}[z_{j}]}{|t_{j}-z_{j}|^{2}}.$

Note that $\mathcal{P}_{n}(\bm{z},\bm{t})>0$ for every
$\bm{z}\in\mathbb{C}^{+n}$ and $\bm{t}\in\mathbb{R}^{n}$. The imaginary part
of the kernel $K_{n}$ is equal to the Poisson kernel $\mathcal{P}_{n}$ plus a
remainder term which can be expressed in terms of the $N_{j}$-factors [13,
Prop. 3.3] and the integral of the remainder with respect to any Nevanlinna
measure is zero.

The following lemma now shows how one can recover the value of the Poison
kernel $\mathcal{P}_{n}$ at some point $\bm{z}\in\mathbb{C}^{+n}$ (and
$\bm{t}\in\mathbb{R}^{n}$) using the values of kernel $K_{n}$ form all of the
connected components of the poly cut-plane
$(\mathbb{C}\setminus\mathbb{R})^{n}$.

######  Lemma 4.1.

Let $n\in\mathbb{N}$, $\bm{z}\in\mathbb{C}^{+n}$ and
$\bm{t}\in\mathbb{R}^{n}$. Then, it holds that

$2\,\mathfrak{i}\,\mathcal{P}_{n}(\bm{z},\bm{t})=\sum_{B\subseteq\\{1,\ldots,n\\}}(-1)^{|B|}K_{n}(\Psi_{B}(\bm{z},\bm{z}),\bm{t}),$

where $\Psi_{B}$ is the selective conjugation map from Section 2.1.

###### Proof.

The proof is done by induction on the dimension $n$. If $n=1$, then

$\sum_{B\subseteq\\{1\\}}(-1)^{|B|}K_{1}(\Psi_{B}(z,z),t)=K_{1}(\Psi_{\emptyset}(z,z),t)+(-1)K_{1}(\Psi_{\\{1\\}}(z,z),t)\\\
=K_{1}(z,t)-K_{1}(\overline{z},t)=2\,\mathfrak{i}\,\mathrm{Im}[K_{1}(z,t)]=2\,\mathfrak{i}\,\mathcal{P}_{1}(z,t),$

as desired.

Assume now that the statement of the lemma holds for all $n=1,2,\ldots,N-1$
for some $N\in\mathbb{N}$. For $n=N$, take $\bm{z}\in\mathbb{C}^{+N}$ and
$\bm{t}\in\mathbb{R}^{N}$ and let $\bm{z}^{\prime}$ and $\bm{t}^{\prime}$
denote the same vectors with the last component removed, i.e.
$\bm{z}^{\prime}:=(z_{1},\ldots,z_{N-1})$ and
$\bm{t}^{\prime}:=(t_{1},\ldots,t_{N-1})$. Furthermore, denote

$A(z,t):=\frac{1}{2\,\mathfrak{i}}\left(\frac{1}{t-z}-\frac{1}{t+\mathfrak{i}}\right).$

Then, we then calculate that

$\displaystyle\sum_{B\subseteq\\{1,\ldots,N\\}}(-1)^{|B|}K_{N}(\Psi_{B}(\bm{z},\bm{z}),\bm{t})$
---
| $\displaystyle=$ | $\displaystyle\sum_{\begin{subarray}{c}B\subseteq\\{1,\ldots,N\\}\\\ N\not\in B\end{subarray}}(-1)^{|B|}K_{N}(\Psi_{B}(\bm{z},\bm{z}),\bm{t})+\sum_{\begin{subarray}{c}B\subseteq\\{1,\ldots,N\\}\\\ N\in B\end{subarray}}(-1)^{|B|}K_{N}(\Psi_{B}(\bm{z},\bm{z}),\bm{t})$
| $\displaystyle=$ | $\displaystyle\sum_{B^{\prime}\subseteq\\{1,\ldots,N-1\\}}(-1)^{|B^{\prime}|}\bigg{[}\mathfrak{i}\bigg{(}2\prod_{j=1}^{N-1}A(\psi_{B^{\prime}}^{j}(z_{j},z_{j}),t_{j})\cdot A(z_{N},t_{N})-\prod_{j=1}^{N}A(\mathfrak{i},t_{j})\bigg{)}\bigg{]}$
|  | $\displaystyle+\sum_{B^{\prime}\subseteq\\{1,\ldots,N-1\\}}(-1)^{|B^{\prime}|+1}\bigg{[}\mathfrak{i}\bigg{(}2\prod_{j=1}^{N-1}A(\psi_{B^{\prime}}^{j}(z_{j},z_{j}),t_{j})\cdot A(\overline{z}_{N},t_{N})-\prod_{j=1}^{N}A(\mathfrak{i},t_{j})\bigg{)}\bigg{]}$
| $\displaystyle=$ | $\displaystyle 2\,\mathfrak{i}\,A(z_{N},t_{N})\sum_{B^{\prime}\subseteq\\{1,\ldots,N-1\\}}(-1)^{|B^{\prime}|}K_{N-1}(\Psi_{B^{\prime}}(\bm{z}^{\prime},\bm{z}^{\prime}),\bm{t}^{\prime})$
|  | $\displaystyle+\,\mathfrak{i}\,\prod_{j=1}^{N-1}A(\mathfrak{i},t_{j})\cdot(A(z_{N},t_{N})-A(\mathfrak{i},t_{N}))\cdot\overbrace{\sum_{B^{\prime}\subseteq\\{1,\ldots,N-1\\}}(-1)^{|B^{\prime}|}}^{=0}$
|  | $\displaystyle-2\,\mathfrak{i}\,A(\overline{z}_{N},t_{N})\sum_{B^{\prime}\subseteq\\{1,\ldots,N-1\\}}(-1)^{|B^{\prime}|}K_{N-1}(\Psi_{B^{\prime}}(\bm{z}^{\prime},\bm{z}^{\prime}),\bm{t}^{\prime})$
|  | $\displaystyle-\,\mathfrak{i}\,\prod_{j=1}^{N-1}A(\mathfrak{i},t_{j})\cdot(A(\overline{z}_{N},t_{N})-A(\mathfrak{i},t_{N}))\cdot\overbrace{\sum_{B^{\prime}\subseteq\\{1,\ldots,N-1\\}}(-1)^{|B^{\prime}|}}^{=0}$
| $\displaystyle=$ | $\displaystyle 2\,\mathfrak{i}\,A(z_{N},t_{N})\,\mathcal{P}_{N-1}(\bm{z}^{\prime},\bm{t}^{\prime})-2\,\mathfrak{i}\,A(\overline{z}_{N},t_{N})\,\mathcal{P}_{N-1}(\bm{z}^{\prime},\bm{t}^{\prime})=2\,\mathfrak{i}\,\mathcal{P}_{N}(\bm{z},\bm{t}),$

finishing the proof. ∎

The Stieltjes inversion for Cauchy-type functions is, thus, the following.

######  Theorem 4.2.

Let $g$ be a Cauchy-type function given by a measure $\mu$. Then, it holds
that

(4.1) $\int_{\mathbb{R}^{n}}\varphi(\bm{t})\mathrm{d}\mu(\bm{t})\\\
=\lim\limits_{\bm{y}\to\bm{0}^{+}}\frac{1}{2\,\mathfrak{i}}\int_{\mathbb{R}^{n}}\varphi(\bm{x})\bigg{[}\sum_{B\subseteq\\{1,\ldots,n\\}}(-1)^{|B|}g(\Psi_{B}(\bm{x}+\mathfrak{i}\,\bm{y},\bm{x}+\mathfrak{i}\,\bm{y}))\bigg{]}\mathrm{d}\bm{x}$

for all $\mathcal{C}^{1}$-functions $\varphi\colon\mathbb{R}^{n}\to\mathbb{R}$
for which there exists a constant $D\geq 0$ such that $|\varphi(\bm{x})|\leq
D\prod_{j=1}^{n}(1+x_{j}^{2})^{-1}$ for all $\bm{x}\in\mathbb{R}^{n}$.

###### Proof.

By the definition of Cauchy-type functions and Lemma 4.1, it holds that

$\sum_{B\subseteq\\{1,\ldots,n\\}}(-1)^{|B|}g(\Psi_{B}(\bm{x}+\mathfrak{i}\,\bm{y},\bm{x}+\mathfrak{i}\,\bm{y}))\\\
=\frac{1}{\pi^{n}}\int_{\mathbb{R}^{n}}\bigg{[}\sum_{B\subseteq\\{1,\ldots,n\\}}(-1)^{|B|}K_{n}(\Psi_{B}(\bm{x}+\mathfrak{i}\,\bm{y},\bm{x}+\mathfrak{i}\,\bm{y}),\bm{t})\bigg{]}\mathrm{d}\mu(\bm{t})\\\
=\frac{2\,\mathfrak{i}}{\pi^{n}}\int_{\mathbb{R}^{n}}\mathcal{P}_{n}(\bm{x}+\mathfrak{i}\,\bm{y},\bm{t})\mathrm{d}\mu(\bm{t}).$

Hence,

$\lim\limits_{\bm{y}\to\bm{0}^{+}}\frac{1}{2\,\mathfrak{i}}\int_{\mathbb{R}^{n}}\varphi(\bm{x})\bigg{[}\sum_{B\subseteq\\{1,\ldots,n\\}}(-1)^{|B|}g(\Psi_{B}(\bm{x}+\mathfrak{i}\,\bm{y},\bm{x}+\mathfrak{i}\,\bm{y}))\bigg{]}\mathrm{d}\bm{x}\\\
=\lim\limits_{\bm{y}\to\bm{0}^{+}}\frac{1}{\pi^{n}}\int_{\mathbb{R}^{n}}\varphi(\bm{x})\bigg{(}\int_{\mathbb{R}^{n}}\mathcal{P}_{n}(\bm{x}+\mathfrak{i}\,\bm{y},\bm{t})\mathrm{d}\mu(\bm{t})\bigg{)}\mathrm{d}\bm{x}\\\
=\lim\limits_{\bm{y}\to\bm{0}^{+}}\frac{1}{\pi^{n}}\int_{\mathbb{R}^{n}}\bigg{(}\int_{\mathbb{R}^{n}}\varphi(\bm{x})\mathcal{P}_{n}(\bm{x}+\mathfrak{i}\,\bm{y},\bm{t})\mathrm{d}\bm{x}\bigg{)}\mathrm{d}\mu(\bm{t}),$

where the assumptions on the function $\varphi$ and condition (2.3) for $\mu$
justify the use of Fubini’s theorem to change the order of integration. The
same assumptions permit for Lebesgue’s dominated convergence to be used,
allowing us to take the limit as $\bm{y}\to\bm{0}^{+}$ before integrating with
respect to the measure $\mu$. Noting that, by e.g. [11, pg. 111],

$\lim\limits_{\bm{y}\to\bm{0}^{+}}\int_{\mathbb{R}^{n}}\varphi(\bm{x})\mathcal{P}_{n}(\bm{x}+\mathfrak{i}\,\bm{y},\bm{t})\mathrm{d}\bm{x}=\pi^{n}\varphi(\bm{t})$

finishes the proof. ∎

As an immediate corollary of the previous theorem, we may now establish that
the correspondence between a Cauchy-type function and its defining measure
$\mu$ is, indeed, a bijection.

######  Corollary 4.3.

Let $\mu_{1},\mu_{2}$ be two positive Borel measures on $\mathbb{R}^{n}$
satisfying the growth condition (2.3). Then,

$\int_{\mathbb{R}^{n}}K_{n}(\bm{z},\bm{t})\mathrm{d}\mu_{1}(\bm{t})=\int_{\mathbb{R}^{n}}K_{n}(\bm{z},\bm{t})\mathrm{d}\mu_{2}(\bm{t})$

for all $\bm{z}\in(\mathbb{C}\setminus\mathbb{R})^{n}$ if and only if
$\mu_{1}\equiv\mu_{2}$.

## Acknowledgments

The author would like to thank Dale Frymark for many enthusiastic discussions
on the subject.

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# Van der Waals type PV diagrams of PNJL matter

Manjeet Seth1<EMAIL_ADDRESS>Kinkar Saha2<EMAIL_ADDRESS>Sudipa
Upadhaya3 Soumitra Maity4 Sabyasachi Ghosh1 1Indian Institute of Technology
Bhilai, GEC Campus, Sejbahar, Raipur - 492015, Chhattisgarh, India  2 Uluberia
College, Howrah, West Bengal, India, 711315  3 Ramsaday College, Howrah, West
Bengal, India, 711401  4 Bose Institute, Block-EN, Sector-V, Salt Lake City,
Kolkata, West Bengal, India, 700091

Study of the systems produced in high- energy heavy ion collision experiments
are of immense interest for various reasons. Here we intend to take an attempt
to discuss the presence of interactions in such systems and their effects to
distinguish them from the ideal scenario under the framework of Polyakov–
Nambu-Jona-Lasinio model. This model en- capsulates two very important aspects
of Quantum Chromo Dynamics(QCD), viz. the chiral and the confienement-
deconfinement phase transitions. With the help of $T$ and $\mu$ dependent
constituent quark mass $M(T,\mu)$, one can calculate net quark number density
$n(T,\mu)$, pressure $P(T,\mu)$:

$\displaystyle n(T,\mu)$ $\displaystyle=$ $\displaystyle
g\int\frac{d^{3}p}{(2\pi)^{3}}(f^{+}_{0}-f^{-}_{0})$ $\displaystyle P(T,\mu)$
$\displaystyle=$ $\displaystyle
g\int\frac{d^{3}p}{(2\pi)^{3}}\Big{(}\frac{p^{2}}{3E}\Big{)}(f^{+}_{0}+f^{-}_{0})~{},$
(1)

where $g=3\times 3\times 2=18$ is degeneracy factor of quark or anti-quark
with energy $E=\sqrt{p^{2}+M^{2}(T,\mu)}$ and
$f^{\pm}_{0}=1/\Big{\\{}e^{\beta(E\mp\mu)}+1\Big{\\}}$ for NJL model case and
for PNJL model, we have to replaced by modified distribution function BI_PNJL1
.

Figure 1: Pressure as a function of volume at various characteristic
temperatures by using PNJL model.

Now let us go for rough estimation number of hadrons or quarks, which can be
produced in heavy ion collision experiments with large range of
$\sqrt{s}=1-200$ GeV. Considering only pion production from this energy, one
can get rough no $\frac{(1-200)}{0.14}\approx 7-1,428$ pions and
$14-2,856\approx 10^{1-3}$. So if we want to restrict 100 net quark by
varrying volume from $V=1$ fm3 to $100$ fm3 with number density $n=100-1$/fm3.
Now for 3 temperatures $T=0.400$ GeV, $0.200$ GeV, $0.120$ GeV, fixing net
quark number as

$\displaystyle N=n(T,\mu)\times V$ $\displaystyle=$ $\displaystyle 100$
$\displaystyle Vg\int\frac{d^{3}p}{(2\pi)^{3}}(f^{+}_{0}-f^{-}_{0})$
$\displaystyle=$ $\displaystyle 100~{},$ (2)

Fig.(1) shows the dependence of Pressure, P on the volume, V of the system,
where V is obtained from Eqn.(2) as function of chemical potential, $\mu$. In
order to obtain the constituent masses from PNJL model BI_PNJL3 ; BI_PNJL4 ;
BI_PNJL5  and thereby the volume, V from Eqn.(2) at various characteristic
temperatures, we have varied $\mu$. The Pressure, P in Fig.(1) show non-
monotonic behaviour with V. The results are portrayers of the quantum level
interactions present in the system among the associated degrees of freedom.
Also the modified potential in Polyakov loop part, mimicking to some extent
the interactions present among gluonic degrees of freedom in Lattice QCD
BI_PNJL2 , also has a considerable effect [comparison with NJL]. From any one
isotherm below T=160 MeV, we can get the idea of the corresponding $\mu$ where
a transition happens indicating a critical point on the phase diagram, using
the V-$\mu$ relation. The transition point is signified by the minima present
in the corresponding isotherms. For T$>$160 MeV, any such minima are absent as
the system then resides entirely in the QGP region no matter what the chemical
potential is. This result is quite commensurate with the QCD phase diagram and
hence T$\sim$160MeV can be considered as the cross-over transition temperature
at $\mu=0$, as extracted from the model study. This agrees quite
satisfactorily with Lattice QCD results at continuum limit LQCD1 ; LQCD2  and
as predicted by our earlier study BI_PNJL2 ; BI_PNJL3 ; BI_PNJL4 ; BI_PNJL5 .

The results appear simpler to explain if we look for isotherms with reduced
Pressure, $P_{r}$ and reduced Volume, $V_{r}$ in Fig.(2) as was done by Van
Der Waal.

Figure 2: Van Der Waal like Plots

The presence of non-monotonicities below $T_{r}$=1 signifies the interactions
present causing transitions occuring in the system at particular chemical
potential for one characteristic temperature. These critical temperatures and
chemical potentials fall on the phase diagram shown in Ref. BI_PNJL2  for six-
quark interactions. Below $T_{r}$=1, the isotherms therefore represent various
states which could be unstable or metastable. Above $T_{r}$=1, the system
comprises of quarks only and hence possibility of phase transition does not
occur. Similar results had also been shown in Ref. NM , for normal nuclear
matter with corrections from Quantum Statistical effects.

## References

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