# On Positivity Preservers with constant Coefficients and their Generators

Philipp J. di Dio  Department of Mathematics and Statistics, University of
Konstanz, Universitätsstraße 10, D-78464 Konstanz, Germany  Zukunftskolleg,
Universtity of Konstanz, Universitätsstraße 10, D-78464 Konstanz, Germany
<EMAIL_ADDRESS>
###### Abstract

In this work we study positivity preservers
$T:\mathds{R}[x_{1},\dots,x_{n}]\to\mathds{R}[x_{1},\dots,x_{n}]$ with
constant coefficients and define their generators $A$ if they exist, i.e.,
$\exp(A)=T$. We use the theory of regular Fréchet Lie groups to show the first
main result. A positivity preserver with constant coefficients has a generator
if and only if it is represented by an infinitely divisible measure (4.37). In
the second main result (4.41) we use the Lévy–Khinchin formula to fully
characterize the generators of positivity preservers with constant
coefficients.

###### keywords:

positivity preserver, generator, constant coefficient, moments

###### MSC:

[2020] Primary 44A60, 47A57, 15A04; Secondary 12D15, 45P05, 47B38.

††journal: ArXiv

##  1 Introduction

Non-negative polynomials are widely applied and studied. A first step to
investigate non-negative polynomials is to study the convex cone

$\mathrm{Pos}(K):=\\{p\in\mathds{R}[x_{1},\dots,x_{n}]\,|\,p(x)\geq 0\
\text{for all}\ x\in K\\}$

of non-negative polynomials on some $K\subseteq\mathds{R}^{n}$.

The second logical step is to study (linear) maps

$T:\mathds{R}[x_{1},\dots,x_{n}]\to\mathds{R}[x_{1},\dots,x_{n}]$

between polynomials, especially when they preserve non-negativity, i.e.,

$T\mathrm{Pos}(K)\subseteq\mathrm{Pos}(K),$

see e.g. [9, 16, 4]. With the multi-index notation
$\partial^{\alpha}=\partial_{1}^{\alpha_{1}}\cdots\partial_{n}^{\alpha_{n}}$
for all $\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathds{N}_{0}^{n}$ with
$n\in\mathds{N}$ the following is long known. An explicit proof can e.g. be
found in [16].

######  Lemma 1.1 (folklore, see e.g. [16, Lem. 2.3]).

Let $n\in\mathds{N}_{0}$ and let

$T:\mathds{R}[x_{1},\dots,x_{n}]\to\mathds{R}[x_{1},\dots,x_{n}]$

be linear. Then for all $\alpha\in\mathds{N}_{0}^{n}$ there exist unique
$q_{\alpha}\in\mathds{R}[x_{1},\dots,x_{n}]$ such that

$T=\sum_{\alpha\in\mathds{N}_{0}^{n}}q_{\alpha}\cdot\partial^{\alpha}.$

The map $T:\mathds{R}[x_{1},\dots,x_{n}]\to\mathds{R}[x_{1},\dots,x_{n}]$ is
called to have constant coefficients if $q_{\alpha}\in\mathds{R}$ for all
$\alpha\in\mathds{N}_{0}^{n}$. The subset with $q_{0}=1$ is denoted by

$\mathfrak{D}:=\left\\{\sum_{\alpha\in\mathds{N}_{0}^{n}}q_{\alpha}\cdot\partial^{\alpha}\,\middle|\,q_{\alpha}\in\mathds{R},\
q_{0}=1\right\\}\quad\subsetneq\mathds{R}[[\partial_{1},\dots,\partial_{n}]].$

Positivity preservers are fully described by their polynomial coefficients. We
have the following.

######  Theorem 1.2 (see e.g. [4, Thm. 3.1]).

Let $n\in\mathds{N}$ and let
$T:\mathds{R}[x_{1},\dots,x_{n}]\to\mathds{R}[x_{1},\dots,x_{n}]$ be linear.
Then the following are equivalent:

  1. (i)

$T\mathrm{Pos}(\mathds{R}^{n})\subseteq\mathrm{Pos}(\mathds{R}^{n})$.

  2. (ii)

$(\alpha!\cdot q_{\alpha}(y))_{\alpha\in\mathds{N}_{0}^{n}}$ is a moment
sequence for all $y\in\mathds{R}^{n}$.

######  Example 1.3.

Let $n=1$. Then

$\exp(t\cdot\partial_{x}^{2})=\sum_{k\in\mathds{N}_{0}}\frac{t^{k}\cdot\partial_{x}^{2k}}{k!}$

is a positivity preserver for all $t\geq 0$. $\circ$

The following is a linear operator $T$ which is not a positivity preserver.

######  Example 1.4.

Let $k\geq 3$ and $a\in\mathds{R}\setminus\\{0\\}$. Then

$\exp(a\partial_{x}^{k}):=\sum_{j\in\mathds{N}_{0}}\frac{a^{j}\cdot\partial_{x}^{j\cdot
k}}{j!}$

is not a positivity preserver since $q_{2k}=a^{2k}\neq 0$ but $q_{2k+2}=0$,
i.e., $(j!\cdot q_{j})_{j\in\mathds{N}_{0}}$ is not a moment sequence. $\circ$

From elementary calculations and measure theory we get the following
additional properties of positivity preservers. Let $T$ be a positivity
preserver with constant coefficients and $\mu$ be a representing measure of
the corresponding moment sequence
$s=(s_{\alpha})_{\alpha\in\mathds{N}_{0}^{n}}=(\alpha!\cdot
q_{\alpha})_{\alpha\in\mathds{N}_{0}^{n}}$, then

$(Tf)(x)=\int_{\mathds{R}^{n}}f(x+y)\leavevmode\nobreak\ \mathrm{d}\mu(y)$

for all $f\in\mathds{R}[x_{1},\dots,x_{n}]$. For short we call $\mu$ also a
_representing measure of $T$_. If follows for two such operators $T$ and
$T^{\prime}$ with representing measures $\mu$ and $\mu^{\prime}$ that

$(TT^{\prime}f)(x)=\int f(x+y)\leavevmode\nobreak\
\mathrm{d}(\mu*\mu^{\prime})(y)$ (1)

where $\mu*\mu^{\prime}$ is the _convolution of the two measures_ $\mu$ and
$\mu^{\prime}$:

$(\mu*\mu^{\prime})(A):=\int_{\mathds{R}^{n}\times\mathds{R}^{n}}\chi_{A}(x+y)\leavevmode\nobreak\
\mathrm{d}\mu(x)\leavevmode\nobreak\ \mathrm{d}\mu^{\prime}(y)\\\
=\int_{\mathds{R}^{n}}\mu(A-y)\leavevmode\nobreak\
\mathrm{d}\mu^{\prime}(y)=\int_{\mathds{R}^{n}}\mu^{\prime}(A-x)\leavevmode\nobreak\
\mathrm{d}\mu(x)$ (2)

for any $A\in\mathfrak{B}(\mathds{R}^{n})$, see e.g. [3, Sect. 3.9]. Here,
$\chi_{A}$ is the characteristic function of the Borel set $A$. For the
supports we have

$\mathrm{supp}\,(\mu*\mu^{\prime})=\mathrm{supp}\,\mu+\mathrm{supp}\,\mu^{\prime}.$
(3)

This can easily be proved from (2) or found in the literature, see e.g. [7,
Prop. 14.5 (ii)] for a special case. We abbreviate

$\mu^{*k}:=\underbrace{\mu*\dots*\mu}_{k\text{-times}}$

for all $k\in\mathds{N}$ and set $\mu^{*0}:=\delta_{0}$ with $\delta_{0}$ the
Dirac measure centered at $0$.

Previously we investigated the heat semi-group (see 1.3 for $n=1$) and its
action on (non-negative) polynomials, see [5, 6]. For $n\in\mathds{N}$ we have
that $\exp(t\Delta)p_{0}$ is the unique solution of the polynomial valued heat
equation

$\partial_{t}p=\Delta p$

with initial values $p_{0}\in\mathds{R}[x_{1},\dots,x_{n}]$. In [6] we
observed the very strange behavior that several non-negative polynomials which
are not sums of squares (e.g. the Motzkin and the Choi–Lam polynomial) become
a sum of squares in finite time under the heat equation. In [6, Thm. 3.20] we
showed that every non-negative $p\in\mathds{R}[x,y,z]_{\leq 4}$ becomes a sum
of squares in finite time. All these observations hold for the heat equation,
i.e., the family $(\exp(t\Delta))_{t\geq 0}$ of positivity preservers with
constant coefficients with the generator
$\Delta=\partial_{1}^{2}+\dots+\partial_{n}^{2}$. So a natural question to
further study these effects is to understand any family $(\exp(tA))_{t\geq 0}$
of positivity preservers and hence to determine all possible generators $A$.
This is the main question of the current work for the constant coefficient
case. It is answered in 4.41.

The paper is structured as follows. To define and work with $\exp(A)$ we
repeat in the preliminaries for the readers convenience the notion of Lie
groups, Lie algebras, and especially their lesser known infinite dimensional
versions of regular Fréchet Lie groups. We will see that $\mathfrak{D}$ is a
regular Fréchet Lie group with a Lie algebra $\mathfrak{d}$. By 1.2 we
transport this property to sequences in Section 3. Section 4 contains the two
main results. In the first 4.37 we show that a positivity preserver with
constant coefficients has a generator if and only if it is represented by an
infinitely divisible measure. Using the Lévy–Khinchin formula (which is also
given in the preliminary, see 2.22) we give in the second 4.41 the full
description of all generators of positivity preservers with constant
coefficients. In Section 5 we discuss a strange action on non-negative
polynomials on $[0,\infty)$ caused by the heat equation with Dirichlet
boundary conditions. We end this paper with a summary and an open question.

##  2 Preliminaries

Lie groups and Lie algebras are standard concepts in mathematics [28].
However, this only applies to the finite dimensional cases. For the readers
convenience we give here the infinite dimensional definitions and examples we
need to make the paper as self-contained as possible. For the sake of
completeness we also include the explicit statement of the Lévy–Khinchin
formula in 2.22.

###  2.1 Lie Groups and their Lie Algebras

######  Definition 2.5.

A group $(G,\,\cdot\,)$ is called a _Lie group_ if $G$ is also a
$n$-dimensional smooth manifold, $n\in\mathds{N}$, such that $G\times G\to
G:(A,B)\mapsto AB^{-1}$ is smooth. The _Lie algebra $\mathfrak{g}$ of $G$_ is
the tangent space $T_{e}G$ at the identity $e\in G$.

The connection between the Lie algebra $\mathfrak{g}$ and the Lie group $G$ is
given by the _exponential map_ $\exp:\mathfrak{g}\to G$. For the special case
$G=\mathrm{Gl}(n,\mathds{C})$ the exponential mapping fulfills the following.

######  Lemma 2.6 (see e.g. [28, p. 134, Ex. 15]).

Let $n\in\mathds{N}$. The following hold:

  1. (i)

The exponential map

$\exp:\mathrm{Gl}(n,\mathds{C})\to\mathrm{Gl}(n,\mathds{C}),\quad A\mapsto\exp
A:=\sum_{k\in\mathds{N}_{0}}\frac{A^{k}}{k!}$

is surjective.

  2. (ii)

Let $\mathrm{id}+N\in\mathrm{Gl}(n,\mathds{C})$ be such that $N$ is nilpotent.
Then

$\log(\mathrm{id}+N)=-\sum_{k\in\mathds{N}}\frac{(-N)^{k}}{k}$

is well-defined and

$\exp(\log(\mathrm{id}+N))=\mathrm{id}+N$

holds.

The proof of 2.6 (ii) follows from formal power series calculations. For more
on Lie groups and Lie algebras see e.g. [28].

The charts $\varphi:U\subseteq\mathds{R}^{n}\to G$ of the $n$-dimensional
smooth manifold $G$ induce the Euclidean topology on the group $G$ and
$(A,B)\mapsto AB^{-1}$ is smooth with respect to this topology. When extending
this to infinite dimensions more than one topology is possible and the choice
of topology on $G$ is important.

###  2.2 Fréchet Spaces

######  Definition 2.7.

A topological vector space $\mathcal{V}$ is called a _Fréchet space_ if the
following three conditions are fulfilled:

  1. (i)

$\mathcal{V}$ is metrizable (i.e., $\mathcal{V}$ is Hausdorff),

  2. (ii)

$\mathcal{V}$ is complete, and

  3. (iii)

$\mathcal{V}$ is locally convex.

It is clear that $\mathds{R}[x_{1},\dots,x_{n}]_{\leq d}$ is a Fréchet space
for all $d\in\mathds{N}_{0}$ since they are finite dimensional. Their Fréchet
topology is unique.

######  Example 2.8 (see e.g. [27, pp. 91–92, Ex. III]).

Let $n\in\mathds{N}$. The vector space $\mathds{R}[[x_{1},\dots,x_{n}]]$ of
formal power series

$p=\sum_{\alpha\in\mathds{N}_{0}^{n}}c_{\alpha}\cdot
x^{\alpha}\qquad\text{with}\ c_{\alpha}\in\mathds{R}$

equipped with the topology induced by the semi-norms

$|p|_{d}:=\sup_{|\alpha|\leq d}|c_{\alpha}|\qquad\text{with}\
d\in\mathds{N}_{0}$

is a Fréchet space. In other words we have the convergence

$p_{i}=\sum_{\alpha\in\mathds{N}_{0}^{n}}c_{i,\alpha}\cdot
x^{\alpha}\quad\to\quad p=\sum_{\alpha\in\mathds{N}_{0}}c_{\alpha}\cdot
x^{\alpha}\qquad\text{for}\ i\to\infty$

if and only if $c_{i,\alpha}\xrightarrow{i\to\infty}c_{\alpha}$ for all
$\alpha\in\mathds{N}_{0}^{n}$. $\circ$

For more on topological vector spaces see e.g. [27, 26].

###  2.3 Regular Fréchet Lie Groups and their Lie Algebras

Already Hideki Omori stated the following, see [17, pp. III-IV]:

> [G]eneral Fréchet manifolds are very difficult to treat. For instance, there
> are some difficulties in the definition of tangent bundles, hence in the
> definition of the concept of $C^{\infty}$-mappings. Of course, there is
> neither an implicit function theorem nor a Frobenius theorem in general.
> Thus, it is difficult to give a theory of general Fréchet Lie groups.

A more detailed study is given by Omori in [18] and the theory successfully
evolved since then, see e.g. [17, 12, 18, 25, 29, 23] and references therein.
We will give here only the basic definitions which will be needed for our
study.

For the definition of $C^{1}(G,\mathfrak{g})$ see e.g. [18, pp. 9–10]. Since
$\mathds{R}[[\partial_{1},\dots,\partial_{n}]]$ has the Fréchet topology in
2.8, i.e., the coordinate-wise convergence, we have that for every
$m\in\mathds{N}_{0}$ a function
$F:\mathds{R}\to\mathds{R}[[\partial_{1},\dots,\partial_{n}]]$,
$F(t)=\sum_{\alpha\in\mathds{N}_{0}^{n}}F_{\alpha}(t)\cdot\partial^{\alpha}$
is $C^{m}$ if and only if every coordinate $F_{\alpha}$ is $C^{m}$.

######  Definition 2.9 (see e.g. [18, p. 63, Dfn. 1.1]).

We call $(G,\,\cdot\,)$ a (_regular_) _Fréchet Lie group_ if the following
conditions are fulfilled:

  1. (i)

$G$ is an infinite dimensional smooth Fréchet manifold.

  2. (ii)

$(G,\,\cdot\,)$ is a group.

  3. (iii)

The map $G\times G\to G$, $(A,B)\mapsto A\cdot B^{-1}$ is smooth.

  4. (iv)

The _Fréchet Lie algebra_ $\mathfrak{g}$ of $G$ is isomorphic to the tangent
space $T_{e}G$ of $G$ at the unit element $e\in G$.

  5. (v)

$\exp:\mathfrak{g}\to G$ is a smooth mapping such that

$\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\exp(tu)=u$

holds for all $u\in\mathfrak{g}$.

  6. (vi)

The space $C^{1}(G,\mathfrak{g})$ of $C^{1}$-curves in $G$ coincides with the
set of all $C^{1}$-curves in $G$ under the Fréchet topology.

For more on infinite dimensional manifolds, differential calculus, Lie groups,
and Lie algebras see e.g. [15, 18, 23].

###  2.4 The Lie Group $\mathfrak{D}_{d}$

######  Definition 2.10.

Let $n\in\mathds{N}$ and $d\in\mathds{N}_{0}$. We define
$\mathfrak{D}_{d}:=\mathfrak{D}|_{\mathds{R}[x_{1},\dots,x_{n}]_{\leq d}}$.

Since

$A\mathds{R}[x_{1},\dots,x_{n}]_{\leq
d}\subseteq\mathds{R}[x_{1},\dots,x_{n}]_{\leq d}$

holds for all $d\in\mathds{N}_{0}$ and $A\in\mathfrak{D}$ the
$\mathfrak{D}_{d}$ are well-defined. From 2.10 we see that $\mathfrak{D}_{d}$
consists only of operators of the form

$\sum_{\alpha\in\mathds{N}_{0}^{n}:|\alpha|\leq
d}c_{\alpha}\cdot\partial^{\alpha}$

with $c_{\alpha}\in\mathds{R}$ and $c_{0}=1$ since on
$\mathds{R}[x_{1},\dots,x_{n}]_{\leq d}$ every operator $\partial^{\beta}$
with $|\beta|>d$ fulfills $\partial^{\beta}p=0$ for all
$p\in\mathds{R}[x_{1},\dots,x_{n}]_{\leq d}$, i.e., $\partial^{\beta}=0$ on
$\mathds{R}[x_{1},\dots,x_{n}]_{\leq d}$.

######  Remark 2.11.

We can also define $\mathfrak{D}_{d}$ by
$\mathfrak{D}/\langle\partial^{\alpha}\,|\,|\alpha|=d+1\rangle$. Both
definitions are almost identical. However, 2.10 has the following advantage.
In $\mathfrak{D}/\langle\partial^{\alpha}\,|\,|\alpha|=d+1\rangle$ we have the
problem that we are working with equivalence classes and hence we can not
calculate $A+B$ for $A\in\mathfrak{D}_{d}$ and $B\in\mathfrak{D}_{e}$ for
$d\neq e$. With 2.10 we can calculate $A+B$ for $A\in\mathfrak{D}_{d}$ and
$B\in\mathfrak{D}_{e}$ with $d\neq e$ since $A+B$ is defined on

$\mathrm{dom}\,(A+B)=\mathrm{dom}\,A\cap\mathrm{dom}\,B=\mathds{R}[x_{1},\dots,x_{n}]_{\leq\min(d,e)}$

as usual for (unbounded) operators [21]. 2.10 can then even be used to
calculate $A+B$ for $A$ on $\mathds{R}[x_{1},\dots,x_{n}]_{\leq d}$ and $B$ on
$\mathds{R}[x_{1},\dots,x_{m}]_{\leq e}$ for $n\neq m$ and $d\neq e$ on

$\mathrm{dom}\,(A+B)=\mathds{R}[x_{1},\dots,x_{\min\\{n,m\\}}]_{\leq\min\\{d,e\\}}.$

######  Example 2.12.

Let $n=1$ and $d=3$. Then

$\mathfrak{D}_{3}=\left\\{1+c_{1}\partial_{x}+c_{2}\partial_{x}^{2}+c_{3}\partial_{x}^{3}\;\middle|\;c_{1},c_{2},c_{3}\in\mathds{R}\right\\}\quad\text{on}\quad\mathds{R}[x]_{\leq
3}.$

Let

$A=1+a_{1}\partial_{x}+a_{2}\partial_{x}^{2}+a_{3}\partial_{x}^{3}\quad\text{and}\quad
B=1+b_{1}\partial_{x}+b_{2}\partial_{x}^{2}+b_{3}\partial_{x}^{3}$

be in $\mathfrak{D}_{3}$. Then

$\begin{split}AB&=BA=(1+a_{1}\partial_{x}+a_{2}\partial_{x}^{2}+a_{3}\partial_{x}^{3})\cdot(1+b_{1}\partial_{x}+b_{2}\partial_{x}^{2}+b_{3}\partial_{x}^{3})\\\
&=1+(a_{1}+b_{1})\partial_{x}+(a_{2}+a_{1}b_{1}+b_{2})\partial_{x}^{2}+(a_{3}+a_{2}b_{1}+a_{1}b_{2}+b_{3})\partial_{x}^{3}\end{split}$
(3)

since derivatives $\partial^{i}$ with $i\geq 4$ are the zero operators on
$\mathds{R}[x]_{\leq 3}$. Hence, $(\mathfrak{D}_{3},\,\cdot\,)$ is a
commutative semi-group with neutral element $\mathds{1}=1$.

We will now see that $\mathfrak{D}_{3}$ is even a commutative group. For that
it is sufficient to find for any $A\in\mathfrak{D}_{3}$ a
$B\in\mathfrak{D}_{3}$ with $AB=\mathds{1}$. By (3) $AB=\mathds{1}$ is
equivalent to

$\displaystyle 0$ $\displaystyle=a_{1}+b_{1}$ $\displaystyle\Rightarrow\quad
b_{1}=-a_{1}$ $\displaystyle 0$ $\displaystyle=a_{2}+a_{1}b_{1}+b_{2}$
$\displaystyle\Rightarrow\quad b_{2}=-a_{2}+a_{1}^{2}$ $\displaystyle 0$
$\displaystyle=a_{3}+a_{2}b_{1}+a_{1}b_{2}+b_{3}$
$\displaystyle\Rightarrow\quad b_{3}=-a_{3}+2a_{2}a_{1}-a_{1}^{3},$

i.e., every $A\in\mathfrak{D}_{3}$ has the unique inverse

$A^{-1}=1-a_{1}\partial_{x}+(-a_{2}+a_{1}^{2})\partial_{x}^{2}+(-a_{3}+2a_{2}a_{1}-a_{1}^{3})\partial_{x}^{3}\quad\in\mathfrak{D}_{3}.$

Hence, $(\mathfrak{D}_{3},\,\cdot\,)$ is a commutative group. $\circ$

We have seen in the previous example that $(\mathfrak{D}_{d},\,\cdot\,)$ for
$n=1$ and $d=3$ is a commutative group. This holds for all $n\in\mathds{N}$
and $d\in\mathds{N}_{0}$.

######  Lemma 2.13.

Let $n\in\mathds{N}$ and $d\in\mathds{N}_{0}$. Then
$(\mathfrak{D}_{d},\,\cdot\,)$ is a commutative group.

###### Proof.

Let $A=\sum_{\alpha:|\alpha|\leq
d}a_{\alpha}\partial^{\alpha},B=\sum_{\beta:|\beta|\leq
d}b_{\beta}\partial^{\beta}\in\mathfrak{D}_{d}$, i.e., $a_{0}=b_{0}=1$. Then

$AB=C=\sum_{\gamma:|\gamma|\leq d}c_{\gamma}\cdot\partial^{\gamma}$

holds with

$c_{\gamma}=\sum_{\alpha,\beta\in\mathds{N}_{0}^{n}:\alpha+\beta=\gamma}a_{\alpha}b_{\beta}.$
(4)

Let
$\alpha=(\alpha_{1},\dots,\alpha_{n})\succeq\beta=(\beta_{1},\dots,\beta_{n})$
on $\mathds{N}_{0}^{n}$ if and only if $\alpha_{i}\geq\beta_{i}$ for all
$i=1,\dots,n$. Then (4) can be solved by induction on $|\gamma|$. For
$|\gamma|=0$ we have $c_{0}=a_{0}\cdot b_{0}$ and $a_{0}=c_{0}=1$, i.e.,
$b_{0}=1$. So assume (4) is solved for all $c_{\gamma}$ with $|\gamma|\leq
d-1$. Then for any $\gamma\in\mathds{N}_{0}$ with $|\gamma|=d$ we have

$b_{\gamma}=a_{0}b_{\gamma}=-\sum_{\alpha\in\mathds{N}_{0}\setminus\\{0\\}:\gamma\succeq\alpha}a_{\alpha}\cdot
b_{\gamma-\alpha},$ (5)

i.e., the system (4) of equations has a unique solution gained by induction.
Hence, for every $A\in\mathfrak{D}_{d}$ there exists a unique
$B\in\mathfrak{D}_{d}$ with $AB=BA=\mathds{1}$. ∎

From 2.13 we have seen that $(\mathfrak{D}_{d},\,\cdot\,)$ for any
$n\in\mathds{N}$ and $d\in\mathds{N}_{0}$ is a commutative group. Let

$\iota_{d}:\\{1\\}\times\mathds{R}^{\binom{n+d}{n}-1}\to\mathfrak{D}_{d},\quad(a_{\alpha})_{\alpha\in\mathds{N}_{0}^{n}:|\alpha|\leq
d}\mapsto\sum_{\alpha\in\mathds{N}_{0}^{n}:|\alpha|\leq
d}a_{\alpha}\cdot\partial^{\alpha}$ (6)

be an affine linear map. Then $\iota_{d}$ in (6) a diffeomorphism and it is a
coordinate map for $\mathfrak{D}_{d}$. The smooth manifold
$\\{1\\}\times\mathds{R}^{\binom{n+d}{n}-1}$ inherits the group structure of
$\mathfrak{D}_{d}$ through $\iota_{d}$, i.e.,

$(\mathfrak{D}_{d},\,\cdot\,)\;\overset{\iota_{d}}{\cong}\;\left(\\{1\\}\times\mathds{R}^{\binom{n+d}{n}-1},\odot\right)$
(7)

Hence, the map $\iota_{d}$ shows the following.

######  Theorem 2.14.

Let $n\in\mathds{N}$ and $d\in\mathds{N}_{0}$. Then
$(\mathfrak{D}_{d},\,\cdot\,)$ is a Lie group.

###### Proof.

The map $\iota_{d}$ in (6) is a diffeomorphism between $\mathfrak{D}_{d}$ and
$\\{1\\}\times\mathds{R}^{\binom{n+d}{n}-1}$. Hence, $\mathfrak{D}_{d}$ is a
differentiable manifold which possesses the group structure
$(\mathfrak{D}_{d},\,\cdot\,)$. By (4) and (5) we have that the map

$\mathfrak{D}_{d}\times\mathfrak{D}_{d}\to\mathfrak{D}_{d},\quad(A,B)\mapsto
AB^{-1}$

is smooth. Hence, $(\mathfrak{D}_{d},\,\cdot\,)$ is a commutative Lie group. ∎

###  2.5 The Lie Algebra $\mathfrak{d}_{d}$ of $\mathfrak{D}_{d}$

Since every $A\in\mathfrak{D}_{d}$ is a linear map

$A:\mathds{R}[x_{1},\dots,x_{n}]_{\leq
d}\to\mathds{R}[x_{1},\dots,x_{n}]_{\leq d}$

between finite-dimensional vector spaces we can choose a basis of
$\mathds{R}[x_{1},\dots,x_{n}]_{\leq d}$ and get a matrix representation
$\tilde{A}$ of $A$. Take the monomial basis of
$\mathds{R}[x_{1},\dots,x_{n}]_{\leq d}$. Then $\tilde{A}$ is an upper
triangular matrix with diagonal entries $1$.

######  Example 2.15 (2.12 continued).

Let $n=1$ and $d=3$. Then every
$A=1+a_{1}\partial_{x}+a_{2}\partial_{x}^{2}+a_{3}\partial_{x}^{3}\in\mathfrak{D}_{3}$
has with the monomial basis $\\{1,x,x^{2},x^{3}\\}$ of $\mathds{R}[x]_{\leq
3}$ the matrix representation

$\tilde{A}=\begin{pmatrix}1&a_{1}&2a_{2}&6a_{3}\\\ 0&1&2a_{1}&6a_{2}\\\
0&0&1&3a_{1}\\\ 0&0&0&1\end{pmatrix}$

and we therefore set

$\tilde{\mathfrak{D}}_{3}:=\left\\{\begin{pmatrix}1&a_{1}&2a_{2}&6a_{3}\\\
0&1&2a_{1}&6a_{2}\\\ 0&0&1&3a_{1}\\\
0&0&0&1\end{pmatrix}\,\middle|\,a_{1},a_{2},a_{3}\in\mathds{R}\right\\}.$

Hence, $(\tilde{A}-\mathrm{id})^{4}=0$ as a matrix and also
$(A-\mathds{1})^{4}=0$ as an operator on $\mathds{R}[x]_{\leq 3}$. From 2.6 we
find that the matrix valued exponential map

$\exp:\mathrm{gl}(4,\mathds{C})\to\mathrm{Gl}(4,\mathds{C}),\quad\tilde{A}\mapsto\exp(\tilde{A}):=\sum_{k\in\mathds{N}_{0}}\frac{\tilde{A}^{k}}{k!}$

is surjective and the logarithm

$\tilde{A}\mapsto\log\tilde{A}:=-\sum_{k\in\mathds{N}}\frac{(\mathrm{id}-\tilde{A})^{k}}{k}$

is well-defined for all $\mathrm{id}+N\in\mathrm{Gl}(4,\mathds{C})$ with $N$
nilpotent. Since $\tilde{\mathfrak{D}}_{3}\subseteq\mathrm{Gl}(4,\mathds{C})$
with $(\mathrm{id}-\tilde{A})^{4}=0$ for all
$\tilde{A}\in\tilde{\mathfrak{D}}_{3}$ we have

$\log:\tilde{\mathfrak{D}}_{3}\to\mathrm{gl}(4,\mathds{C}),\tilde{A}\mapsto\log\tilde{A}=-\sum_{k=1}^{3}\frac{(\mathrm{id}-\tilde{A})^{k}}{k}.$
(8)

Since also $(A-\mathds{1})^{4}=0$ for all $A\in\mathfrak{D}_{3}$ we can use
(8) also for the differential operators in $\mathfrak{D}_{3}$:

$\log:\mathfrak{D}_{3}\to\left\\{d_{0}+d_{1}\partial_{x}+d_{2}\partial_{x}^{2}+d_{3}\partial_{x}^{3}\,\middle|\,d_{0},\dots,d_{3}\in\mathds{R}\right\\},\quad
A\mapsto-\sum_{k=1}^{3}\frac{(\mathds{1}-A)^{k}}{k}.$

To determine the image $\log\mathfrak{D}_{3}$ recall that also $\log$ is an
injective map by 2.6 and hence $\log\mathfrak{D}_{3}$ is $3$-dimensional with
$d_{0}=0$, i.e., we have

$\log\mathfrak{D}_{3}=\left\\{d_{1}\partial_{x}+d_{2}\partial_{x}^{2}+d_{3}\partial_{x}^{3}\,\middle|\,d_{1},d_{2},d_{3}\in\mathds{R}\right\\}=:\mathfrak{d}_{3}.$

In summary, since $A^{4}=0$ for all $A\in\mathfrak{d}_{3}$ we have that

$\exp:\mathfrak{d}_{3}\to\mathfrak{D}_{3},\quad
A\mapsto\sum_{k=0}^{3}\frac{A^{k}}{k!}$ (9)

is surjective with inverse

$\log:\mathfrak{D}_{3}\to\mathfrak{d}_{3},\quad
A\mapsto-\sum_{k=1}^{3}\frac{(\mathds{1}-A)^{k}}{k}.$ (10)

Therefore, $\mathfrak{d}_{3}$ is the Lie algebra of $\mathfrak{D}_{3}$ and
$\exp$ in (9) is the exponential map between the Lie algebra
$\mathfrak{d}_{3}$ and its Lie group $\mathfrak{D}_{3}$ with inverse $\log$ in
(10). $\circ$

The previous example of the Lie algebra $\mathfrak{d}_{3}$ of the Lie group
$\mathfrak{D}_{3}$ holds for all $n\in\mathds{N}$ and $d\in\mathds{N}_{0}$. We
define the following.

######  Definition 2.16.

Let $n\in\mathds{N}$ and $d\in\mathds{N}_{0}$. We define

$\mathfrak{d}_{d}:=\left\\{\sum_{\alpha\in\mathds{N}_{0}^{n}\setminus\\{0\\}:|\alpha|\leq
d}d_{\alpha}\cdot\partial^{\alpha}\,\middle|\,d_{\alpha}\in\mathds{R}\
\text{for all}\ \alpha\in\mathds{N}_{0}^{n}\setminus\\{0\\}\ \text{with}\
|\alpha|\leq d\right\\}.$

It is clear that $(\mathfrak{d}_{d},\,\cdot\,,+)$ is an algebra on
$\mathds{R}[x_{1},\dots,x_{n}]_{\leq d}$ and we have the following.

######  Theorem 2.17.

Let $n\in\mathds{N}$ and $d\in\mathds{N}_{0}$. Then
$(\mathfrak{d}_{d},\,\cdot\,,+)$ is the Lie algebra of the Lie group
$(\mathfrak{D}_{d},\,\cdot\,)$ with exponential map

$\exp:\mathfrak{d}_{d}\to\mathfrak{D}_{d},\quad
A\mapsto\sum_{k=0}^{d}\frac{A^{k}}{k!}$

with inverse

$\log:\mathfrak{D}_{d}\to\mathfrak{d}_{d},\quad
A\mapsto-\sum_{k=1}^{d}\frac{(\mathds{1}-A)^{k}}{k}.$

###### Proof.

Follows from 2.6 similar to 2.15. ∎

###  2.6 The Regular Fréchet Lie Group $\mathfrak{D}$ and its Lie Algebra
$\mathfrak{d}$

In Section 2.4 and 2.5 we have seen that $(\mathfrak{D}_{d},\,\cdot\,)$ is a
Lie group with Lie algebra $(\mathfrak{d}_{d},\,\cdot\,,+)$ for all
$d\in\mathds{N}_{0}$. Hence, similar to 2.16 we define the following.

######  Definition 2.18.

Let $n\in\mathds{N}$. We define

$\mathfrak{d}:=\left\\{\sum_{\alpha\in\mathds{N}_{0}^{n}\setminus\\{0\\}}d_{\alpha}\cdot\partial^{\alpha}\,\middle|\,d_{\alpha}\in\mathds{R}\
\text{for all}\ \alpha\in\mathds{N}_{0}^{n}\setminus\\{0\\}\right\\}.$

For $\mathfrak{D}$ and $\mathfrak{d}$ we have the following.

######  Theorem 2.19.

Let $n\in\mathds{N}$. Then the following hold:

  1. (i)

$(\mathfrak{d},\,\cdot\,,+)$ is a commutative algebra.

  2. (ii)

$(\mathfrak{D},\,\cdot\,)$ is a commutative group.

  3. (iii)

The map

$\exp:\mathfrak{d}\to\mathfrak{D},\quad
A\mapsto\sum_{k\in\mathds{N}_{0}}\frac{A^{k}}{k!}$

is bijective.

  4. (iv)

The map

$\log:\mathfrak{D}\to\mathfrak{d},\quad
A\mapsto-\sum_{k\in\mathds{N}}\frac{(\mathds{1}-A)^{k}}{k}$

is bijective.

  5. (v)

The maps $\exp:\mathfrak{d}\to\mathfrak{D}$ and
$\log:\mathfrak{D}\to\mathfrak{d}$ are inverse to each other.

###### Proof.

(i): That is clear.

(ii): That $A\cdot B=B\cdot A$ holds for all $A,B\in\mathfrak{D}$ is clear.
The inverse of $A\in\mathfrak{D}$ is uniquely determined by solving (4) to get
(5) for all $\gamma\in\mathds{N}_{0}^{n}$. This is a formal power series
argument with coordinate-wise convergence (as in the Fréchet topology, 2.8).

(iii): At first we show that $\exp:\mathfrak{d}\to\mathfrak{D}$ is well-
defined. To see this note that for any $A\in\mathfrak{d}$ we have

$A^{k}=\sum_{\alpha\in\mathds{N}_{0}^{n}:|\alpha|\geq
k}c_{\alpha}\cdot\partial^{\alpha},$

i.e., $A^{k}$ contains no differential operators of order $\leq k-1$. Hence,
the sum

$\sum_{k=0}^{K}\frac{A^{k}}{k!}=\sum_{\alpha\in\mathds{N}_{0}^{n}}c_{K,\alpha}\cdot\partial^{\alpha}$

converges coefficient-wise to

$\exp
A=\sum_{k\in\mathds{N}_{0}}\frac{A^{k}}{k!}=\sum_{\alpha\in\mathds{N}_{0}^{n}}c_{\alpha}\cdot\partial^{\alpha},$

i.e., in the Fréchet topology of
$\mathfrak{D}\subsetneq\mathds{R}[[\partial_{1},\dots,\partial_{n}]]\cong\mathds{R}[[x_{1},\dots,x_{n}]]$,
see 2.8. In other words, the coefficients $c_{\alpha}$ depend only on $A^{k}$
for $k=0,\dots,|\alpha|$, we therefore have $c_{K,\alpha}=c_{\alpha}$ for all
$K>|\alpha|$, and hence $\exp A\in\mathfrak{D}$ is well-defined. With that we
have $\exp\mathfrak{d}\subseteq\mathfrak{D}$. For equality we give the inverse
map in (v).

(iv): To show that $\log:\mathfrak{D}\to\mathfrak{d}$ is well-defined the same
argument as in (iii) holds for $(\mathds{1}-A)^{k}$ with $A\in\mathfrak{D}$.
It shows that $\log A\in\mathfrak{d}$ for all $A\in\mathfrak{D}$ is well-
defined and we have $\log\mathfrak{D}\subseteq\mathfrak{d}$.

(v): To prove that $\exp$ and $\log$ are inverse to each other we remark

$\mathds{R}[x_{1},\dots,x_{n}]=\bigcup_{d\in\mathds{N}_{0}}\mathds{R}[x_{1},\dots,x_{n}]_{\leq
d}.$

Define

$\exp_{d}A:=\sum_{k=0}^{d}\frac{A^{k}}{k!}\quad\text{and}\quad\log_{d}A:=-\sum_{k=1}^{d}\frac{(\mathds{1}-A)^{k}}{k}.$

Then for every $p\in\mathds{R}[x_{1},\dots,x_{n}]$ with $d=\deg p$ we have

$\exp(\log A)p=\exp_{d}(\log_{d}A)p=Ap$

for all $A\in\mathfrak{d}$ by 2.17, i.e., $\exp(\log A)=A$ for all
$A\in\mathfrak{D}$. Similarly, we have $\log(\exp A)p=Ap$ for all
$A\in\mathfrak{d}$. This also shows the remaining assertions
$\exp(\mathfrak{d})=\mathfrak{D}$ and $\log\mathfrak{D}=\mathfrak{d}$ from
(iii) and (iv). ∎

For
$\mathds{R}[[\partial_{1},\dots,\partial_{n}]]\supseteq\mathfrak{d},\mathfrak{D}$
being a Fréchet space with the before mentioned topology (of coordinate-wise
convergence, 2.8) we have the following.

######  Corollary 2.20.

Let $n\in\mathds{N}$ and let
$\mathfrak{d},\mathfrak{D}\subseteq\mathds{R}[[\partial_{1},\dots,\partial_{n}]]$
be Fréchet spaces (equipped with the coordinate-wise convergence). Then the
following hold:

  1. (i)

$\mathfrak{D}\times\mathfrak{D}\to\mathfrak{D}$, $(A,B)\mapsto AB^{-1}$ is
smooth.

  2. (ii)

$\exp:\mathfrak{d}\to\mathfrak{D}$ is smooth and

$\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\exp(tu)=u$

holds for all $u\in\mathfrak{g}$.

  3. (iii)

$\log:\mathfrak{d}\to\mathfrak{D}$ is smooth.

###### Proof.

(i): Let $A=\sum_{\alpha\in\mathds{N}_{0}^{n}}a_{\alpha}\partial^{\alpha}$ and
$B=\sum_{\alpha\in\mathds{N}_{0}^{n}}b_{\alpha}\partial^{\alpha}$ with
$a_{0}=b_{0}=1$. From (4) we see that the multiplication is smooth since every
coordinate $c_{\gamma}$ of the product
$AB=\sum_{\alpha\in\mathds{N}_{0}^{n}}c_{\alpha}\partial^{\alpha}$ is a
polynomial in $a_{\alpha}$ and $b_{\alpha}$ with $|\alpha|\leq|\gamma|$. The
inverse $B^{-1}=\sum_{\alpha\in\mathds{N}_{0}^{n}}d_{\alpha}\partial^{\alpha}$
is smooth because of (5), i.e., also the coefficients $d_{\gamma}$ of the
inverse depend polynomially on the coefficients $b_{\alpha}$ with
$|\alpha|\leq|\gamma|$.

(ii): In the proof of 2.19 (iii) we have already seen that the coefficients
$c_{\gamma}$ of $\exp
A=\sum_{\alpha\in\mathds{N}_{0}^{n}}c_{\alpha}\partial^{\alpha}$ depend
polynomially on the coefficients $a_{\alpha}$ of
$A=\sum_{\alpha\in\mathds{N}_{0}^{n}\setminus\\{0\\}}a_{\alpha}\partial^{\alpha}$
with $|\alpha|\leq|\gamma|$.

The condition

$\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\exp(tu)=u$

then follows by direct calculations.

(iii): Follows like (ii) from 2.19 (iv). ∎

It is easy to see that $\mathfrak{d}$ and $\mathfrak{D}$ are both infinite
dimensional smooth (Fréchet) manifolds. Hence, summing everything up we have
the following.

######  Theorem 2.21.

Let $n\in\mathds{N}$. Then $(\mathfrak{D},\,\cdot\,)$ as a Fréchet space is a
commutative regular Fréchet Lie group with the commutative Fréchet Lie algebra
$(\mathfrak{d},\,\cdot\,,+)$. The exponential map

$\exp:\mathfrak{d}\to\mathfrak{D},\quad
A\mapsto\sum_{k\in\mathds{N}_{0}}\frac{A^{k}}{k!}$

is smooth and bijective with the smooth and bijective inverse

$\log:\mathfrak{D}\to\mathfrak{d},\quad
A\mapsto-\sum_{k\in\mathds{N}}\frac{(\mathds{1}-A)^{k}}{k}.$

###### Proof.

We have that $\mathfrak{D}$ is an infinite dimensional smooth manifold,
$\mathfrak{D}$ is a Fréchet space (with the coefficient-wise convergence
topology, 2.8) and by 2.19 (ii) we also have that $(\mathfrak{D},\,\cdot\,)$
is a commutative group. By 2.20 (i) we have that $(A,B)\mapsto AB^{-1}$ is
continuous in the Fréchet topology. Hence, $(\mathfrak{D},\,\cdot\,)$ is an
infinite dimensional commutative Fréchet Lie group.

The properties about $\exp$ and $\log$ are 2.19 (iii)–(v).

We now prove the regularity condition (vi) in 2.9. Let
$F:\mathds{R}\to\mathfrak{D}$ be a $C^{1}$-differentiable function, i.e.,

$\lim_{n\to\infty}\left(F\\!\left(t+\frac{s}{n}\right)\cdot F(t)^{-1}\right)$
(11)

converges uniformly on each compact interval to a one-parameter subgroup

$\exp(sf(t))$

where $f:\mathds{R}\to\mathfrak{d}$ is the derivative $\dot{F}(t)$ of $F(t)$
at $t\in\mathds{R}$, see [18, p. 10]. But we can take the logarithm of $F$

$\tilde{f}(t):=\log F(t)$

for all $t\in\mathds{R}$ to see that $\tilde{f}$ is $C^{1}$ since $\log$ is
smooth by 2.20 (iii). By 2.20 (ii) we have $f=\tilde{f}$. Hence,
$C^{1}(\mathfrak{D},\mathfrak{d})$ coincides with the set of all
$C^{1}$-curves in $\mathfrak{D}$ under the Fréchet topology of $\mathfrak{D}$.
∎

In the previous proof we can also replace (11) by the fact that a function
$F:\mathds{R}\to\mathds{R}[[\partial_{1},\dots,\partial_{n}]]$, $t\mapsto
F(t)=\sum_{\alpha\in\mathds{N}_{0}^{n}}F_{\alpha}(t)\cdot\partial^{\alpha}$ is
$C^{m}$ for some $m\in\mathds{N}_{0}$ if and only if every coordinate function
$F_{\alpha}$ is $C^{m}$.

###  2.7 Lévy–Khinchin Formula

A measure $\mu$ on $\mathds{R}^{n}$ is called _divisible by $k\in\mathds{N}$_
if there exists a measure $\nu$ such that $\mu=\nu^{*k}$. A measure $\mu$ on
$\mathds{R}^{n}$ is called _infinitely divisible_ if it is divisible by any
$k\in\mathds{N}$. Infinitely divisible measures are fully characterized by the
_Lévy–Khinchin formula_.

######  Theorem 2.22 (Lévy–Khinchin, see e.g. [13, Cor. 15.8] or [14, Satz
16.17]).

Let $n\in\mathds{N}$ and $\mu$ be a measure on $\mathds{R}^{n}$. The following
are equivalent:

  1. (i)

$\mu$ is infinitely divisible.

  2. (ii)

There exist a vector $b\in\mathds{R}^{n}$, a symmetric matrix
$\Sigma\in\mathds{R}^{n\times n}$ with $\Sigma\succeq 0$, and a measure $\nu$
on $\mathds{R}^{n}$ such that

$\log\int e^{itx}\leavevmode\nobreak\
\mathrm{d}\mu(x)=itb-\frac{1}{2}t^{T}\Sigma
t+\int(e^{itx}-1-itx\cdot\chi_{\|x\|_{2}<1})\leavevmode\nobreak\
\mathrm{d}\mu(x)$

holds for the characteristic function of $\mu$.

##  3 The Regular Fréchet Lie Group Structure of Sequences

We define appropriate sets
$\mathfrak{S},\mathfrak{s}\subsetneq\mathds{R}^{\mathds{N}_{0}^{n}}$ of
sequences with convolution $*$. From these definitions we will see that the
set of sequences $\mathds{R}^{\mathds{N}_{0}^{n}}$, and therefore also
$\mathfrak{S}$ and $\mathfrak{s}$, inherit the Fréchet topology (2.8) from
$\mathds{R}[[x_{1},\dots,x_{n}]]$.

######  Definition 3.23.

Let $n\in\mathds{N}$. We define

$\mathfrak{S}:=\left\\{(s_{\alpha})_{\alpha\in\mathds{N}_{0}^{n}}\in\mathds{R}^{\mathds{N}_{0}^{n}}\,\middle|\,s_{0}=1\right\\}\quad\text{and}\quad\mathfrak{s}:=\left\\{(s_{\alpha})_{\alpha\in\mathds{N}_{0}^{n}}\in\mathds{R}^{\mathds{N}_{0}^{n}}\,\middle|\,s_{0}=0\right\\}.$

Define the linear and bijective map

$D:\mathds{R}^{\mathds{N}_{0}^{n}}\to\mathds{R}[[\partial_{1},\dots,\partial_{n}]],\quad
s=(s_{\alpha})_{\alpha\in\mathds{N}_{0}^{n}}\mapsto
D(s):=\sum_{\alpha\in\mathds{N}_{0}^{n}}\alpha!\cdot
s_{\alpha}\cdot\partial^{\alpha}$ (12)

and on $\mathds{R}^{\mathds{N}_{0}^{n}}$ the convolution $*$ as

$*:\mathds{R}^{\mathds{N}_{0}^{n}}\times\mathds{R}^{\mathds{N}_{0}^{n}}\to\mathds{R}^{\mathds{N}_{0}^{n}},\quad(s,t)\mapsto
s*t:=D^{-1}(D(s)D(t)).$

We abbreviate

$s^{0}=\mathds{1}:=(\delta_{0,\alpha})_{\alpha\in\mathds{N}_{0}^{n}}\qquad\text{and}\qquad
s^{*k}:=\underbrace{s*\cdots*s}_{k\text{-times}}$

for all $k\in\mathds{N}$ and $s\in\mathds{R}^{\mathds{N}_{0}^{n}}$.

The map $D$ in (12) is of course a Fréchet space isomorphism. The map $D$ also
equips $\mathfrak{S}$ and $\mathfrak{s}$ with the Fréchet topology of
$\mathfrak{D}$ and $\mathfrak{d}$, respectively. In summary, we have the
following.

######  Corollary 3.24.

Let $n\in\mathds{N}$. Then the following hold:

  1. (i)

$s*t\in\mathfrak{S}$ holds for all $s,t\in\mathfrak{S}$.

  2. (ii)

$D|_{\mathfrak{S}}:(\mathfrak{S},*)\to(\mathfrak{D},\,\cdot\,)$ is a Fréchet
group isomorphisms.

  3. (iii)

$s*t\in\mathfrak{s}$ and $s+t\in\mathfrak{s}$ hold for all
$s,t\in\mathfrak{s}$.

  4. (iv)

$D|_{\mathfrak{s}}:(\mathfrak{s},*,+)\to(\mathfrak{d},\,\cdot\,,+)$ is a
Fréchet algebra isomorphisms.

######  Theorem 3.25.

Let $n\in\mathds{N}$. Then $(\mathfrak{S},*)$ is a commutative regular Fréchet
Lie group with the commutative Lie algebra $(\mathfrak{s},*,+)$. The
exponential map

$\exp:\mathfrak{s}\to\mathfrak{S},\quad
s\mapsto\exp(s):=\sum_{k\in\mathds{N}_{0}}\frac{s^{*k}}{k!}$

is smooth and bijective with the smooth and bijective inverse

$\log:\mathfrak{S}\to\mathfrak{s},\quad
s\mapsto\log(s):=-\sum_{k\in\mathds{N}}\frac{(\mathds{1}-s)^{*k}}{k}.$

###### Proof.

From 3.24 we see that the map
$D:\mathds{R}^{\mathds{N}_{0}^{n}}\to\mathds{R}^{\mathds{N}_{0}^{n}}$ induces
the Fréchet Lie group isomorphism
$D|_{\mathfrak{S}}:(\mathfrak{S},*)\to(\mathfrak{D},\,\cdot\,)$ and the
Fréchet Lie algebra isomorphism
$D|_{\mathfrak{s}}:(\mathfrak{s},*,+)\to(\mathfrak{d},\,\cdot\,,+)$. Apply
both isomorphisms to 2.21 and the assertion is proved. ∎

We want to point out that Hirschman and Widder [11] extensively investigated
the inversion theory of convolution of measures and functions. But from 3.25
we see that the convolution of their moments is trivial when we allow signed
moment sequences.

For $*$ on $\mathfrak{S}$ we find from (2) the following.

######  Corollary 3.26.

Let $n\in\mathds{N}$. If $s\in\mathfrak{S}$ is represented by the signed
measure $\mu$ and $t\in\mathfrak{S}$ is represented by the signed representing
measure $\nu$ then $s*t$ is represented by the measure $\mu*\nu$.

Note, that every sequence $s\in\mathds{R}^{\mathds{N}_{0}^{n}}$ is represented
by a signed measure, see e.g. [19, 2, 24, 20]. It is even possible to restrict
the support to $[0,\infty)^{n}$, see e.g. [2, 24] or to use only linear
combinations of Dirac measures [1].

Because of the connection of positivity preservers to moment sequences by 1.2
we define the following.

######  Definition 3.27.

Let $n\in\mathds{N}$. We define

$\mathfrak{S}_{+}:=\\{s\in\mathfrak{S}\,|\,s\ \text{is a moment sequence}\\}.$

Then $\mathfrak{S}_{+}$ is a base of the moment cone.

######  Remark 3.28.

For a general moment sequence $s=(s_{\alpha})_{\alpha\in\mathds{N}_{0}^{n}}$
we only have $s_{0}>0$. But, scaling $\tilde{s}=s_{0}^{-1}\cdot
s\in\mathfrak{S}$ gives $s=\exp(\ln s_{0}+\log\tilde{s})$. $\circ$

The Fréchet topology on $\mathfrak{S}$ and 3.26 imply the following.

######  Corollary 3.29.

Let $n\in\mathds{N}$. The following hold:

  1. (i)

The set $\mathfrak{S}_{+}$ is convex and closed.

  2. (ii)

For all $s,t\in\mathfrak{S}_{+}$ we have $s*t\in\mathfrak{S}_{+}$.

###### Proof.

(i): Convexity is clear. It suffices to prove that $\mathfrak{S}_{+}$ is
closed in the Fréchet topology.

Let
$s^{(n)}=(s_{\alpha}^{(n)})_{\alpha\in\mathds{N}_{0}^{n}}\in\mathfrak{S}_{+}$
for all $n\in\mathds{N}_{0}$ be such that $s^{(n)}\to s$ in the Fréchet
topology (2.8), i.e., $s_{\alpha}^{(n)}\to s_{\alpha}$ for all
$\alpha\in\mathds{N}_{0}^{n}$. Define
$L_{s}:\mathds{R}[x_{1},\dots,x_{n}]\to\mathds{R}$ by
$L_{s}(x^{\alpha}):=s_{\alpha}$ for all $\alpha\in\mathds{N}_{0}^{n}$ and
extend it linearly to $\mathds{R}[x_{1},\dots,x_{n}]$. Let
$p\in\mathrm{Pos}(\mathds{R}^{n})$ then $0\leq L_{s^{(n)}}(p)\to L_{s}(p)\geq
0$ shows that $L_{s}$ is $\mathrm{Pos}(\mathds{R}^{n})$-positive, i.e., by
Haviland’s Theorem [10] $L_{s}$ is a moment functional and
$s\in\mathcal{S}_{+}$.

(ii): Since $s,t\in\mathfrak{S}_{+}$ we have that $s$ is represented by $\mu$
and $t$ is represented by $\nu$. Therefore, $s*t$ is represented by $\mu*\nu$
by 3.26. ∎

The following is another trivial consequence of (2).

######  Corollary 3.30.

Let $n\in\mathds{N}$ and $s,t\in\mathfrak{S}_{+}$. If $s$ or $t$ is
indeterminate then $s*t\in\mathfrak{S}_{+}$ is indeterminate.

##  4 Generators of Positivity Preservers with Constant Coefficients

In Section 2.6 we proved that $\mathfrak{D}$ is a Fréchet Lie group with
Fréchet Lie algebra $\mathfrak{d}$. With the smooth and bijective
$\exp:\mathfrak{d}\to\mathfrak{D}$ with the inverse
$\log:\mathfrak{D}\to\mathfrak{d}$ we can now easily go from positivity
preservers with constant coefficients to their generators.

######  Definition 4.31.

Let $n\in\mathds{N}$. We define the set

$\mathfrak{D}_{+}:=\\{A\in\mathfrak{D}\,|\,A\ \text{is a positivity
preserver}\\}$

of all _positivity preservers with constant coefficients_ and we define the
set

$\mathfrak{d}_{+}:=\left\\{A\in\mathfrak{d}\,\middle|\,\exp(tA)\in\mathfrak{D}_{+}\
\text{for all}\ t\geq 0\right\\}$

of all _generators of positivity preservers with constant coefficients_.

From 1.2 we see that
$D|_{\mathfrak{S}_{+}}:\mathfrak{S}_{+}\to\mathfrak{D}_{+}$ in (12) is an
isomorphism.

######  Examples 4.32.

Let $n\in\mathds{N}$. Then we have the following:

  1. (a)

$c\cdot\Delta=c\cdot(\partial_{1}^{2}+\dots+\partial_{n}^{2})\in\mathfrak{d}_{+}$
for all $c\geq 0$ since $\Delta$ generates the heat equation/kernel.

  2. (b)

$c_{1}\cdot\partial_{1}+\dots+c_{n}\cdot\partial_{n}\in\mathfrak{d}_{+}$ for
all $c_{1},\dots,c_{n}\in\mathds{R}$ since $\partial_{i}$ generates the
translation group in the direction $(c_{1},\dots,c_{n})\in\mathds{R}^{n}$.
$\circ$

######  Example 4.33.

Let $k\in\mathds{N}$ with $k\geq 3$. Then
$\partial_{i}^{k}\not\in\mathfrak{d}_{+}$, see 1.4. $\circ$

The following result shows that the cases in 4.32 are the only generators of
positivity preserves of finite rank.

######  Lemma 4.34.

Let $A=\sum_{j=1}^{k}a_{j}\partial_{x}^{j}\in\mathfrak{d}_{+}$. Then $k\leq
2$.

###### Proof.

Let $k\geq 3$ and $a_{k}=1$. By 1.4 (and 4.33) we have that
$\exp(\partial_{x}^{k})\not\in\mathfrak{D}_{+}$, i.e., it is not a positivity
preserver. Hence, there exists a $f_{0}\in\mathds{R}[x]$ with $f_{0}\geq 0$
and $x_{0}\in\mathds{R}$ such that $[\exp(\partial_{x}^{k})f_{0}](x_{0})=-1$.

Assume to the contrary that $A\in\mathfrak{d}_{+}$. By scaling $x$ and $A$ we
have that

$A_{\lambda}:=\sum_{j=1}^{k}\lambda^{k-j}a_{j}\partial_{x}^{j}\quad\in\mathfrak{d}_{+}$
(13)

holds for all $\lambda>0$. By 2.21 we have that
$[\exp(A_{\lambda})f_{0}](x_{0})$ is continuous in $\lambda$. Since
$(\exp(A_{0})f_{0})(x_{0})=-1$ there exists a $\lambda_{0}>0$ such that
$[\exp(A_{\lambda_{0}})f_{0}](x_{0})<0$, i.e., $A\not\in\mathfrak{d}_{+}$ and
therefore we have $k\leq 2$. ∎

It is easy to see that the previous result also holds for $n\geq 2$. To see
this let $\alpha\in\mathds{N}_{0}^{n}$ with $|\alpha|\geq 3$. Then from 1.2 it
follows that $\exp(\partial^{\alpha})\not\in\mathfrak{D}_{+}$. Choosing the
same scaling argument (13) we find $\partial^{\alpha}\not\in\mathfrak{d}_{+}$.

For $\mathfrak{D}_{+}$ and $\mathfrak{d}_{+}$ the following holds.

######  Corollary 4.35.

Let $n\in\mathds{N}$. Then the following hold:

  1. (i)

$\mathfrak{D}_{+}$ is a closed and convex set.

  2. (ii)

$\mathfrak{d}_{+}$ is a non-trivial, closed, and convex cone.

###### Proof.

(i): Follows from 3.29 (i) with the bijective and linear map $D$ from 3.23 eq.
(12).

(ii): For the non-triviality we have the non-trivial 4.32.

For the convexity let $A,B\in\mathfrak{d}_{+}$. Since
$(\mathfrak{d},\,\cdot\,,+)$ is a commutative algebra we have that
$\exp(t(A+B))=\exp(tA)\exp(tB)$ and since the product of two positivity
preservers is again a positivity preserver we have $A+B\in\mathfrak{d}_{+}$.

For the closeness let $A_{n}\in\mathfrak{d}_{+}$ for all $n\in\mathds{N}_{0}$
with $A_{n}\to A$ in the Fréchet topology of
$\mathds{R}[[\partial_{1},\dots,\partial_{n}]]$, see 2.8. By 3.25 we have that
$\exp:\mathfrak{d}\to\mathfrak{D}$ is smooth, i.e., especially continuous.
Hence,

$\mathfrak{D}_{+}\ni\exp(tA_{n})\;\to\;\exp(tA)\in\mathfrak{D}_{+}$

for all $t\geq 0$ since $\mathfrak{D}_{+}$ is closed by (i). Hence,
$A\in\mathfrak{d}_{+}$.

For the cone property let $A\in\mathfrak{d}_{+}$ then also
$cA\in\mathfrak{d}_{+}$ for all $c\geq 0$. ∎

######  Corollary 4.36.

Let $n\in\mathds{N}$ and $A\in\mathfrak{d}$. Then the following are
equivalent:

  1. (i)

$A\in\mathfrak{d}_{+}$.

  2. (ii)

The unique solution $p_{t}$ of

$\partial_{t}p=Ap$ (14)

for any initial value $p_{0}\in\mathrm{Pos}(\mathds{R}^{n})$ fulfills
$p_{t}\in\mathrm{Pos}(\mathds{R}^{n})$ for all $t\geq 0$.

###### Proof.

Since $p_{t}=\exp(tA)p_{0}$ is the unique solution of the time evolution (14)
we have that (i) $\Leftrightarrow$ $\exp(tA)$ is a positivity preserver for
all $t\geq 0$ $\Leftrightarrow$ (ii). ∎

While we have $\mathfrak{d}_{+}\subseteq\log\mathfrak{D}_{+}$ equality does
not hold as we will see in 4.40. The existence of a positivity preserver is
equivalent to the existence of an infinitely divisible representing measure as
the following result shows.

######  Main Theorem 4.37.

Let $n\in\mathds{N}$. Then the following are equivalent:

  1. (i)

$A\in\mathfrak{d}_{+}$.

  2. (ii)

$\exp A$ has an infinitely divisible representing measure.

  3. (iii)

$\exp(tA)$ has an infinitely divisible representing measure for some $t>0$.

  4. (iv)

$\exp(tA)$ has an infinitely divisible representing measure for all $t>0$.

###### Proof.

(i) $\Rightarrow$ (ii): Let $A\in\mathfrak{d}_{+}$, i.e.,
$\exp(tA)\in\mathfrak{D}_{+}$ has a representing measure $\mu_{t}$ for all
$t\in[0,\infty)$. Set $\nu_{k}:=(\mu_{1/k!})^{*k!}$. Then $\nu_{k}$ is a
representing measure of $\exp A$ for all $k\in\mathds{N}$. Since
$\mathds{R}[x_{1},\dots,x_{n}]$ is an adapted space
$(\nu_{k})_{k\in\mathds{N}}$ is vaguely compact by [22, Thm. 1.19] and there
exists a subsequence $(k_{i})_{i\in\mathds{N}}$ such that $\nu_{k_{i}}\to\nu$
and $\nu$ is a representing measure of $\exp A$.

It remains to show that $\nu$ is infinitely divisible, i.e., for every
$l\in\mathds{N}$ there exists a measure $\omega_{l}$ with
$\omega_{l}^{*l}=\nu$.

Let $l\in\mathds{N}$. For $i\geq l$ we define

$\omega_{l,i}:=(\mu_{1/k_{i}!})^{*k_{i}!/l}$

i.e., $\omega_{l,i}$ is a representing measure of $\exp(A/l)$. Again, since
$\mathds{R}[x_{1},\dots,x_{n}]$ is an adapted space by [22, Thm. 1.19] there
exists a subsequence $(i_{j})_{j\in\mathds{N}}$ such that $\omega_{l,i_{j}}$
converges to some $\omega_{l}$, i.e.,
$\omega_{l,i_{j}}\xrightarrow{j\to\infty}\omega_{l}$. Hence,

$(\omega_{l})^{*l}=\lim_{j\to\infty}(\omega_{l,i_{j}})^{*l}=\lim_{j\to\infty}\nu_{k_{i_{j}}}=\nu,$

i.e., $\nu$ is divisible by all $l\in\mathds{N}$ and hence $\nu$ is an
infinitely divisible representing measure of $\exp A$.

(ii) $\Rightarrow$ (i): Let $\mu_{1}$ be an infinitely divisible representing
measure of $\exp A$. Then

$\mu_{q}:=\mu_{1}^{*q}$

exists for all $q\in\mathds{Q}\cap[0,\infty)$ and it is a representing measure
of $\exp(qA)$, i.e., $\exp(qA)\in\mathfrak{D}_{+}$ for all
$q\in[0,\infty)\cap\mathds{Q}$. Since by 2.21
$\exp:\mathfrak{d}\to\mathfrak{D}$ is continuous and by 4.35 (i)
$\mathfrak{D}_{+}$ is closed we have that $\exp(qA)\in\mathfrak{D}_{+}$ for
all $q\geq 0$. Hence, we have $A\in\mathfrak{d}_{+}$.

(iv) $\Rightarrow$ (iii): Clear.

(iii) $\Rightarrow$ (i): By “(ii) $\Leftrightarrow$ (i)” we have that
$qtA\in\mathfrak{d}_{+}$ for $t>0$ and all $q\in[0,\infty)\cap\mathds{Q}$.
Since $\mathfrak{d}_{+}$ is closed by 4.35 (ii) we have $q_{i}tA\to
A\in\mathfrak{d}_{+}$ for $q_{i}\in\mathds{Q}$ with $q_{i}\to t^{-1}$ as
$i\to\infty$.

(i) $\Rightarrow$ (iv): Since $A\in\mathfrak{d}_{+}$ and $\mathfrak{d}_{+}$ is
a closed convex cone by 4.35 (ii) we have that $tA\in\mathfrak{d}_{+}$ for all
$t>0$ and hence by “(i) $\Leftrightarrow$ (ii)” we have that $\exp(tA)$ has an
infinitely divisible representing measure for all $t>0$. ∎

######  Example 4.38 (Example 4.32 (b) continued).

Let $n=1$ and $a\in\mathds{R}$. Then

$\exp(r\partial_{x})=\sum_{k\in\mathds{N}_{0}}\frac{a^{k}}{k!}\cdot\partial_{x}^{k}$

is represented by $\mu=\delta_{a}$ since $\delta_{a}$ is the representing
measure of the moment sequence $(a^{k})_{k\in\mathds{N}_{0}}$. For any $r>0$
we have $\delta_{a/r}^{*r}=\delta_{a}$, i.e., $\delta_{a}$ is infinitely
divisible. In fact, $\delta_{a}$ are the only compactly supported infinite
divisible measures, see e.g. [14, p. 316]. Hence, by 4.37 we have
$\partial_{x}\in\mathfrak{d}_{+}$. $\circ$

######  Example 4.39.

Let $A\in\mathfrak{D}_{+}$ be the positivity preserver represented by the
measure

$\mathrm{d}\mu=\chi_{[0,1]^{n}}\leavevmode\nobreak\ \mathrm{d}\lambda$

where $\lambda$ is the $n$-dimensional Lebesgue measure and $\chi_{[0,1]^{n}}$
is the characteristic function of $[0,1]^{n}$. Since $\mathrm{supp}\,\mu$ is
compact $\mu$ is unique.

It is known that the only infinitely divisible measures with compact support
are $\delta_{x}$ for $x\in\mathds{R}^{n}$, see e.g. [14, p. 316]. Therefore,
we have that $\mu$ is not infinitely divisible and hence $\log
A\not\in\mathfrak{d}_{+}$. $\circ$

The previous example implies that the inclusion
$\mathfrak{d}_{+}\subseteq\log\mathfrak{D}_{+}$ is proper.

######  Corollary 4.40.

Let $n\in\mathds{N}$. Then $\mathfrak{d}_{+}\subsetneq\log\mathfrak{D}_{+}$.

###### Proof.

We have $\log\mathfrak{D}_{+}\setminus\mathfrak{d}_{+}\neq\emptyset$ by 4.39.
∎

We have seen in 4.37 the one-to-one correspondence between a positivity
preserver $A\in\mathfrak{D}_{+}$ having an infinitely divisible representing
measure and $A\in\mathfrak{D}_{+}$ having a generator. The infinitely
divisible measures are fully characterized by the Lévy–Khinchin formula, see
2.22. The Lévi–Khinchin formula is used in the following result to fully
characterize the generators $\mathfrak{d}_{+}$ of the positivity preservers
$\mathfrak{D}_{+}$.

######  Main Theorem 4.41.

Let $n\in\mathds{N}$. The following are equivalent:

  1. (i)

$A=\sum_{\alpha\in\mathds{N}_{0}^{n}\setminus\\{0\\}}\frac{a_{\alpha}}{\alpha!}\cdot\partial^{\alpha}\in\mathfrak{d}_{+}$.

  2. (ii)

There exists a symmetric matrix
$\Sigma=(\sigma_{i,j})_{i,j=1}^{n}\in\mathds{R}^{n}$ with $\Sigma\succeq 0$, a
vector $b=(b_{1},\dots,b_{n})^{T}\in\mathds{R}^{n}$, and a measure $\nu$ on
$\mathds{R}^{n}$ with

$\int_{\|x\|_{2}\geq 1}|x_{i}|\leavevmode\nobreak\
\mathrm{d}\nu(x)<\infty\qquad\text{and}\qquad\int_{\mathds{R}^{n}}|x^{\alpha}|\leavevmode\nobreak\
\mathrm{d}\nu(x)<\infty$

for all $i=1,\dots,n$ and $\alpha\in\mathds{N}_{0}^{n}$ with $|\alpha|\geq 2$
such that

$\displaystyle a_{e_{i}}$ $\displaystyle=b_{i}+\int_{\|x\|_{2}\geq
1}x_{i}\leavevmode\nobreak\ \mathrm{d}\nu(x)$ $\displaystyle\text{for all}\
i=1,\dots,n,$ $\displaystyle a_{e_{i}+e_{j}}$
$\displaystyle=\sigma_{i,j}+\int_{\mathds{R}^{n}}x^{e_{i}+e_{j}}\leavevmode\nobreak\
\mathrm{d}\nu(x)$ $\displaystyle\text{for all}\ i,j=1,\dots,n,$ and
$\displaystyle a_{\alpha}$
$\displaystyle=\int_{\mathds{R}^{n}}x^{\alpha}\leavevmode\nobreak\
\mathrm{d}\nu(x)$ $\displaystyle\text{for all}\ \alpha\in\mathds{N}_{0}^{n}\
\text{with}\ |\alpha|\geq 3.$

###### Proof.

By 4.37 “(i) $\Leftrightarrow$ (ii)” we have that (i) $A\in\mathfrak{d}_{+}$
if and only if $\exp A$ has an infinitely divisible representing measure
$\mu$, i.e., by 1.2 we have

$\exp
A=\sum_{\alpha\in\mathds{N}_{0}^{n}}\frac{1}{\alpha!}\cdot\int_{\mathds{R}^{n}}x^{\alpha}\leavevmode\nobreak\
\mathrm{d}\mu(x)\cdot\partial^{\alpha}.$ (15)

By 2.21 we can take the logarithm and hence (15) is equivalent to

$A=\sum_{\alpha\in\mathds{N}_{0}^{n}\setminus\\{0\\}}\frac{a_{\alpha}}{\alpha!}\cdot\partial^{\alpha}=\log\left(\sum_{\alpha\in\mathds{N}_{0}^{n}}\frac{1}{\alpha!}\cdot\int_{\mathds{R}^{n}}x^{\alpha}\leavevmode\nobreak\
\mathrm{d}\mu(x)\cdot\partial^{\alpha}\right).$ (16)

With the isomorphism

$\mathds{C}[[\partial_{1},\dots,\partial_{n}]]\to\mathds{C}[[t_{1},\dots,t_{n}]],\
\partial_{1}\mapsto it_{1},\ \dots,\ \partial_{n}\mapsto it_{n}$

we have that (16) is equivalent to

$\displaystyle\sum_{\alpha\in\mathds{N}_{0}^{n}\setminus\\{0\\}}\frac{a_{\alpha}}{\alpha!}\cdot(it)^{\alpha}$
$\displaystyle=\log\left(\sum_{\alpha\in\mathds{N}_{0}^{n}}\frac{1}{\alpha!}\cdot\int_{\mathds{R}^{n}}x^{\alpha}\leavevmode\nobreak\
\mathrm{d}\mu(x)\cdot(it)^{\alpha}\right).$ (17) But the right hand side of
(17) is now the characteristic function $\displaystyle=\log\int
e^{itx}\leavevmode\nobreak\ \mathrm{d}\mu(x)$ of the $\mu$. Hence, by the
Lévy–Khinchin formula (see 2.22) we have
$\displaystyle=ibt-\frac{1}{2}t^{T}\Sigma
t+\int(e^{itx}-1-itx\cdot\chi_{\|x\|_{2}<1})\leavevmode\nobreak\
\mathrm{d}\nu(x).$ (18)

After a formal power series expansion of $e^{itx}$ in the Fréchet topology of
$\mathds{C}[[x_{1},\dots,x_{n}]]$, see 2.8, and a comparison of coefficients
we have that (18) is equivalent to (ii) which ends the proof. ∎

Form the previous result we see that the difference between $\mathfrak{d}_{+}$
and a moment sequence is that the representing (Lévy) measure $\nu$ in (18)
can have a singularity of order $\leq 2$ at the origin.

##  5 A Strange Action on $\mathrm{Pos}([0,\infty))$

Before we end this work we want to discuss an example of a strange positivity
action on $[0,\infty)$. We take the following example.

######  Example 5.42.

Let $\Delta=\partial_{x}^{2}$ on $L^{2}([0,\infty),\mathds{R})$ with Dirichlet
boundary conditions

$\displaystyle\partial_{t}u$ $\displaystyle=\Delta u$ $\displaystyle u(0,t)$
$\displaystyle=0\quad\text{for all}\ t\geq 0$ $\displaystyle u(\,\cdot\,,0)$
$\displaystyle=u_{0}\in L^{2}([0,\infty),\mathds{R}).$

Then

$u(x,t)=\int_{0}^{\infty}k_{t}(x,y)u_{0}(y)\leavevmode\nobreak\ \mathrm{d}y$

with

$k_{t}(x,y)=\frac{1}{\sqrt{4\pi t}}\cdot
e^{-(x-y)^{2}/4}\cdot\left[1-e^{xy/t}\right]$

for all $x,y\geq 0$ and $t>0$, see [8, Exm. 4.1.1].

For $u_{0}\in\mathcal{S}([0,\infty),\mathds{R})$ a Schwartz function on
$[0,\infty)$ we have that $k_{t}*u_{0}\in\mathcal{S}([0,\infty),\mathds{R})$
for all $t>0$ and hence we look at the time-dependent moments

$s_{j}(t)=\int_{K}x^{j}\cdot u(x,t)\leavevmode\nobreak\
\mathrm{d}x=\int_{[0,\infty)^{2}}x^{j}\cdot k_{t}(x,y)\cdot
u_{0}(y)\leavevmode\nobreak\ \mathrm{d}x\leavevmode\nobreak\ \mathrm{d}y$

for all $j\in\mathds{N}_{0}$ and $t>0$.

The action on $\mathds{R}[x]$ is then given by
$(T_{t}p)(x)=\int_{0}^{\infty}p(y)\cdot k_{t}(x,y)\leavevmode\nobreak\
\mathrm{d}y$. We find for $p=1$ that

$(T_{t}1)(x)=\int_{0}^{\infty}k_{t}(x,y)\leavevmode\nobreak\ \mathrm{d}y$

is a continuous, non-decreasing function in $x\in[0,\infty)$ with
$(T_{t}1)(0)=0$ and $\lim_{x\to\infty}(T_{t}1)(x)=1$, i.e.,
$T_{t}1\notin\mathds{R}[x]$ for any $t\in(0,\infty)$. While for $p=x^{j}$ with
$j\in\mathds{N}$ we get for the time-dependent moments by partial integration

$\displaystyle\partial_{t}s_{j}(t)$
$\displaystyle=\partial_{t}\int_{0}^{\infty}x^{j}\cdot
u(x,t)\leavevmode\nobreak\ \mathrm{d}x$
$\displaystyle=\int_{0}^{\infty}x^{j}\cdot
u(x,t)^{\prime\prime}\leavevmode\nobreak\ \mathrm{d}x$
$\displaystyle=\underbrace{x^{j}\cdot u(x,t)^{\prime}\leavevmode\nobreak\
\Big{|}_{x=0}^{\infty}}_{=0}-\int_{0}^{\infty}j\cdot x^{j-1}\cdot
u(x,t)^{\prime}\leavevmode\nobreak\ \mathrm{d}x$
$\displaystyle=\underbrace{-j\cdot x^{j-1}\cdot u(x,t)\leavevmode\nobreak\
\Big{|}_{x=0}^{\infty}}_{=0}+\int_{0}^{\infty}j\cdot(j-1)\cdot x^{j-2}\cdot
u(x,t)\leavevmode\nobreak\ \mathrm{d}x$ $\displaystyle=j\cdot(j-1)\cdot
s_{j-2}(t).$

These are the recursive relations we already encountered with the heat
equation on $\mathds{R}$, see [5, 6]. Therefore, for _odd_ polynomials
$p\in\mathds{R}[x]$ we have

$(T_{t}p)(x)=\int_{0}^{\infty}p(y)\cdot k_{t}(x,y)\leavevmode\nobreak\
\mathrm{d}y=(e^{t\partial_{x}^{2}}p)(x)\in\mathds{R}[x]$

with $T_{t}x^{2d+1}=\mathfrak{p}_{2d+1}(x,t)$ for all $d\in\mathds{N}_{0}$ in
[6, Dfn. 3.1] and if additionally $p\geq 0$ on $[0,\infty)$, then $T_{t}p\geq
0$ on $[0,\infty)$. On the other hand for _even_ polynomials
$p\in\mathds{R}[x]$ we find that $T_{t}p\notin
C^{\infty}([0,\infty))\setminus\mathds{R}[x]$ but at least $T_{t}p\geq 0$ on
$[0,\infty)$. The reason for this strange behavior is of course the Dirichlet
boundary condition and the resulting reflection principle. This effect needs
further investigations. $\circ$

This example shall also serve as a warning. Just because the operator (symbol)
is $\partial_{x}^{2}$ does not mean that the positivity preservers are
$\exp(t\partial_{x}^{2})$. The domain and the corresponding boundary
conditions have to be taken into account as is done and is well-known in the
partial differential equation literature and semi-group theory. Besides the
6.43 below this is another direction where further studies have to be done.

##  6 Summary and Open Question

If

$A=\sum_{\alpha\in\mathds{N}_{0}^{n}}q_{\alpha}\cdot\partial^{\alpha}\quad\text{with}\quad
q_{\alpha}\in\mathds{R}[x_{1},\dots,x_{n}]_{\leq|\alpha|}$ (19)

then

$A\mathds{R}[x_{1},\dots,x_{n}]_{\leq
d}\subseteq\mathds{R}[x_{1},\dots,x_{n}]_{\leq d}\quad\text{for all}\quad
d\in\mathds{N}_{0},$ (20)

i.e., $A$ is called _degree preserving_. In fact for any linear
$A:\mathds{R}[x_{1},\dots,x_{n}]\to\mathds{R}[x_{1},\dots,x_{n}]$ we have (19)
$\Leftrightarrow$ (20). (19) $\Rightarrow$ (20) is clear. For (20)
$\Rightarrow$ (19) take in (19) the smallest $\alpha$ with respect to the lex-
order such that $\deg q_{\alpha}>|\alpha|$. Then $\deg Ax^{\alpha}>|\alpha|$
which is a contradiction to (20).

Similar to 4.36 we have that the partial differential equation

$\partial_{t}p(x,t)=Ap(x,t)\quad\text{with}\quad
p(x,0)=p_{0}(x)\in\mathds{R}[x_{1},\dots,x_{n}]$

and $A$ as in (19) is equivalent to a linear system of ordinary differential
equations in the coefficients of $p$. Hence, it has a unique solution
$p(\,\cdot\,,t)=\exp(tA)p_{0}$ for all $t\in\mathds{R}$, i.e., $\exp(tA)$ is
well-defined for any $A$ as in (19) and $t\in\mathds{R}$. If additionally $A$
is a positivity preserver then $\exp(tA)$ is a degree preserving positivity
preserver with polynomial coefficients for all $t\geq 0$. In 4.41 we have
already seen for the positivity preservers with constant coefficients that
additional $A$ appear. The restriction that $A$ is degree preserving is a
natural restriction since otherwise
$\exp(tA)\mathds{R}[x_{1},\dots,x_{n}]\not\subseteq\mathds{R}[x_{1},\dots,x_{n}]$
for any $t>0$. So we arrive at the following open problem.

######  Open Problem 6.43.

What is the description of the set

$\left\\{A=\sum_{\alpha\in\mathds{N}_{0}^{n}}q_{\alpha}\cdot\partial^{\alpha}\;\middle|\;\exp(tA)\
\text{is a positivity preserver for all}\ t\geq 0\right\\}$

of all positivity generators, i.e., not necessarily with constant
coefficients?

The constant coefficient case is completely solved by 4.41. Part of the non-
constant coefficient cases are covered by the previous case that $A$ is a
degree preserving positivity preserver. For the non-constant coefficient cases
note that we are then in the framework of a non-commutative infinite
dimensional Lie group and its Lie algebra. Here the theory is much richer, see
e.g. [17, 12, 18, 25, 29, 23] and references therein.

## Funding

The author and this project are supported by the Deutsche
Forschungsgemeinschaft DFG with the grant DI-2780/2-1 and his research
fellowship at the Zukunfskolleg of the University of Konstanz, funded as part
of the Excellence Strategy of the German Federal and State Government.

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11institutetext: Yamaguchi University, Yamaguchi, Japan 11email:
<EMAIL_ADDRESS>22institutetext: Shinshu
University, Nagano, Japan 22email<EMAIL_ADDRESS>33institutetext: corresponding author

# An Integrated Web Platform for the Mizar Mathematical Library

Hideharu Furushima 11 Daichi Yamamichi 11 Seigo Shigenaka 11 Kazuhisa Nakasho
1133 0000-0003-1110-4342 Katsumi Wasaki 22

###### Abstract

This paper reports on the development of a Web platform to host the Mizar
Mathematical Library (MML). In recent years, the size of formalized
mathematical libraries has been drastically increasing, and this has led to a
growing demand for tools that support efficient and comprehensive browsing,
searching, and annotation of these libraries. This platform implements a Wiki
function to add comments to the HTMLized MML, three types of search function
(article, symbol, and theorem), and a function to show the dependency graph of
the MML. This platform is designed with consistency, scalability, and
interoperability as top priorities for long-term use.

###### Keywords:

mathematical knowledge management Mizar Mathematical Library QED manifesto Web
service.

##  1 Introduction

Many systems have been proposed to improve the browsability and searchability
of formalized mathematical libraries. The MathWiki
Project111https://www.nwo.nl/en/projects/612066825 aims to improve the
readability of formalized mathematical libraries and make them accessible to
wider communities. A Wiki for Mizar [17], Large Formal Wikis for Coq/CoRN [2],
and Agora System for Flyspeck Project [16] were proposed in the MathWiki
project. However, since the MathWiki Project was terminated in 2014, its
contents cannot follow the libraries’ updates. ProofWiki [19] and the Lean
Mathematical Library [8] accumulate mathematical libraries and convert them
into highly readable HTML documents. However, these two systems do not
collaborate with advanced search engines or graphical tools to visualize
library dependencies.

We developed the emwiki system, a Wiki service that hosts the Mizar
Mathematics Library (MML) [4] while addressing issues with existing systems.
The emwiki system is a Web platform based on the Django
framework222https://www.djangoproject.com/, featuring extensibility and
interoperability. Vue.js333https://vuejs.org/ is used as the front-end
framework, making it a partial single-page application. This eliminates extra
rendering to reduce user stress. Vuetify444https://vuetifyjs.com/ is used as
the user interface library, which allows for intuitive operation. Currently,
this service is available at https://em1.cs.shinshu-u.ac.jp/emwiki/release/.

##  2 Wiki Function

The Wiki feature is implemented to embed additional comments to read and
understand articles in the MML. Although users cannot edit mathematical
statements written in the formal language itself, they can add comments to
theorems and definitions. We reused the HTMLized MML [18] to improve the
convenience of the MML. The HTMLized MML is a document in which the MML is
converted to HTML format, and the reference relationships are expressed as
hyperlinks. We adopted the LaTeX syntax for mathematical expressions in the
comments, which is familiar to mathematicians. MathJax [7] is used as a
mathematical expression rendering engine. A real-time preview feature is also
implemented to check rendering results while editing.

The version tracking feature manages the history of comments. This function is
achieved by using Git, a distributed version control system, as its backend.
Moreover, it is necessary to maintain the linkage between theorems/definitions
and comments during library updates. Our system exploits the Git merge
function for this purpose. As theorems/definitions do not have persistent
identifiers in the Mizar language, the most effective way to identify
theorems/definitions before and after a library update is to track text
differences. Our system embeds comments directly into the MML. Since the
comments are written just before the theorems/definitions, administrators can
maintain the consistency of the linkage between the theorems/definitions and
comments using the three-way merge function during library updates.

The user management function allows administrators to track and block
malicious users. The Wiki function stores a history of editors and their
revisions, and when a comment is accidentally rewritten, it is possible to
contact the user who wrote it and roll back it. Also, if a malicious comment
is found, it will be deleted, and the user who wrote it will be blocked. The
user management function is available in any component on our platform.

The screenshot of the Wiki feature is shown in Fig. 1.

Figure 1: emwiki

##  3 Search Function

The emwiki system has three types of search components. The two incremental
search components accept the name of articles and symbols as input. As of
2018, the MML contains 1,290 articles and 8,852 symbols [4], and being able to
search for these articles and symbols efficiently is critical for users. MML
Reference [13] generates HTML documents from the MML to help users understand
its symbols. Each HTML page contains symbol definitions, referrers, and
references. It also has an incremental search function that enables users to
search with a fast and intuitive operation. We integrated MML Reference into
emwiki and linked this search function with the Wiki function.

We also developed a flexible theorem search component. For a long time, grep
has been used for full-text search in the MML [12]. MML Query [5], developed
by G. Bancerek et al. in 2001, has dramatically improved the efficiency of the
search of the MML. However, MML Query is not easy for beginners, because it
has its own query language. MML Query is a pattern-matching-based search
system. Therefore, it is not good at searching modified (but logically
equivalent) theorems. The Alcor system [6], developed by P. Carins et al.,
provides an Latent Semantic Indexing (LSI)-based search function for the Mizar
Mathematical Library. We have implemented a similar LSI-based search component
into the emwiki system. The search component does not require its own query
language, and the user can perform searches by inputting the desired theorems
in the Mizar language. We have also implemented a function to collect user
evaluations by displaying a “good” button on the search results. In the
future, we plan to use the evaluations as training data for machine learning.

##  4 Visualization of Library Dependencies

The dependency graph component is designed to encourage library maintainers to
refactor their libraries by visualizing the dependencies of files in the MML.
The MML has been maintained for the past 30 years by the University of
Bialystok [9] and is continuously refactored using specialized tools to
minimize the dependencies of the articles and eliminate the cyclical
dependencies between groups [14]. However, some of the Mizar language
specifications hinder the refactoring of the MML [15]. The Mizar language does
not have namespace or package features. Also, the length of the Mizar file
name is limited to 8 characters due to the legacy of MS-DOS constraints.
Therefore, all of the 1,290 articles with ambiguous file names are currently
arranged in a flat structure in the MML. Furthermore, the Mizar language has a
constructor overload feature and a context-sensitive grammar in which the last
imported constructor takes precedence. These language specifications should be
improved in the future, but we must continue to maintain our vast library of
assets in the meantime.

J. Alama (2011) [1] created a database of reference relations among theorems,
definitions, and notations in the MML and provided a way to access the
dependency graph through a Web interface. J. Heras et al. (2014) [10] provides
a tool to visualize the dependency graph of the HoTT library written in Coq.
J. Alama et al. (2012) [3] extracted the dependencies in the Coq and Mizar
libraries and performed a quantitative comparative analysis of the library
features. R. Marcus et al. (2020) [11] proposed the TGView3D system, which
renders the dependencies of formalized mathematical libraries as 3D graphs in
a hybrid of force-directed and hierarchical layouts.

In this study, we adopted the hierarchical graph. The hierarchical graph is
intended to clarify the flow of dependencies and utilize refactoring, such as
minimizing dependencies and eliminating circular references among groups. In
the Mizar language, reference relationships between articles are described in
the environment part at the beginning of an article. Since Mizar does not
recursively load external files written in environment sections, a transitive
reduction is performed to cut redundant edges before constructing the
dependency graph. The data structure is saved in dot and sfdp formats of
Graphviz and is drawn in a Web browser using Cytoscape.js library. The
component also provides functions such as highlighting nearby nodes, searching
nodes, moving nodes, hyperlinking to articles, and zooming in/out. A detailed
usage of the dependency graph component can be found in the Help menu in the
upper right corner.

##  5 Conclusions and Future Work

In this paper, we have developed the emwiki system as a Web platform for
hosting the MML. This system is differentiated from existing services hosting
mathematical libraries in consistency, extendability, and interoperability.
One of the most severe difficulties in the long-term use of a service that
hosts formalized mathematical libraries is keeping up with library updates. In
this study, we attempted to solve this problem using Git and its merge
function to maintain reference relationships during version updates. Since
this platform is built using Django, a new component development is stylized
as adding Django applications. Also, all components are allowed to access
their version-controlled library, user management function, and database. The
components implemented on the platform share Web pages and data with each
other. For example, MML Reference, now integrated into the system, shares a
Wiki function and Web pages. Ease of operation is also improved through
automated testing during development and deployment using containers.

For the dependency graph component, several issues remain. While our
hierarchical graph could properly place the dependencies of the articles, it
did not classify the articles into meaningful groups. Mizar articles have
metadata such as the Mathematics Subject Classification
(MSC)555https://msc2020.org/. We consider taking this information into account
for graph layout and including it as additional information for article
search. We also think it will be effective to analyze the vocabulary contained
in the articles and reflect the semantic classification in the layout
algorithm. Another beneficial enhancement is drawing theorem graphs. However,
handling as many nodes as the theorems in the MML requires a more powerful
graph database and drawing library than the conservative library we currently
use.

This project aims to improve the browsability, searchability, and
comprehensive understandability of formalized mathematical libraries. Although
the system currently supports only the Mizar language, it could be extended to
other languages. It is also significant to incorporate other components,
including a Web IDE.

It is also important to establish a cooperative framework for the long-term
operation of the system. In particular, we need to seek the cooperation of the
core team that maintains the MML and the Mizar system to create a flow to
ensure consistency between the MML versions. Also, we would like to invite a
wide range of developers from the Mizar community to work with us in defining
requirements and developing new functions.

## Acknowledgments

This work was supported by JSPS KAKENHI Grant Number JP20K19863.

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