($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$) also as
($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$).

To support an engineer, we characterise whether a tuple
($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$) is sufficient for a
given setting. The basic idea is: If a system is an _optimal_ autonomous
system, then its formed beliefs conserve the relevant aspects of
${{W}}_{\\!D}$. Hence the
($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$) is sufficient if a
relevance conserving belief formation exists. To answer whether
($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$) are relevant, we test
whether it is possible to build an optimal autonomous system with less
knowledge, observations or beliefs.

For the following we consider a doxastic model $D$ to be given with
$({{W}}_{\\!D},\psi,\mathit{L}_{\mathbb{K}},\mathcal{O},$
$\mathbb{B},\mathcal{B})$ with a knowledge-consistent belief formation
$\mathcal{B}$ and a system $\mathcal{S}=(D,{\textrm{s}_{d}})$ with a doxastic
strategy.

###  6.1 Conservation of the Relevant

We first define when the relevant is conserved. Therefore we compare Ego’s
(doxastic and autonomous-decisive) performance with the performance that Ego
could have when it would access the ground-truth, ${{W}}_{\\!D}$.

We first develop a notion of relevance conservation for doxastic systems in
order to highlight that the requirements for autonomous systems are more
demanding.

We say that the belief formation $\mathcal{B}$ conserves the relevant of
${{W}}_{\\!D}$, if $D$ can perform based on its beliefs as successful as it
could when directly and truthfully observing the ground-truth ${{W}}_{\\!D}$.

######  Definition (Relevance Conservation for Doxastic Systems).

Let $\mathcal{O}_{D}\subseteq\textit{{AP}}$ be a set of propositions. The
belief formation $\mathcal{B}$ of a doxastic model
$D=({{W}}_{\\!D},\psi,\mathit{L}_{\mathbb{K}},\mathcal{O},\mathbb{B},\mathcal{B})$
_conserves the relevant of a $\mathcal{O}$-observable ${{W}}_{\\!D}$_, if
there exists a doxastic strategy ${\textrm{s}_{d}}$ for $D$ that is dominant
w.r.t. all AP-observing strategies ${\textsf{s}_{t}}$.

When $\mathcal{B}$ is conserving the relevant of completely observable design-
time model ${{W}}_{\\!D}$. then Ego could –by implementing ${\textrm{s}_{d}}$
of Def. Definition– perform as well as possible when the ground-truth
${{W}}_{\\!D}$ would be completely observable. Def. Definition captures this
aspect by comparing the performance of Ego that is observing $\mathcal{O}$
with the performance on the ground-truth ${{W}}_{\\!D}$ that is observable via
AP.

But what does it mean that a belief formation $\mathcal{B}$ conserves the
relevant? Intuitively, it means that $\mathcal{B}$ preserves ${{W}}_{\\!D}$ in
sufficient detail to map the history of beliefs “somehow” to the best action.
The choice of action does not have to be plausible w.r.t. the content of a
system’s beliefs though. It is up to the engineer to choose which strategy
${\textrm{s}_{d}}$ will be implemented by the system.

The choice of action must be plausible w.r.t. the belief content though, when
it comes to autonomously-decisive systems. An autonomous-decisive system
chooses at all times actions act that are justified in the respective current
belief $B$, i.e. $\textit{act}\in\small\textit{Act}(B)\normalsize$ (cf. Def.
Definition). We hence say that the belief formation conserves the relevant for
autonomous-decisiveness, if at all times the “best actions” w.r.t. the belief
content are chosen.

######  Definition (Relevance Conservation for Autonomous-Decisiveness).

Let ${\textsf{S}_{a}}$ be the set of autonomous strategies that exist for
$D=({{W}}_{\\!D},\psi,\mathit{L}_{\mathbb{K}},\mathcal{O},\mathbb{B},\mathcal{B})$.

The belief formation $\mathcal{B}$ of $D$ _conserves the relevant of a
$\mathcal{O}$-observable design-time world ${{W}}_{\\!D}$ for autonomous-
decisiveness_, if all ${\textrm{s}_{a}}\in{\textsf{S}_{a}}$ are dominant
w.r.t. AP-observing strategies ${\textsf{s}_{t}}$ on ${{W}}_{\\!D}$.

Note, that we assume (cf. Ass. Definition, p. Definition) that the belief
formation $\mathcal{B}$ forms only beliefs $B$ in which a dominant current-
state decisive strategy exists. Hence ${\textsf{S}_{a}}$ contains at least one
strategy.

If $\mathcal{B}$ conserves the relevant for autonomous-decisiveness as defined
in Def. Definition, then any of the autonomous strategies ${\textrm{s}_{a}}$
of $D$ observing ${{W}}_{\\!D}$ via $\mathcal{O}$ will perform as successful
as possible when directly accessing ${{W}}_{\\!D}$ via $\mathcal{O}^{\prime}$.
It may seem surprising, that Def. Definition refers to _all_ autonomous
strategies ${\textrm{s}_{a}}\in{\textsf{S}_{a}}$. The reason is, that the
final decision on the chosen action lies with the autonomous system.

######  Example 18.

(Conservation of the Relevant) As an example of a belief formation that
conserves the relevant for autonomous-decisiveness, we refer the reader back
to Exp. 17 on page 17. There we sketched a setting where the sensors are
initially broken but when the decision has to taken the sensors provide the
relevant information. The resulting belief formation
$\mathcal{B}_{\textsf{auton}}$ allows a system to perform as well as when
knowing the ground-truth, i.e. not having initially disturbed sensor readings.
In Exp. 9 on page 9 we saw an example of a belief formation that conserves the
relevant for doxastic systems but not relevance for autonomous-decisiveness.
In the example, Ego’s sensors are permanently switching colours and Ego has a
knowledge base that forces it to believe that a read car is fast and a slow
car is blue. Consequently, Ego cannot autonomously determine what is best to
do in ${{W}}_{\\!D}$. But the belief-formation is such that an engineer can
choose a strategy for Ego that deals with the color readings appropriately,
i.e. “switch them back”. $\blacksquare$

Conserving the relevant for autonomous-decisiveness is stronger than
conserving the relevant for doxastic systems:

######  Proposition (Relevance Conservation).

  1. 1.

If $\mathcal{B}$ conserves the relevant for autonomous-decisiveness, then
$\mathcal{B}$ conserves the relevant for doxastic systems.

  2. 2.

If $\mathcal{B}$ conserves the relevant for doxastic systems, then
$\mathcal{B}$ does not necessarily conserve the relevant for autonomous-
decisiveness.

######  Proof Sketch (Prop. Proposition).

Prop. Proposition(1) follows directly from Def. Definition and Def.
Definition. Prop. Proposition(2) follows from the example 18. $\square$

The next proposition is concerned with systems, where the belief formation is
captured via a set of rules. Such autonomous systems still play an important
role especially in safety critical applications, although artificial
intelligence systems, that intransparently build their beliefs, gain more and
more importance.

Since the resources of a system $\mathcal{S}$ are limited, we consider belief
formation functions that can be represented by a finite number of regular
expressions.

######  Definition (Regular Belief Formation).

We say $\mathcal{B}$ is regular, if $\mathcal{B}$ can be defined via a finite
number $n$ of regular expressions $\rho_{i}$, i.e., for all observable
histories $h\in\mathbb{H}_{\mathcal{O}}$ of ${{W}}_{\\!D}$ it holds, that
there is an $i,1\leq i\leq n$ such that $\mathcal{B}(h)=B_{i}$ iff
$h\models\rho_{i}$.

Given a doxastic model with a regular belief formation $\mathcal{B}$, we can
decide whether $\mathcal{B}$ conserves the relevant for autonomous-
decisiveness:

######  Proposition (Conservation of Relevance).

Given a regular belief formation $\mathcal{B}$, we can decide whether
$\mathcal{B}$ conserves the relevant for autonomous-decisiveness.

######  Proof (Prop. Proposition).

We first determine the maximal priority $n_{\textit{m}}$ up to which the goal
list $\psi$ can be achieved on ${{W}}_{\\!D}$ by applying iteratively strategy
synthesis for LTL properties [36] starting with the maximum goal list and then
iteratively decreasing $n_{\textit{m}}$. $\mathcal{B}$ of $D$ conserves the
relevant, if all autonomous strategies ${\textrm{s}_{a}}\in{\textsf{S}_{a}}$
achieve at least $n_{\textit{m}}$ (cf. Def. Definition). We then construct an
automaton $\mathcal{A}_{\textit{Act}()\times{{W}}_{\\!D}}$, in which the
environment is unconstrained and Ego chooses its actions from
$\textit{Act}(\mathcal{B}(h))$ after observing history $h$. It holds that iff
$\mathcal{A}_{\textit{Act}()\times{{W}}_{\\!D}}$ satisfies $\psi$ up to
$n_{\textit{m}}$, then $\mathcal{B}$ conserves the relevant for autonomous
systems.

Construction of $\mathcal{A}_{\textit{Act}()\times{{W}}_{\\!D}}$: For each
belief $B\in\mathbb{B}$, we can determine the current-state choices
$\textit{bestAct}(B)$ (Prop. Proposition). Thus, the belief formation
$\mathcal{B}$ can be considered as an $\mathcal{O}$-observing strategy
assigning $\textit{Act}(B)$ to an observed history $h$ with
$B=\mathcal{B}(h)$: Since $\mathcal{B}$ is regular, we can build a mealy
automaton $\mathcal{A}_{\textit{Act}()}$ that determines
$\textit{Act}(\mathcal{B}(h))$ for an observed history $h$. When
$\mathcal{A}_{\textit{Act}()}$ transitions to an accepting state because of
$h$, this transition gets labelled with the current-state choices
$\textit{Act}(\mathcal{B}(h))$. We derive a composed automaton
${\mathcal{A}_{\textit{Act}()\times{{W}}_{\\!D}}}$ by parallel composition of
the design-time world ${{W}}_{\\!D}$ and $\mathcal{A}_{\textit{Act}()}$. In
$\mathcal{A}_{\textit{Act}()\times{{W}}_{\\!D}}$, Ego can take an action act
only if $\mathcal{A}_{\textit{Act}()}$ allows this, i.e. it is a current-state
choice for the observed history. If Ego may not take $\textit{act}_{1}$ in
state s, the combined action
$\textit{act}=(\textit{act}_{1},\textit{act}_{2})$ for all
$\textit{act}_{2}\in\textit{Act}_{\textit{env}}$, leads to the state
$\textit{s}_{\textsf{undef}}$. $\square$

In the next section we will characterise what knowledge, observations and
beliefs are relevant. Thereby we turn to questions like “Can we do with less
observations?”, “Can we do with less detailed beliefs?” or “Can we compensate
missing observations by adding knowledge?”.

###  6.2 Relevance of ($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$)

Our notion of _relevance conservation_ characterises combinations of
($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$) that allow a system to
form beliefs that are _sufficiently precise_ for the system to be optimal. In
this section we define that
($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$) is _relevant_ , if it
conserves the relevant (i.e. is sufficient), and in additional also
_“minimal”_.

The three dimensions of ($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$)
are of course interrelated. Intuitively, knowledge ($\mathit{L}_{\mathbb{K}}$)
about the world can replace observations that a system $\mathcal{S}$ needs
otherwise. Having more resources for the representation of the inner world
model ($\mathbb{B}$) allows a system $\mathcal{S}$ to store more of the made
observations and allows it to make finer predictions. More observations
($\mathcal{O}$) vice versa allow $\mathcal{S}$ to forget more and thus have a
simpler model of the history and future or to have less knowledge. We hence
expect that often several incomparable minima can be determined.

To define a _minimal($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$)_,
we first define partial order relations on the set of knowledge labeling
functions, the set of observations and the set of possible beliefs. We then
infer a partial order to order tuples
($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$).

We chose the partial orders to reflect the decisions an engineer has to make
during the design:

  1. (PO1)

$\mathcal{O}\leq\mathcal{O}^{\prime}$ $:\Leftrightarrow$
$\mathcal{O}\subseteq\mathcal{O}^{\prime}$
For this paper we assume that a greater set of observations means that more
sensors are necessary. We are hence interested in determining the minimal set
of required observations.

  2. (PO2)

$\mathbb{B}\leq\mathbb{B}^{\prime}$ $:\Leftrightarrow$
$\mathbb{B}\subseteq\mathbb{B}^{\prime}$
For the design of a $\mathcal{S}$ the size of the set of possible beliefs
$\mathbb{B}$ corresponds to the resources that are necessary to encode the
beliefs.

  3. (PO3)

$\mathit{L}_{\mathbb{K}}\leq\mathit{L}_{\mathbb{K}}^{\prime}$
$:\Leftrightarrow$
$\forall\textit{s}\in\textit{S}:\mathit{L}_{\mathbb{K}}(\textit{s})\leq\mathit{L}_{\mathbb{K}}^{\prime}(\textit{s})$
$:\Leftrightarrow$
$\forall\textit{s}\in\textit{S}:[\mathit{L}_{\mathbb{K}}^{\prime}(\textit{s})]\subseteq[\mathit{L}_{\mathbb{K}}(\textit{s})]$,
where [$\mathcal{K}$] denotes the set of traces on all possible worlds,
${\mathbb{W}}$, that satisfy the believed knowledge $\mathcal{K}$. As we deal
with knowledge-consistent belief formations here,
$\mathcal{K}\leq\mathcal{K}^{\prime}$ means that $\mathcal{K}$’ constrains the
beliefs that can be formed less. In other words, the system $\mathcal{S}$
knows less since it has more uncertainty.

$\mathit{L}_{\mathbb{K}}\leq\mathit{L}_{\mathbb{K}}^{\prime}$ means that
$\mathit{L}_{\mathbb{K}}^{\prime}$ declares more knowledge at least at one
state of ${{W}}_{\\!D}$ and it declares not less knowledge than
$\mathit{L}_{\mathbb{K}}$ in all other states. An engineer can provide prior
knowledge, e.g. she can hard-code the believed knowledge into $\mathcal{S}$,
and she can implement the knowledge labelling, i.e. ensure that mechanisms are
in place that will update the knowledge base during $\mathcal{S}$’s missions.

  4. (PO4)

$\small\normalsize{({\mathit{L}_{\mathbb{K}}},{\mathcal{O}},{\mathbb{B}})}\normalsize\leq\small\normalsize{({\mathit{L}_{\mathbb{K}}^{\prime}},{\mathcal{O}^{\prime}},{\mathbb{B}^{\prime}})}\normalsize$
$:\Leftrightarrow$ (PO1)-(PO3) hold.

By (PO4) we now define the notion of _weak relevance_. A tuple
($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$) is weak relevant, if we
cannot find a strictly smaller tuple
($\mathit{L}_{\mathbb{K}}$’,$\mathcal{O}$’,$\mathbb{B}$’). We call this _weak_
, since there can be other tuples
($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$) that are incomparable
with ($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$). Hence the
question “Is ($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$) relevant”
does not have a definite answer. Nevertheless the notion of weak relevance
allows to answer, whether a system $\mathcal{S}$ can do with more observations
in exchange for less knowledge or fewer possible beliefs or whether more
knowledge allows $\mathcal{S}$ to have fewer possible beliefs or less
observations and so on.

######  Definition (Weak Relevance).

Let a design-time world ${{W}}_{\\!D}$ and a prioritised list of goals $\psi$
be given.

($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$) is weakly relevant for
$({{W}}_{\\!D},\psi)$, if

  1. 1.

there is a belief formation $\mathcal{B}$ of
$D:=({{W}}_{\\!D},\psi,\mathit{L}_{\mathbb{K}},\mathcal{O},\mathbb{B},\mathcal{B})$
that conserves the relevant for autonomous systems and

  2. 2.

for all
$\small\normalsize{({\mathit{L}_{\mathbb{K}}^{\prime}},{\mathcal{O}^{\prime}},{\mathbb{B}^{\prime}})}\normalsize\not=\small\normalsize{({\mathit{L}_{\mathbb{K}}},{\mathcal{O}},{\mathbb{B}})}\normalsize$
with $\mathit{L}_{\mathbb{K}}^{\prime}\leq\mathit{L}_{\mathbb{K}}$,
$\mathcal{O}^{\prime}\leq\mathcal{O}$ and $\mathbb{B}^{\prime}\leq\mathbb{B}$
$\small\normalsize{({\mathit{L}_{\mathbb{K}}^{\prime}},{\mathcal{O}^{\prime}},{\mathbb{B}^{\prime}})}\normalsize$
there is no knowledge-consistent belief formation $\mathcal{B}$’ of
$D^{\prime}:=({{W}}_{\\!D},\psi,\mathit{L}_{\mathbb{K}}^{\prime},\mathcal{O}^{\prime},\mathbb{B}^{\prime},\mathcal{B}^{\prime})$
that conserves the relevant for autonomous systems.

$\mathit{L}_{\mathbb{K}}$ is weakly relevant if there are $\mathcal{O}$ and
$\mathbb{B}$, such that ($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$,$\mathbb{B}$)
is weakly relevant. Analogously we define that $\mathcal{O}$ ($\mathbb{B}$) is
weakly relevant if there are $\mathcal{B}$, $\mathit{L}_{\mathbb{K}}$ and
$\mathbb{B}$ ($\mathcal{O}$).

To illustrate the notion, we consider an example.

######  Example 19.

(Weak Relevance) Let us assume Ego observes its position pos, a time stamp $t$
and its speed $v$. Its goal is to determine its past average acceleration
acc.121212We assume finite domains and hence finite encodings of numerical
values. The computations will be rounded appropriately. Ego’s actions are
computation steps. Moreover, let us assume that the perception of position is
flawed when it is raining while the speed is still correctly measured. Then
only $\\{v,t\\}$ is weakly relevant, that is, they suffice to determine the
average acceleration. Neither the set $\\{\textrm{pos},v,t\\}$ is weakly
relevant nor the set $\\{\textrm{pos},t\\}$. The further is not minimal, the
latter does not conserve the relevant, since acc cannot be determined while it
is raining.

Given Ego has the knowledge “it will not rain” both $\\{\textrm{pos},t\\}$ and
$\\{v,t\\}$ are weakly relevant. $\blacksquare$

Let us now turn to questions like “Is $\mathcal{O}$ relevant, given
$\mathit{L}_{\mathbb{K}}$ and $\mathbb{B}$?”, i.e. we assume tow component of
the triple are known. The question of relevance ten might have a definite
answer, but not necessarily. We hence consider it an interesting notion. In
Def. Definition we define $\mathit{L}_{\mathbb{K}}$ ($\mathcal{O}$,
$\mathbb{B}$) to be relevant, if there is no alternative minimal choice, i.e.,
the system $\mathcal{S}$ has to have $\mathit{L}_{\mathbb{K}}$ ($\mathcal{O}$,
$\mathbb{B}$) in order to be able to perform autonomously optimal.

######  Definition (Relevance).

$\mathcal{O}$ is relevant for (${{W}}_{\\!D}$,$\psi$) with
($\mathit{L}_{\mathbb{K}}$,$\mathbb{B}$) iff

  1. 1.

$\mathcal{O}$ is weakly relevant and

  2. 2.

there is no other $\mathcal{O}$’ that is weakly relevant.

Likewise we define $\mathit{L}_{\mathbb{K}}$ and $\mathbb{B}$ are relevant for
(${{W}}_{\\!D}$,$\psi$) with ($\mathcal{O}$,$\mathbb{B}$) and respectively
($\mathit{L}_{\mathbb{K}}$,$\mathcal{O}$).

######  Theorem 2 (Relevance).

Given a doxastic model
$D=({{W}}_{\\!D},\psi,\mathcal{K},\mathcal{O},\mathcal{B})$ of $\mathcal{S}$
within its environment, we can decide whether $(\mathcal{K},\mathcal{O})$ is
(weakly) relevant for $\mathcal{B}$ in $D$.

######  Proof (Thm. 2).

To show item 1 of Def. Definition we check whether there is a
$\mathcal{O}$-observing strategy in ${{W}}_{\\!D}$. To check item 2 of Def.
Definition we build the “lesser” pairs
$(\mathcal{K}^{\prime},\mathcal{O}^{\prime})$, i.e.
$\mathcal{K}^{\prime}\leq\mathcal{K}$, $\mathcal{O}^{\prime}\leq\mathcal{O}$
and
$(\mathcal{K},\mathcal{O})\not=(\mathcal{K}^{\prime},\mathcal{O}^{\prime})$,
and check whether there is a belief labelling $\mathcal{B}$’ that conserves
the relevant again by checking whether there a
$\mathcal{O}^{\prime}$-observing strategy in ${{W}}_{\\!D}$. $\square$

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# Some criteria of rational infinite divisibility for distribution functions

A. A. Khartov111Laboratory for Approximation Problems of Probability, Smolensk
State University, 4 Przhevalsky st., 214000 Smolensk, Russia, e-mail:
<EMAIL_ADDRESS>
###### Abstract

We study distribution functions $F$ that have the property of rational
infinite divisibility: there exist some infinitely divisible distribution
functions $F_{1}$ and $F_{2}$ such that $F_{1}=F*F_{2}$. We study a class
$\boldsymbol{Q}$ of such distribution functions, which is a wide natural
extension of the well-known class of infinitely divisible distribution
functions. We are interested in general criteria of membership of
$\boldsymbol{Q}$. The obtained conditions are formulated in terms of
characteristic functions and they seem to be convenient for the application.

Keywords and phrases: distribution functions, characteristic functions,
infinite divisibility, rational infinite divisibility, quasi-infinite
divisibility, the Lévy–Khinchine formula.

##  1 Introduction

Let $F$ be a distribution function on the real line $\mathbb{R}$ with the
characteristic function

$\displaystyle f(t)\colonequals\int\limits_{\mathbb{R}}e^{itx}\,dF(x),\quad
t\in\mathbb{R}.$ (1)

Recall that $F$ is called infinitely divisible if for every positive integer
$n$ there exists a distribution function $F_{1/n}$ such that
$F=(F_{1/n})^{*n}$, where “$*$” is the convolution, i.e. $F$ is $n$-fold
convolution power of $F_{1/n}$. It is well known that $F$ is infinitely
divisible if and only if the characteristic function $f$ is represented by the
Lévy–Khinchine formula:

$\displaystyle
f(t)=\exp\biggl{\\{}it\gamma+\int\limits_{\mathbb{R}}\bigl{(}e^{itx}-1-it\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x)\biggr{\\}},\quad
t\in\mathbb{R},$ (2)

with some shift parameter $\gamma\in\mathbb{R}$, and with a bounded non-
decreasing spectral function $G:\mathbb{R}\to\mathbb{R}$ that is assumed to be
right-continuous with condition $G(-\infty)=0$. We use $x\mapsto\sin(x)$ as
the “centering function” in the integral in (2) following to Zolotarev [19]
and [20], but there are also some other variants (see [16], p. 38). It is well
known that the spectral pair $(\gamma,G)$ is uniquely determined by $f$ and
hence by $F$. The Lévy–Khinchine formula is very important for probability
theory and for it’s applications in related fields (see [2] and [17]).

There is a natural extension for the class of infinitely divisible
distribution functions. We call a distribution function $F$ rationally
infinitely divisible if there exist some infinitely divisible distribution
functions $F_{1}$ and $F_{2}$ such that $F_{1}=F*F_{2}$. This equality may be
written in terms of characteristic functions: $f(t)=f_{1}(t)/f_{2}(t)$,
$t\in\mathbb{R}$, where $f_{1}$ and $f_{2}$ denote characteristic functions of
$F_{1}$ and $F_{2}$, respectively. It is not difficult to show in this case
that $f$ admits representation (2) with $\gamma=\gamma_{1}-\gamma_{2}$ and
$G=G_{1}-G_{2}$, where $(\gamma_{1},G_{1})$ and $(\gamma_{2},G_{2})$ denote
the spectral pairs of $F_{1}$ and $F_{2}$, respectively. Note that the
function $G$ is of bounded variation on $\mathbb{R}$ (non-monototic in
general), right-continuous and $G(-\infty)=0$. The class of all functions
satisfying these properties will be denoted by $\boldsymbol{V}$. The pair
$(\gamma,G)$ is uniquely determined by $f$ and by $F$ as for infinitely
divisible distribution functions (see [10] p. 31-32, or [7] p. 80). Now
suppose that $f$ is represented by formula (2) with some $\gamma\in\mathbb{R}$
and $G\in\boldsymbol{V}$. Following Lindner and Sato [11], the corresponding
distribution function $F$ is called quasi-infinitely divisible. Due to the
Hahn–Jordan decomposition for $G$, it is not difficult to show that $F$ is
rationally infinitely divisibile. Various forms of definition and the first
detailed analysis of the class of such distribution function on $\mathbb{R}$
was performed in [12]. There are interesting applications in theory of
stochastic processes (see [11] and [14]), number theory (see [13]), physics
(see [5] and [6]), and insurance mathematics (see [18]).

The class of all rationally infinitely divisible distribution functions we
will denote by $\boldsymbol{Q}$. By definition, it is clear that this class
contains the class $\boldsymbol{I}$ of all infinitely divisible distribution
functions and, moreover, it is essentially wider than the latter. So
$\boldsymbol{Q}$ includes all discrete distribution functions $F$ whose
characteristic functions $f$ are separeted from zero, i.e.
$|f(t)|\geqslant\mu$ for some constant $\mu>0$ and for any $t\in\mathbb{R}$
(see [1] and [8]). The class $\boldsymbol{Q}$ is also contains mixtures of
discrete and absolutely continuous distribution functions, where the discrete
part is non-zero and its characteristic function is separeted from zero (see
[3] and [4]). Note that a rational infinitely divisible distribution may well
have a bounded support that is impossible for non-degenerate infinitely
divisible distributions. There are also some interesting particular sufficient
conditions to belong to $\boldsymbol{Q}$ (see [12]). However, the problem
about general criteria for membership of $\boldsymbol{Q}$ is not completely
solved. In this work, we propose some new results in this direction. The
criteria are formulated in terms of characteristic functions and these
conditions seem to be convenient for the application.

The statements of the results are given in Section 2 and the proofs are
provided in Section 3. Throughout the paper, we use the following notation.
For any characteristic function $f$ such that $f(t)\neq 0$, $t\in\mathbb{R}$,
the function $\mathrm{Ln}f(\cdot)$ will denote the distinguished logarithm,
i.e. $\mathrm{Ln}f(t)=\ln|f(t)|+i\mathrm{Arg}f(t)$, $t\in\mathbb{R}$, where
the argument $\mathrm{Arg}f$ is uniquely defined on $\mathbb{R}$ by continuity
with the condition $\mathrm{Arg}f(0)=0$. The function $\ln(\cdot)$ will as
usual denote the principal value of the logarithm. For every
$G\in\boldsymbol{V}$ its total variation on $\mathbb{R}$ will be denoted by
$\|G\|$ and the total variation on $(-\infty,x]$ — by $|G|(x)$,
$x\in\mathbb{R}$. So we have $|G(x)|\leqslant|G|(x)\leqslant\|G\|$,
$x\in\mathbb{R}$, and $|G|(+\infty)=\|G\|$. We denote by $\mathbb{N}$ the sets
of positive integers and let
$\mathbb{N}_{0}\colonequals\mathbb{N}\cup\\{0\\}$. The quantity
$\mathds{1}(A)$ equals zero or one if $A$ is false or true, respectively. Let
$\lfloor\cdot\rfloor$ be the floor function, i.e. for any $x\in\mathbb{R}$ we
have $\lfloor x\rfloor=k$ with integer $k$ such that $k\leqslant x<k+1$. We
always set for the sums $\sum_{j=n}^{m}a_{j}=0$ if $n>m$.

##  2 Results

We consider a characteristic function $f$ of some distribution function $F$,
i.e. we have (1). We always assume that $f(t)\neq 0$, $t\in\mathbb{R}$.
Therefore the distinguished logarithm $\mathrm{Ln}f$ is defined and we may
introduce the following function

$\displaystyle\Delta^{2}_{h}\mathrm{Ln}f(t)\colonequals\mathrm{Ln}f(t-h)+\mathrm{Ln}f(t+h)-2\mathrm{Ln}f(t),\quad
t,h\in\mathbb{R}.$ (3)

So this is a finite difference of the second order for $\mathrm{Ln}f$. It
should be noted that $\Delta^{2}_{h}\mathrm{Ln}f$ is often rather simpler and
easier handling than $\mathrm{Ln}f$. It is also used for establishing some
properties of $f$ (see [8] p. 3, [9] p. 119, and [15] p. 34). The following
theorem shows that $\Delta^{2}_{h}\mathrm{Ln}f$ contains in fact full
information about the existence of the property of rational infinite
divisibility for the corresponding distribution function.

###### Theorem 1

$a)$ Suppose that $F\in\boldsymbol{Q}$ and $f$ admits representation (2) with
some $\gamma\in\mathbb{R}$ and $G\in\boldsymbol{V}$. Then

$\displaystyle\Delta^{2}_{h}\mathrm{Ln}f(t)=\int\limits_{\mathbb{R}}e^{itx}\bigl{(}\cos(hx)-1\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x)\quad\text{for
all}\quad t,h\in\mathbb{R}.$ (4)

$b)$ Suppose that for some $G\in\boldsymbol{V}$ we have

$\displaystyle\Delta^{2}_{h}\mathrm{Ln}f(kh)=2\int\limits_{\mathbb{R}}e^{ikhx}\bigl{(}\cos(hx)-1\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x)+a_{k}(h),\quad
k\in\mathbb{N}_{0},\,\,h>0,$ (5)

with the quantities $a_{k}(h)$ satisfying

$\displaystyle\sum_{k=0}^{n_{t,l}-1}\bigl{(}n_{t,l}-k\bigr{)}|a_{k}(h_{l})|\to
0,\quad l\to\infty,\quad\text{for any}\quad t>0,$ (6)

for some sequence $(h_{l})_{l\in\mathbb{N}}$ of positive reals such that
$h_{l}\to 0$, $l\to\infty$, and with $n_{t,l}\colonequals\lfloor
t/h_{l}\rfloor$, $t>0$. Then $F\in\boldsymbol{Q}$ and $f$ admits
representation (2) with with the function $G$ and some $\gamma\in\mathbb{R}$.

###### Remark 1

If (4) is true for $t>0$ and $h>0$, then (6) obviously holds $($here
$a_{k}(h)=0$, $k\in\mathbb{N}_{0}$, $h>0$$)$.

###### Corollary 1

Suppose that for any $t>0$ and $h>0$ the formula

$\displaystyle\Delta^{2}_{h}\mathrm{Ln}f(t)=2\int\limits_{\mathbb{R}}e^{itx}\bigl{(}\cos(hx)-1\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x)$

holds with some function $G\in\boldsymbol{V}$. Then $F\in\boldsymbol{Q}$ and
$f$ admits representation (2) with the function $G$ and some
$\gamma\in\mathbb{R}$.

It is sometimes more convenient to work with the following functions

$\displaystyle\exp\bigl{\\{}\pm\Delta^{2}_{h}\mathrm{Ln}f(t)\bigr{\\}}-1=\biggl{(}\dfrac{f(t-h)f(t+h)}{f(t)^{2}}\biggr{)}^{\pm
1}-1,\quad t,h\in\mathbb{R}.$

Using these functions, one can formulate the sufficient condition of
membership of $\boldsymbol{Q}$.

###### Theorem 2

Suppose that for some $G\in\boldsymbol{V}$ we have

$\displaystyle\exp\bigl{\\{}\pm\Delta^{2}_{h}\mathrm{Ln}f(kh)\bigr{\\}}-1=\pm
2\int\limits_{\mathbb{R}}e^{ikhx}\bigl{(}\cos(hx)-1\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x)+b_{k}(h),\quad
k\in\mathbb{N}_{0},\,\,h>0,$ (7)

with the quantities $b_{k}(h)$ satisfying

$\displaystyle\sum_{k=0}^{n_{t,l}-1}\bigl{(}n_{t,l}-k\bigr{)}|b_{k}(h_{l})|\to
0,\quad l\to\infty,\quad\text{for any}\quad t>0,$ (8)

and also

$\displaystyle\sum_{k=0}^{n_{t,l}-1}\bigl{(}n_{t,l}-k\bigr{)}\bigl{|}\exp\bigl{\\{}\pm\Delta^{2}_{h_{l}}\mathrm{Ln}f(kh_{l})\bigr{\\}}-1\bigr{|}^{2}\to
0,\quad l\to\infty,\quad\text{for any}\quad t>0,$ (9)

for some sequence $(h_{l})_{l\in\mathbb{N}}$ of positive reals such that
$h_{l}\to 0$, $l\to\infty$, and with $n_{t,l}\colonequals\lfloor
t/h_{l}\rfloor$, $t>0$. Then $F\in\boldsymbol{Q}$ and $f$ admits
representation (2) with the function $G$ and some $\gamma\in\mathbb{R}$.

Let us return to the function (3). We saw that admission of representation (4)
is criterion for rational infinite divisibility of the distribution function.
Let us write this in the following form:

$\displaystyle\dfrac{\Delta^{2}_{h}\mathrm{Ln}f(t)}{h^{2}}=-\int\limits_{\mathbb{R}}e^{itx}\cdot\dfrac{\sin^{2}\bigl{(}\tfrac{hx}{2}\bigr{)}}{\bigl{(}\tfrac{hx}{2}\bigr{)}^{2}}\cdot(1+x^{2})\,dG(x).$

If $h\to 0$ then the left-hand side formally tends to
$(\mathrm{Ln}f)^{\prime\prime}(t)$ and the fraction formally dissapears from
the integrand in the right-hand side. This idea leads us to the following
result.

###### Theorem 3

$a)$ Suppose that $F\in\boldsymbol{Q}$ and $f$ admits representation (2) with
some $\gamma\in\mathbb{R}$ and $G\in\boldsymbol{V}$ satisfying

$\displaystyle\int\limits_{\mathbb{R}}(1+x^{2})\,d|G|(x)<\infty.$ (10)

Then for any $t\in\mathbb{R}$ there exists $(\mathrm{Ln}f)^{\prime\prime}(t)$
that admits the following representation

$\displaystyle(\mathrm{Ln}f)^{\prime\prime}(t)=-\int\limits_{\mathbb{R}}e^{itx}(1+x^{2})\,dG(x),\quad
t\in\mathbb{R}.$ (11)

$b)$ Suppose that the formula (11) holds with some function
$G\in\boldsymbol{V}$ satisfying (10). Then $F\in\boldsymbol{Q}$ and $f$ admits
representation (2) with the function $G$ and some $\gamma\in\mathbb{R}$.

Theorems 1–3 may help to prove the infinite divisibility of a given
distribution function and to find the corresponding spectral function.

###### Remark 2

If the function $G$ turned out to be non-decreasing in Theorems 1–3, then
$F\in\boldsymbol{I}$.

Indeed, if $G$ is non-decreasing function, then (2) is the canonical
Lévy–Khinchine formula and hence $F$ is infinitely divisible, i.e.
$F\in\boldsymbol{I}$.

##  3 Proofs

Proof of Theorem 1. a) Using (2) in (3), for any $t,h\in\mathbb{R}$ we have

$\displaystyle\Delta^{2}_{h}\mathrm{Ln}f(t)$ $\displaystyle=$ $\displaystyle
i2t\gamma+\int\limits_{\mathbb{R}}\Bigl{(}e^{itx}\bigl{(}e^{-ihx}+e^{ihx}\bigr{)}-2-i2t\sin(x)\Bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x)$
$\displaystyle{}-2\biggl{(}it\gamma+\int\limits_{\mathbb{R}}\Bigl{(}e^{itx}-1-it\sin(x)\Bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x)\biggr{)}$
$\displaystyle=$ $\displaystyle
2\int\limits_{\mathbb{R}}e^{itx}\bigl{(}\cos(hx)-1\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x),$

as required.

b) We first show that for any $n\in\mathbb{N}_{0}$ and $h\in\mathbb{R}$ the
quantity $\mathrm{Ln}f(nh)-in\mathrm{Arg}f(h)$ is expressed through the values
$\Delta^{2}_{h}\mathrm{Ln}f(jh)$, $j\in\mathbb{N}_{0}$. We fix such $n$ and
$h$. Observe that for any $k\in\mathbb{N}$

$\displaystyle\sum_{j=1}^{k-1}\Delta^{2}_{h}\mathrm{Ln}f(jh)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{k-1}\bigl{(}\mathrm{Ln}f((j-1)h)+\mathrm{Ln}f((j+1)h)-2\mathrm{Ln}f(jh)\bigr{)}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{k-1}\bigl{(}\mathrm{Ln}f((j-1)h)-\mathrm{Ln}f(jh)\bigr{)}+\sum_{j=1}^{k-1}\bigl{(}\mathrm{Ln}f((j+1)h)-\mathrm{Ln}f(jh)\bigr{)}$
$\displaystyle=$
$\displaystyle\mathrm{Ln}f(0)-\mathrm{Ln}f((k-1)h)+\mathrm{Ln}f(kh)-\mathrm{Ln}f(h)$
$\displaystyle=$
$\displaystyle\mathrm{Ln}f(kh)-\mathrm{Ln}f((k-1)h)-\mathrm{Ln}f(h),$

and

$\displaystyle\sum_{j=0}^{k-1}\Delta^{2}_{h}\mathrm{Ln}f(jh)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{k-1}\Delta^{2}_{h}\mathrm{Ln}f(jh)+\Delta^{2}_{h}\mathrm{Ln}f(0)$
$\displaystyle=$
$\displaystyle\bigl{(}\mathrm{Ln}f(kh)-\mathrm{Ln}f((k-1)h)-\mathrm{Ln}f(h)\bigr{)}+\mathrm{Ln}f(-h)+\mathrm{Ln}f(h)-2\mathrm{Ln}f(0)$
$\displaystyle=$
$\displaystyle\mathrm{Ln}f(kh)-\mathrm{Ln}f((k-1)h)+\mathrm{Ln}f(-h).$

Hence for any $k\in\mathbb{N}$

$\displaystyle\sum_{j=1}^{k-1}\Delta^{2}_{h}\mathrm{Ln}f(jh)+\tfrac{1}{2}\Delta^{2}_{h}\mathrm{Ln}f(0)$
$\displaystyle=$
$\displaystyle\mathrm{Ln}f(kh)-\mathrm{Ln}f((k-1)h)-\tfrac{1}{2}\bigl{(}\mathrm{Ln}f(h)-\mathrm{Ln}f(-h)\bigr{)}.$

Let us consider

$\displaystyle\sum_{k=1}^{n}\biggl{(}\sum_{j=1}^{k-1}\Delta^{2}_{h}\mathrm{Ln}f(jh)+\tfrac{1}{2}\Delta^{2}_{h}\mathrm{Ln}f(0)\biggr{)}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\bigl{(}\mathrm{Ln}f(kh)-\mathrm{Ln}f((k-1)h)\bigr{)}-\sum_{k=1}^{n}\tfrac{1}{2}\bigl{(}\mathrm{Ln}f(h)-\mathrm{Ln}f(-h)\bigr{)}$
$\displaystyle=$
$\displaystyle\mathrm{Ln}f(nh)-\mathrm{Ln}f(0)-n\cdot\tfrac{1}{2}\bigl{(}\mathrm{Ln}f(h)-\mathrm{Ln}f(-h)\bigr{)}$
$\displaystyle=$
$\displaystyle\mathrm{Ln}f(nh)-n\cdot\tfrac{1}{2}\bigl{(}\mathrm{Ln}f(h)-\mathrm{Ln}f(-h)\bigr{)}.$

Since $\ln|f(\cdot)|$ and $\mathrm{Arg}f(\cdot)$ are respectively even and odd
functions, we have

$\displaystyle\mathrm{Ln}f(h)-\mathrm{Ln}f(-h)$ $\displaystyle=$
$\displaystyle\ln|f(h)|+i\mathrm{Arg}f(h)-\ln|f(-h)|-i\mathrm{Arg}f(-h)$
$\displaystyle=$
$\displaystyle\ln|f(h)|+i\mathrm{Arg}f(h)-\ln|f(h)|+i\mathrm{Arg}f(h)$
$\displaystyle=$ $\displaystyle 2i\mathrm{Arg}f(h).$

Thus we obtain

$\displaystyle\mathrm{Ln}f(nh)-in\mathrm{Arg}f(h)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\Bigl{(}\sum_{j=1}^{k-1}\Delta^{2}_{h}\mathrm{Ln}f(jh)+\tfrac{1}{2}\Delta^{2}_{h}\mathrm{Ln}f(0)\Bigr{)},\quad
n\in\mathbb{N}_{0},\,\,h\in\mathbb{R},$

as required. By the way, we observe that for any $n\in\mathbb{N}_{0}$,
$h\in\mathbb{R}$, and $m\in\mathbb{N}$:

$\displaystyle\mathrm{Ln}f(nh)-i\tfrac{n}{m}\mathrm{Arg}f(mh)$
$\displaystyle=$
$\displaystyle\mathrm{Ln}f(nh)-i\tfrac{n}{m}\mathrm{Im}\,\\{\mathrm{Ln}f(mh)\\}$
$\displaystyle=$
$\displaystyle\mathrm{Ln}f(nh)-in\mathrm{Arg}f(h)-i\tfrac{n}{m}\mathrm{Im}\,\bigl{\\{}\mathrm{Ln}f(mh)-im\mathrm{Arg}f(h)\bigr{\\}}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\Bigl{(}\sum_{j=1}^{k-1}\Delta^{2}_{h}\mathrm{Ln}f(jh)+\tfrac{1}{2}\Delta^{2}_{h}\mathrm{Ln}f(0)\Bigr{)}$
$\displaystyle{}-i\tfrac{n}{m}\mathrm{Im}\,\biggl{\\{}\sum_{k=1}^{m}\Bigl{(}\sum_{j=1}^{k-1}\Delta^{2}_{h}\mathrm{Ln}f(jh)+\tfrac{1}{2}\Delta^{2}_{h}\mathrm{Ln}f(0)\Bigr{)}\biggr{\\}}.$

Thus $\mathrm{Ln}f(nh)-i\tfrac{n}{m}\mathrm{Arg}f(mh)$ is also expressed
through the values $\Delta^{2}_{h}\mathrm{Ln}f(jh)$, $j\in\mathbb{N}_{0}$.

Let us consider the integrals from (5). We define

$\displaystyle I(kh,h)\colonequals
2\int\limits_{\mathbb{R}}e^{ikhx}\bigl{(}\cos(hx)-1\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x)=\int\limits_{\mathbb{R}}e^{ikhx}\,dR_{h}(x)\quad\text{for
all}\quad k\in\mathbb{N}_{0},\,\,h\in\mathbb{R},$

where

$\displaystyle
R_{h}(x)=2\int\limits_{(-\infty,x]}\bigl{(}\cos(hu)-1\bigr{)}\tfrac{1+u^{2}}{u^{2}}\,dG(u),\quad
x\in\mathbb{R}.$ (12)

So $a_{k}(h)=\Delta^{2}_{h}\mathrm{Ln}f(kh)-I(kh,h)$, $k\in\mathbb{N}_{0}$,
$h\in\mathbb{R}$. Let us represent

$\displaystyle\mathrm{Ln}f(nh)-i\tfrac{n}{m}\mathrm{Arg}f(mh)$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\Bigl{(}\sum_{j=1}^{k-1}I(jh,h)+\tfrac{1}{2}I(0,h)\Bigr{)}-i\tfrac{n}{m}\mathrm{Im}\,\biggl{\\{}\sum_{k=1}^{m}\Bigl{(}\sum_{j=1}^{k-1}I(jh,h)+\tfrac{1}{2}I(0,h)\Bigr{)}\biggr{\\}}$
(13) $\displaystyle{}+\Delta(n,m,h),\quad
n\in\mathbb{N}_{0},\,\,m\in\mathbb{N},\,\,h\in\mathbb{R},$

where $\Delta(n,m,h)$ admits the representation:

$\displaystyle\Delta(n,m,h)=\sum_{k=1}^{n}\Bigl{(}\sum_{j=1}^{k-1}a_{j}(h)+\tfrac{1}{2}a_{0}(h)\Bigr{)}-i\tfrac{n}{m}\mathrm{Im}\,\biggl{\\{}\sum_{k=1}^{m}\Bigl{(}\sum_{j=1}^{k-1}a_{j}(h)+\tfrac{1}{2}a_{0}(h)\Bigr{)}\biggr{\\}}.$

For this we have the estimates:

$\displaystyle|\Delta(n,m,h)|$ $\displaystyle\leqslant$
$\displaystyle\sum_{k=1}^{n}\Bigl{(}\sum_{j=1}^{k-1}|a_{j}(h)|+\tfrac{1}{2}|a_{0}(h)|\Bigr{)}+\tfrac{n}{m}\sum_{k=1}^{m}\Bigl{(}\sum_{j=1}^{k-1}|a_{j}(h)|+\tfrac{1}{2}|a_{0}(h)|\Bigr{)}$
(14) $\displaystyle\leqslant$
$\displaystyle\sum_{k=1}^{n}\sum_{j=0}^{k-1}|a_{j}(h)|+\tfrac{n}{m}\sum_{k=1}^{m}\sum_{j=0}^{k-1}|a_{j}(h)|$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n-1}(n-k)|a_{k}(h)|+\tfrac{n}{m}\sum_{k=0}^{m-1}(m-k)|a_{k}(h)|.$

We now fix any $n\in\mathbb{N}_{0}$, $m\in\mathbb{N}$, $h>0$, and we consider
the sums

$\displaystyle\sum_{k=1}^{n}\Bigl{(}\sum_{j=1}^{k-1}I(jh,h)+\tfrac{1}{2}I(0,h)\Bigr{)}=\sum_{k=1}^{n}\Bigl{(}\sum_{j=0}^{k-1}I(jh,h)-\tfrac{1}{2}I(0,h)\Bigr{)}=\sum_{k=1}^{n}\sum_{j=0}^{k-1}I(jh,h)-\tfrac{n}{2}I(0,h).$

For any $k\in\mathbb{N}$

$\displaystyle\sum_{j=0}^{k-1}I(jh,h)=\sum_{j=0}^{k-1}\int\limits_{\mathbb{R}}e^{ijhx}\,dR_{h}(x)=\int\limits_{\mathbb{R}}\sum_{j=0}^{k-1}e^{ijhx}\,dR_{h}(x)=\int\limits_{\mathbb{R}}\dfrac{e^{ikhx}-1}{e^{ihx}-1}\,dR_{h}(x).$

Next, we write

$\displaystyle\sum_{k=1}^{n}\sum_{j=0}^{k-1}I(jh,h)=\sum_{k=1}^{n}\int\limits_{\mathbb{R}}\dfrac{e^{ikhx}-1}{e^{ihx}-1}\,dR_{h}(x)=\int\limits_{\mathbb{R}}\sum_{k=1}^{n}\dfrac{e^{ikhx}-1}{e^{ihx}-1}\,dR_{h}(x)=\int\limits_{\mathbb{R}}\dfrac{\sum_{k=1}^{n}e^{ikhx}-n}{e^{ihx}-1}\,dR_{h}(x).$

Observe that

$\displaystyle\sum_{k=1}^{n}e^{ikhx}=\dfrac{e^{i(n+1)hx}-1}{e^{ihx}-1}-1.$

Hence

$\displaystyle\dfrac{\sum_{k=1}^{n}e^{ikhx}-n}{e^{ihx}-1}=\dfrac{e^{i(n+1)hx}-1}{(e^{ihx}-1)^{2}}-\dfrac{n+1}{e^{ihx}-1}=\dfrac{e^{i(n+1)hx}-1-(n+1)(e^{ihx}-1)}{(e^{ihx}-1)^{2}}.$

Thus

$\displaystyle\sum_{k=1}^{n}\sum_{j=0}^{k-1}I(jh,h)=\int\limits_{\mathbb{R}}\biggl{(}\dfrac{e^{i(n+1)hx}-1-(n+1)(e^{ihx}-1)}{(e^{ihx}-1)^{2}}\biggr{)}\,dR_{h}(x).$

Since

$\displaystyle I(0,h)=\int\limits_{\mathbb{R}}\,dR_{h}(x),$

we have

$\displaystyle\sum_{k=1}^{n}\sum_{j=0}^{k-1}I(jh,h)-\tfrac{n}{2}I(0,h)=\int\limits_{\mathbb{R}}\biggl{(}\dfrac{e^{i(n+1)hx}-1-(n+1)(e^{ihx}-1)}{(e^{ihx}-1)^{2}}-\dfrac{n}{2}\biggr{)}\,dR_{h}(x).$

Let us simplify the expression in the integral. We first observe that

$\displaystyle|e^{ihx}-1|^{2}=(1-\cos(hx))^{2}+(\sin(hx))^{2}=(1-\cos(hx))^{2}+1-(\cos(hx))^{2}=2(1-\cos(hx)).$

Next, we write

$\displaystyle|e^{ihx}-1|^{2}=(e^{ih}-1)(e^{-ih}-1)=e^{-ihx}(e^{ihx}-1)(1-e^{ihx})=-e^{-ihx}(e^{ihx}-1)^{2}.$

Therefore

$\displaystyle(e^{ihx}-1)^{2}=-e^{ihx}\cdot 2(1-\cos(hx)).$

So we write

$\displaystyle\dfrac{e^{i(n+1)hx}-1-(n+1)(e^{ihx}-1)}{(e^{ihx}-1)^{2}}$
$\displaystyle=$
$\displaystyle-\dfrac{\bigl{(}e^{i(n+1)hx}-1-(n+1)(e^{ihx}-1)\bigr{)}e^{-ihx}}{2(1-\cos(hx))}$
$\displaystyle=$
$\displaystyle-\dfrac{e^{inhx}-e^{-ihx}-(n+1)(1-e^{-ihx})}{2(1-\cos(hx))}$
$\displaystyle=$
$\displaystyle-\dfrac{e^{inhx}-1-n(1-e^{-ihx})}{2(1-\cos(hx))}.$

Here $1-e^{-ihx}=1-\cos(hx)+i\sin(hx)$. Then

$\displaystyle-\dfrac{e^{inhx}-1-n(1-e^{-ihx})}{2(1-\cos(hx))}=-\dfrac{e^{inhx}-1-in\sin(hx)}{2(1-\cos(hx))}+\dfrac{n}{2},$

and

$\displaystyle\dfrac{e^{i(n+1)hx}-1-(n+1)(e^{ihx}-1)}{(e^{ihx}-1)^{2}}-\dfrac{n}{2}=-\dfrac{e^{inhx}-1-in\sin(hx)}{2(1-\cos(hx))}.$

Thus

$\displaystyle\sum_{k=1}^{n}\sum_{j=0}^{k-1}I(jh,h)-\tfrac{n}{2}I(0,h)=\int\limits_{\mathbb{R}}\biggl{(}-\dfrac{e^{inhx}-1-in\sin(hx)}{2(1-\cos(hx))}\biggr{)}\,dR_{h}(x).$

Returning to (13), we analogously write

$\displaystyle\sum_{k=1}^{m}\sum_{j=0}^{k-1}I(jh,h)-\tfrac{m}{2}I(0,h)=\int\limits_{\mathbb{R}}\biggl{(}-\dfrac{e^{imhx}-1-im\sin(hx)}{2(1-\cos(hx))}\biggr{)}\,dR_{h}(x),$

and also

$\displaystyle\mathrm{Im}\,\biggl{\\{}\sum_{k=1}^{m}\sum_{j=0}^{k-1}I(jh,h)-\tfrac{m}{2}I(0,h)\biggr{\\}}=\int\limits_{\mathbb{R}}\biggl{(}-\dfrac{\sin(mhx)-m\sin(hx)}{2(1-\cos(hx))}\biggr{)}\,dR_{h}(x).$

Hence

$\displaystyle\mathrm{Ln}f(nh)-i\tfrac{n}{m}\mathrm{Arg}f(mh)$
$\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}}\biggl{(}-\dfrac{e^{inhx}-1-in\sin(hx)}{2(1-\cos(hx))}\biggr{)}\,dR_{h}(x)$
$\displaystyle{}\quad-i\tfrac{n}{m}\cdot\int\limits_{\mathbb{R}}\biggl{(}-\dfrac{\sin(mhx)-m\sin(hx)}{2(1-\cos(hx))}\biggr{)}\,dR_{h}(x)+\Delta(n,m,h)$
$\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}}\biggl{(}-\dfrac{e^{inhx}-1-i\tfrac{n}{m}\sin(mhx)}{2(1-\cos(hx))}\biggr{)}\,dR_{h}(x)+\Delta(n,m,h).$

According to (12), we obtain the representation

$\displaystyle\mathrm{Ln}f(nh)-i\tfrac{n}{m}\mathrm{Arg}f(mh)=\int\limits_{\mathbb{R}}\bigl{(}e^{inhx}-1-i\tfrac{n}{m}\sin(mhx)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x)+\Delta(n,m,h)$
(15)

for any $n\in\mathbb{N}_{0}$, $m\in\mathbb{N}$, and $h>0$.

Let us choose the sequence $(h_{l})_{l\in\mathbb{N}}$ and integers $n_{t,l}$
as in the statement of the theorem, i.e. $h_{l}\to 0$ as $l\to\infty$,
$n_{t,l}\colonequals\lfloor t/h_{l}\rfloor\in\mathbb{N}_{0}$, $t>0$,
$l\in\mathbb{N}$, and also (6) holds. Without loss of generality we can assume
that $h_{l}\leqslant\tfrac{1}{2}$ for all $l\in\mathbb{N}$. Let us define
$m_{l}\colonequals n_{1,l}=\lfloor 1/h_{l}\rfloor\in\mathbb{N}$,
$l\in\mathbb{N}$. We fix $t>0$. Observe that $0\leqslant t-n_{t,l}h_{l}<h_{l}$
and $0\leqslant 1-m_{l}h_{l}<h_{l}$ for any $l\in\mathbb{N}$. Since
$n_{t,l}h_{l}\to t$ and $m_{l}h_{l}\to 1$ as $l\to\infty$, we have

$\displaystyle\mathrm{Ln}f(t)-it\mathrm{Arg}f(1)$ $\displaystyle=$
$\displaystyle\lim_{l\to\infty}\biggl{(}\mathrm{Ln}f(n_{t,l}h_{l})-in_{t,l}h_{l}\mathrm{Arg}f(1)\biggr{)}$
$\displaystyle=$
$\displaystyle\lim_{l\to\infty}\biggl{(}\mathrm{Ln}f(n_{t,l}h_{l})-in_{t,l}h_{l}\,\dfrac{\mathrm{Arg}f(m_{l}h_{l})}{m_{l}h_{l}}\biggr{)}$
$\displaystyle=$
$\displaystyle\lim_{l\to\infty}\Bigl{(}\mathrm{Ln}f(n_{t,l}h_{l})-i\tfrac{n_{t,l}}{m_{l}}\,\mathrm{Arg}f(m_{l}h_{l})\Bigr{)}.$

Using (15), we write

$\displaystyle\mathrm{Ln}f(t)-it\mathrm{Arg}f(1)=\lim_{l\to\infty}\Biggl{(}\,\int\limits_{\mathbb{R}}\bigl{(}e^{in_{t,l}h_{l}x}-1-\tfrac{in_{l}}{m_{l}}\sin(m_{l}h_{l}x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x)+\Delta(n_{t,l},m_{l},h_{l})\Biggr{)}.$

We first consider the sequence $\Delta(n_{t,l},m_{l},h_{l})$,
$l\in\mathbb{N}$. Applying inequality (14), we get

$\displaystyle|\Delta(n_{t,l},m_{l},h_{l})|\leqslant\sum_{k=0}^{n_{t,l}-1}(n_{t,l}-k)|a_{k}(h_{l})|+\tfrac{n_{t,l}}{m_{l}}\sum_{k=0}^{m_{l}-1}(m_{l}-k)|a_{k}(h_{l})|.$

Due to (6), the first sum goes to 0 as $l\to\infty$. The second sum goes to 0
as $l\to\infty$ too, because $m_{l}=n_{1,l}$ and we again apply (6) with
$t=1$. Also $\tfrac{n_{t,l}}{m_{l}}=\tfrac{n_{t,l}h_{l}}{m_{l}h_{l}}\to t$ as
$l\to\infty$. Thus we showed that $\Delta(n_{t,l},m_{l},h_{l})\to 0$ as
$l\to\infty$.

We now have

$\displaystyle\mathrm{Ln}f(t)-it\gamma_{1}=\lim_{l\to\infty}\int\limits_{\mathbb{R}}\bigl{(}e^{in_{t,l}h_{l}x}-1-\tfrac{in_{l}}{m_{l}}\sin(m_{l}h_{l}x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x).$
(16)

where we set $\gamma_{1}=\mathrm{Arg}f(1)$. It is easily seen that

$\displaystyle\lim_{l\to\infty}\Bigl{[}\bigl{(}e^{in_{t,l}h_{l}x}-1-\tfrac{in_{l}}{m_{l}}\sin(m_{l}h_{l}x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\Bigr{]}=\bigl{(}e^{itx}-1-it\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\quad\text{for
any}\quad x\in\mathbb{R}.$ (17)

We will dominate the integrand in (16). Let $r$ be a positive real number. For
any $x\in\mathbb{R}$ satisfying $|x|>r$ we have the estimate

$\displaystyle\bigl{|}e^{in_{t,l}h_{l}x}-1-\tfrac{in_{l}}{m_{l}}\sin(m_{l}h_{l}x)\bigr{|}\tfrac{1+x^{2}}{x^{2}}\leqslant\Bigl{(}\bigl{|}e^{in_{t,l}h_{l}x}\bigr{|}+1+\bigl{|}\tfrac{in_{l}}{m_{l}}\sin(m_{l}h_{l}x)\bigr{|}\Bigr{)}\tfrac{1+x^{2}}{x^{2}}\leqslant\bigl{(}2+\tfrac{n_{l}}{m_{l}}\bigr{)}\tfrac{1+r^{2}}{r^{2}}.$

Since $n_{l}h_{l}\leqslant t$ and $m_{l}h_{l}>1-h_{l}\geqslant\tfrac{1}{2}$,
we have $\tfrac{n_{l}}{m_{l}}\leqslant\tfrac{n_{l}h_{l}}{m_{l}h_{l}}\leqslant
2t$. Thus

$\displaystyle\bigl{|}e^{in_{t,l}h_{l}x}-1-\tfrac{in_{l}}{m_{l}}\sin(m_{l}h_{l}x)\bigr{|}\tfrac{1+x^{2}}{x^{2}}\leqslant
2(1+t)\tfrac{1+r^{2}}{r^{2}},\quad|x|>r,\quad l\in\mathbb{N}.$ (18)

For the case $|x|\leqslant r$ we will use the representation

$\displaystyle
e^{in_{t,l}h_{l}x}-1-\tfrac{in_{l}}{m_{l}}\sin(m_{l}h_{l}x)=\bigl{(}e^{in_{t,l}h_{l}x}-1-in_{t,l}h_{l}x\bigr{)}-\tfrac{in_{l}}{m_{l}}\bigl{(}\sin(m_{l}h_{l}x)-m_{l}h_{l}x\bigr{)}.$

Then we have the estimate

$\displaystyle\bigl{|}e^{in_{t,l}h_{l}x}-1-\tfrac{in_{l}}{m_{l}}\sin(m_{l}h_{l}x)\bigr{|}\leqslant\bigl{|}e^{in_{t,l}h_{l}x}-1-in_{t,l}h_{l}x\bigr{|}+\tfrac{n_{l}}{m_{l}}\bigl{|}\sin(m_{l}h_{l}x)-m_{l}h_{l}x\bigr{|}.$

Applying the well-known inequalities $|e^{iy}-1-iy|\leqslant\tfrac{y^{2}}{2!}$
and $|\sin y-y|\leqslant\tfrac{|y|^{3}}{3!}$, $y\in\mathbb{R}$, we get

$\displaystyle\bigl{|}e^{in_{t,l}h_{l}x}-1-in_{t,l}h_{l}x\bigr{|}\leqslant\dfrac{(n_{t,l}h_{l}x)^{2}}{2!},\qquad\bigl{|}\sin(m_{l}h_{l}x)-m_{l}h_{l}x\bigr{|}\leqslant\dfrac{|m_{l}h_{l}x|^{3}}{3!}.$

Therefore

$\displaystyle\bigl{|}e^{in_{t,l}h_{l}x}-1-\tfrac{in_{l}}{m_{l}}\sin(m_{l}h_{l}x)\bigr{|}\tfrac{1+x^{2}}{x^{2}}\leqslant\Bigl{(}\tfrac{(n_{t,l}h_{l}x)^{2}}{2!}+\tfrac{|m_{l}h_{l}x|^{3}}{3!}\Bigr{)}\tfrac{1+x^{2}}{x^{2}}\leqslant\Bigl{(}\tfrac{(n_{t,l}h_{l})^{2}}{2!}+\tfrac{(m_{l}h_{l})^{3}|x|}{3!}\Bigr{)}(1+x^{2}).$

Since $|x|\leqslant r$, $n_{t,l}h_{l}\leqslant t$, and $m_{l}h_{l}\leqslant
1$, we obtain

$\displaystyle\bigl{|}e^{in_{t,l}h_{l}x}-1-\tfrac{in_{l}}{m_{l}}\sin(m_{l}h_{l}x)\bigr{|}\tfrac{1+x^{2}}{x^{2}}\leqslant\Bigl{(}\tfrac{t^{2}}{2!}+\tfrac{r}{3!}\Bigr{)}(1+r^{2}),\quad|x|\leqslant
r,\quad l\in\mathbb{N}.$ (19)

Thus (18) and (19) together yield the needed domination of the intagrand in
(16):

$\displaystyle\bigl{|}e^{in_{t,l}h_{l}x}-1-\tfrac{in_{l}}{m_{l}}\sin(m_{l}h_{l}x)\bigr{|}\tfrac{1+x^{2}}{x^{2}}\leqslant
C_{t,r},\quad x\in\mathbb{R},\quad l\in\mathbb{N},$

with the following constant (for fixed $t$ and $r$) majorant

$\displaystyle
C_{t,r}\colonequals\max\Bigl{\\{}\tfrac{2(1+t)}{r^{2}},\tfrac{t^{2}}{2!}+\tfrac{r}{3!}\Bigr{\\}}\cdot(1+r^{2}).$

We now return to (16). Due to the Jordan decomposition, we decompose
$G(x)=G^{+}(x)-G^{-}(x)$, $x\in\mathbb{R}$, with some non-decreasing functions
$G^{+}$ and $G^{-}$ from $\boldsymbol{V}$. So we have

$\displaystyle\mathrm{Ln}f(t)-it\gamma_{1}$ $\displaystyle=$
$\displaystyle\lim_{l\to\infty}\Biggl{(}\int\limits_{\mathbb{R}}\bigl{(}e^{in_{t,l}h_{l}x}-1-\tfrac{in_{l}}{m_{l}}\sin(m_{l}h_{l}x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG^{+}(x)$
$\displaystyle{}\qquad\qquad-\int\limits_{\mathbb{R}}\bigl{(}e^{in_{t,l}h_{l}x}-1-\tfrac{in_{l}}{m_{l}}\sin(m_{l}h_{l}x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG^{-}(x)\Biggr{)}.$

Due to the convergence (17) and the prove domination, by Lebesgue’s dominated
convergence theorem, we obtain

$\displaystyle\mathrm{Ln}f(t)-it\gamma_{1}$ $\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}}\bigl{(}e^{itx}-1-it\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG^{+}(x)-\int\limits_{\mathbb{R}}\bigl{(}e^{itx}-1-it\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG^{-}(x)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\bigl{(}e^{itx}-1-it\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x).$

where $t>0$ was fixed arbitrarily.

Let us show that the obtained formula holds for $t\leqslant 0$ too. Since
$\mathrm{Ln}f(0)=0$, it is easily seen that this is true for $t=0$. If $t<0$
then $-t>0$ and we can write

$\displaystyle\mathrm{Ln}f(-t)-i(-t)\gamma_{1}=\int\limits_{\mathbb{R}}\bigl{(}e^{-itx}-1-i(-t)\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x).$

Hence, according to the properties of the components $\ln|f(\cdot)|$ and
$\mathrm{Arg}f(\cdot)$ of the logarithm $\mathrm{Ln}f(\cdot)$, we get

$\displaystyle\mathrm{Ln}f(t)-it\gamma_{1}=\overline{\mathrm{Ln}f(-t)-i(-t)\gamma_{1}}=\int\limits_{\mathbb{R}}\bigl{(}e^{itx}-1-it\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x).$

Thus we obtain that

$\displaystyle\mathrm{Ln}f(t)=it\gamma_{1}+\int\limits_{\mathbb{R}}\bigl{(}e^{itx}-1-it\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x),\quad
t\in\mathbb{R},$

and, in particular, $F\in\boldsymbol{Q}$. $\Box$

Proof of Theorem 2. Let us define

$\displaystyle\varphi_{\pm}(t,h)\colonequals\exp\bigl{\\{}\pm\Delta^{2}_{h}\mathrm{Ln}f(t)\bigr{\\}}-1,\quad
t\in\mathbb{R},\,\,h\in\mathbb{R}.$

So we have

$\displaystyle\Delta^{2}_{h}\mathrm{Ln}f(t)=\pm\mathrm{Ln}\bigl{(}1+\varphi_{\pm}(t,h)\bigr{)},\quad
t\in\mathbb{R},\,\,h\in\mathbb{R}.$

Next, we define quantities $a_{k}(h)$ by the formula (5) with the same
$G\in\boldsymbol{V}$:

$\displaystyle
a_{k}(h)=\Delta^{2}_{h}\mathrm{Ln}f(kh)-2\int\limits_{\mathbb{R}}e^{ikh_{l}x}\bigl{(}\cos(hx)-1\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x),\quad
k\in\mathbb{N}_{0},\,\,h>0,$

On account of (7), for any $k\in\mathbb{N}_{0}$ and $h>0$ we accordingly have

$\displaystyle
a_{k}(h)=\Delta^{2}_{h}\mathrm{Ln}f(kh)\mp\bigl{(}\varphi_{\pm}(kh,h)-b_{k}(h)\bigr{)}=\pm\mathrm{Ln}\bigl{(}1+\varphi_{\pm}(t,h)\bigr{)}\mp\varphi_{\pm}(kh,h)\pm
b_{k}(h).$

Then

$\displaystyle|a_{k}(h)|\leqslant\bigl{|}\mathrm{Ln}\bigl{(}1+\varphi_{\pm}(kh,h)\bigr{)}-\varphi_{\pm}(kh,h)\bigr{|}+|b_{k}(h)|,\quad
k\in\mathbb{N}_{0},\,\,h\in\mathbb{R}.$

The first modulus less than $|\varphi_{\pm}(kh,h)|^{2}$ when
$|\varphi_{\pm}(kh,h)|\leqslant\tfrac{1}{2}$. Indeed, under this inequality,
we have

$\displaystyle\mathrm{Ln}\bigl{(}1+\varphi_{\pm}(kh,h)\bigr{)}=\ln\bigl{(}1+\varphi_{\pm}(kh,h)\bigr{)}=\sum_{k=1}^{\infty}\tfrac{(-1)^{k-1}}{k}\,\varphi_{\pm}(kh,h)^{k},$

and, consequently,

$\displaystyle\bigr{|}\mathrm{Ln}\bigl{(}1+\varphi_{\pm}(kh,h)\bigr{)}-\varphi_{\pm}(kh,h)\bigr{|}\leqslant\dfrac{1}{2}\sum_{k=2}^{\infty}|\varphi_{\pm}(kh,h)|^{k}=\dfrac{1}{2}\cdot\dfrac{|\varphi_{\pm}(kh,h)|^{2}}{1-|\varphi_{\pm}(kh,h)|}\leqslant|\varphi_{\pm}(kh,h)|^{2}.$

Thus we get the estimate

$\displaystyle|a_{k}(h)|\leqslant|\varphi_{\pm}(kh,h)|^{2}+|b_{k}(h)|,\quad
k\in\mathbb{N}_{0},\,\,h>0,$

under the condition
$\bigl{|}\varphi_{\pm}(kh,h)\bigr{|}\leqslant\tfrac{1}{2}$.

Let us choose the sequence $(h_{l})_{l\in\mathbb{N}}$ as in the statement of
the theorem, i.e. $h_{l}\to 0$ as $l\to\infty$, (8) and (9) hold. We also
arbitrarily fix $t>0$ and define the integers $n_{t,l}\colonequals\lfloor
t/h_{l}\rfloor\in\mathbb{N}_{0}$, $l\in\mathbb{N}$. So we have
$\varphi_{\pm}(kh_{l},h_{l})\to 0$, $l\to\infty$, and, in particular,
$|\varphi_{\pm}\bigl{(}kh_{l},h_{l})\bigr{|}\leqslant\tfrac{1}{2}$ for all
sufficiently large $l$ uniformly over $k=0,\ldots,n_{t,l}$. This is seen due
to the expression $\varphi_{\pm}$ through the values of $f$ on account of the
uniform continuity of $f$ and of the assumption $f(t)\neq 0$,
$t\in\mathbb{R}$. So for all sufficiently large $l\in\mathbb{N}$ we have

$\displaystyle\bigl{|}a_{k}(h_{l})|\leqslant|\varphi_{\pm}(kh_{l},h_{l})|^{2}+|b_{k}(h_{l})\bigr{|}\quad\text{for
all}\quad k=0,\ldots,n_{t,l}.$

This estimate together with (8) and (9) yield (6). Then, by Theorem 1, the
statement of Theorem 2 hold. $\Box$

Proof of Theorem 3. a) Under the assumptions of the theorem, we first show
that for any $t\in\mathbb{R}$ there exists $(\mathrm{Ln}f)^{\prime}(t)$. Let
us fix $t\in\mathbb{R}$ and consider the relation

$\displaystyle\dfrac{\mathrm{Ln}f(t+h)-\mathrm{Ln}f(t)}{h}=i\gamma+\int\limits_{\mathbb{R}}\biggl{(}\dfrac{e^{i(t+h)x}-e^{itx}}{h}-i\sin(x)\biggr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x),$
(20)

where $0<|h|<1$. We will propose majorant for the integrated function. Let $r$
be a positive real number. For any $x\in\mathbb{R}$ satisfying $|x|>r$ we have
the estimate

$\displaystyle\biggl{|}\dfrac{e^{i(t+h)x}-e^{itx}}{h}-i\sin(x)\biggr{|}\leqslant\biggl{|}\dfrac{e^{i(t+h)x}-e^{itx}}{h}\biggr{|}+|i\sin(x)|=\biggl{|}\dfrac{e^{ihx}-1}{h}\biggr{|}+|\sin(x)|.$

In the last sum, each term is bounded by $|x|$. So we have

$\displaystyle\biggl{|}\dfrac{e^{i(t+h)x}-e^{itx}}{h}-i\sin(x)\biggr{|}\leqslant
2|x|,\quad|x|>r.$

For the case $|x|\leqslant r$ we will use the representation

$\displaystyle\dfrac{e^{i(t+h)x}-e^{itx}}{h}-i\sin(x)=\dfrac{e^{i(t+h)x}-e^{itx}-ihx}{h}-i(\sin(x)-x).$

So we have the inequality

$\displaystyle\biggl{|}\dfrac{e^{i(t+h)x}-e^{itx}}{h}-i\sin(x)\biggr{|}\leqslant\biggl{|}\dfrac{e^{i(t+h)x}-e^{itx}-ihx}{h}\biggl{|}+|\sin(x)-x|.$

For the last term we have

$\displaystyle|\sin(x)-x|\leqslant\dfrac{|x|^{3}}{3!}\leqslant\dfrac{rx^{2}}{3!},\quad|x|\leqslant
r.$

Next, we represent

$\displaystyle\dfrac{e^{i(t+h)x}-e^{itx}-ihx}{h}=\dfrac{ix}{h}\int\limits_{t}^{t+h}(e^{iux}-1)\,du=\dfrac{ix}{h}\int\limits_{0}^{h}(e^{i(v+t)x}-1)\,dv.$

Since
$\bigl{|}e^{i(v+t)x}-1\bigr{|}\leqslant\bigl{|}(v+t)x\bigr{|}\leqslant\bigl{(}|v|+|t|\bigr{)}|x|$,
we get

$\displaystyle\Biggl{|}\int\limits_{0}^{h}(e^{i(v+t)x}-1)\,dv\Biggr{|}\leqslant\Biggl{|}\int\limits_{0}^{h}\bigl{|}e^{i(v+t)x}-1\bigr{|}\,dv\Biggr{|}\leqslant\Biggl{|}\int\limits_{0}^{h}\bigl{(}|v|+|t|\bigr{)}|x|\,dv\Biggr{|}=|x|\int\limits_{0}^{|h|}\bigl{(}|v|+|t|\bigr{)}\,dv.$

Therefore

$\displaystyle\biggl{|}\dfrac{e^{i(t+h)x}-e^{itx}-ihx}{h}\biggr{|}=\Biggl{|}\dfrac{x}{h}\int\limits_{0}^{h}\bigl{(}e^{i(v+t)x}-1\bigr{)}\,du\Biggr{|}\leqslant\dfrac{x^{2}}{|h|}\int\limits_{0}^{|h|}\bigl{(}|v|+|t|\bigr{)}\,dv\leqslant\bigl{(}|h|+|t|\bigr{)}x^{2}.$

Since $0<|h|<1$, we have

$\displaystyle\biggl{|}\dfrac{e^{i(t+h)x}-e^{itx}}{h}-i\sin(x)\biggr{|}\leqslant
C_{t,r}x^{2},\quad|x|\leqslant r,$

with $C_{t,r}=1+|t|+\tfrac{r}{3!}$.

The obtained upper estimates are integrable with $\tfrac{1+x^{2}}{x^{2}}$ by
$|G|(\cdot)$ over the coresponding sets :

$\displaystyle\int\limits_{|x|>r}2|x|\,\tfrac{1+x^{2}}{x^{2}}\,d|G|(x)=2\int\limits_{|x|>r}\tfrac{1+x^{2}}{|x|}\,d|G|(x)\leqslant\dfrac{2}{r}\int\limits_{|x|>r}(1+x^{2})\,d|G|(x)<\infty,$

and

$\displaystyle\int\limits_{|x|\leqslant
r}C_{t,r}x^{2}\,\tfrac{1+x^{2}}{x^{2}}\,d|G|(x)=C_{t,r}\int\limits_{|x|\leqslant
r}(1+x^{2})\,d|G|(x)<\infty.$

Thus the integrand in (20) has the integrable majorant.

Next, we observe that the integrand in (20) converges for any $x\neq 0$:

$\displaystyle\lim_{h\to
0}\biggl{(}\dfrac{e^{i(t+h)x}-e^{itx}}{h}-i\sin(x)\biggr{)}\tfrac{1+x^{2}}{x^{2}}=\bigl{(}ixe^{itx}-i\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}.$
(21)

Defining the right-hand side by continuity at $x=0$, it is seen that the
equality is true for this point too. Indeed, due to the well-known expansions

$\displaystyle e^{itx}=1+itx+O(x^{2}),\quad\sin(x)=x-O(x^{3}),\quad x\to 0,$

we have

$\displaystyle\bigl{(}ixe^{itx}-i\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}=\bigl{(}(ix)^{2}t+O(x^{3})\bigr{)}\tfrac{1+x^{2}}{x^{2}}\to-t,\quad
x\to 0.$

By the following well-known convention

$\displaystyle\Bigl{(}\bigl{(}e^{itx}-1-it\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\Bigr{)}\biggl{|}_{x=0}\colonequals\lim\limits_{x\to
0}\Bigl{(}\bigl{(}e^{itx}-1-it\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\Bigr{)}=-\dfrac{t^{2}}{2},$

we have

$\displaystyle\lim_{h\to
0}\biggl{(}\dfrac{e^{i(t+h)x}-e^{itx}}{h}-i\sin(x)\biggr{)}\tfrac{1+x^{2}}{x^{2}}=\lim_{h\to
0}\biggl{\\{}\dfrac{1}{h}\biggl{(}-\dfrac{(t+h)^{2}}{2}+\dfrac{t^{2}}{2}\biggr{)}\biggr{\\}}=\lim_{h\to
0}\biggl{\\{}\dfrac{1}{h}\biggl{(}-th-\dfrac{h^{2}}{2}\biggr{)}\biggr{\\}}=-t.$

Thus (21) holds for all $x\in\mathbb{R}$.

Due to the Jordan decomposition, we decompose $G(x)=G^{+}(x)-G^{-}(x)$,
$x\in\mathbb{R}$, with some non-decreasing functions $G^{+}$ and $G^{-}$ from
$\boldsymbol{V}$ that satisfy

$\displaystyle\int\limits_{\mathbb{R}}(1+x^{2})\,d|G^{+}|(x)<\infty,\quad\text{and}\quad\int\limits_{\mathbb{R}}(1+x^{2})\,d|G^{-}|(x)<\infty.$

For example, $G^{+}(x)\colonequals\tfrac{1}{2}\bigl{(}|G|(x)+G(x)\bigr{)}$ and
$G^{-}(x)\colonequals\tfrac{1}{2}\bigl{(}|G|(x)-G(x)\bigr{)}$,
$x\in\mathbb{R}$. So we write

$\displaystyle\int\limits_{\mathbb{R}}\biggl{(}\dfrac{e^{i(t+h)x}-e^{itx}}{h}-i\sin(x)\biggr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x)$
$\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}}\biggl{(}\dfrac{e^{i(t+h)x}-e^{itx}}{h}-i\sin(x)\biggr{)}\tfrac{1+x^{2}}{x^{2}}\,dG^{+}(x)$
$\displaystyle{}\qquad-\int\limits_{\mathbb{R}}\biggl{(}\dfrac{e^{i(t+h)x}-e^{itx}}{h}-i\sin(x)\biggr{)}\tfrac{1+x^{2}}{x^{2}}\,dG^{-}(x).$

Due to (21) and the proved domination, we now apply Lebesgue’s dominated
convergence theorem to the integrals in the right-hand side as $h\to 0$:

$\displaystyle\lim_{h\to
0}\int\limits_{\mathbb{R}}\biggl{(}\dfrac{e^{i(t+h)x}-e^{itx}}{h}-i\sin(x)\biggr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x)$
$\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}}\bigl{(}ixe^{itx}-i\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG^{+}(x)$
$\displaystyle{}\qquad-\int\limits_{\mathbb{R}}\bigl{(}ixe^{itx}-i\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG^{-}(x)$
$\displaystyle=$
$\displaystyle\int\limits_{\mathbb{R}}\bigl{(}ixe^{itx}-i\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x).$

Thus, according to formula (20), for any $t\in\mathbb{R}$ we have

$\displaystyle(\mathrm{Ln}f)^{\prime}(t)=\lim_{h\to
0}\dfrac{\mathrm{Ln}f(t+h)-\mathrm{Ln}f(t)}{h}=i\gamma+\int\limits_{\mathbb{R}}\bigl{(}ixe^{itx}-i\sin(x)\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x).$

We now prove that $(\mathrm{Ln}f)^{\prime\prime}(t)$ exists for any
$t\in\mathbb{R}$ and it is represented by formula (11). Let us fix
$t\in\mathbb{R}$ and consider the relation

$\displaystyle\dfrac{(\mathrm{Ln}f)^{\prime}(t+h)-(\mathrm{Ln}f)^{\prime}(t)}{h}=\int\limits_{\mathbb{R}}\biggl{(}ix\cdot\dfrac{e^{i(t+h)x}-e^{itx}}{h}\biggr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x),\quad
h\in\mathbb{R}\setminus\\{0\\}.$

Observe that

$\displaystyle\lim_{h\to
0}\Biggl{[}\biggl{(}ix\cdot\dfrac{e^{i(t+h)x}-e^{itx}}{h}\biggr{)}\dfrac{1+x^{2}}{x^{2}}\Biggr{]}=ixe^{itx}\lim_{h\to
0}\biggl{(}\dfrac{e^{ihx}-1}{h}\biggr{)}\cdot\dfrac{1+x^{2}}{x^{2}}=(ix)^{2}e^{itx}\,\dfrac{1+x^{2}}{x^{2}}=-e^{itx}(1+x^{2}),$

with the following domination uniformly over $h\in\mathbb{R}\setminus\\{0\\}$:

$\displaystyle\Biggl{|}\biggl{(}ix\cdot\dfrac{e^{i(t+h)x}-e^{itx}}{h}\biggr{)}\dfrac{1+x^{2}}{x^{2}}\Biggr{|}=|x|\cdot\biggl{|}\dfrac{e^{ihx}-1}{h}\biggr{|}\cdot\dfrac{1+x^{2}}{x^{2}}\leqslant|x|\cdot|x|\cdot\dfrac{1+x^{2}}{x^{2}}=1+x^{2}.$

Due to (10), the analogous application of Lebesgue’s dominated convergence
theorem is valid and it yields

$\displaystyle\lim_{h\to
0}\int\limits_{\mathbb{R}}\biggl{(}ix\cdot\dfrac{e^{i(t+h)x}-e^{itx}}{h}\biggr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x)=-\int\limits_{\mathbb{R}}e^{itx}(1+x^{2})\,dG(x).$

Thus for any $t\in\mathbb{R}$ we obtain the required equality

$\displaystyle(\mathrm{Ln}f)^{\prime\prime}(t)=\lim_{h\to
0}\dfrac{(\mathrm{Ln}f)^{\prime}(t+h)-(\mathrm{Ln}f)^{\prime}(t)}{h}=-\int\limits_{\mathbb{R}}e^{itx}(1+x^{2})\,dG(x).$

b) We now assume that (11) holds with some function $G\in\boldsymbol{V}$ that
satisfies (10). We first will show that the function
$(\mathrm{Ln}f)^{\prime\prime}$ is uniformly continuous on $\mathbb{R}$. For
any $h\in\mathbb{R}$ we have

$\displaystyle\sup_{t\in\mathbb{R}}\bigl{|}(\mathrm{Ln}f)^{\prime\prime}(t+h)-(\mathrm{Ln}f)^{\prime\prime}(t)\bigr{|}$
$\displaystyle=$
$\displaystyle\sup_{t\in\mathbb{R}}\biggl{|}\int\limits_{\mathbb{R}}(e^{i(t+h)x}-e^{itx})(1+x^{2})\,dG(x)\biggr{|}$
$\displaystyle\leqslant$
$\displaystyle\sup_{t\in\mathbb{R}}\int\limits_{\mathbb{R}}\bigr{|}e^{i(t+h)x}-e^{itx}\bigl{|}(1+x^{2})\,d|G|(x)$
$\displaystyle\leqslant$
$\displaystyle\int\limits_{\mathbb{R}}|e^{ihx}-1|(1+x^{2})\,d|G|(x).$

Hence for any $r>0$ the following estimate holds

$\displaystyle\sup_{t\in\mathbb{R}}\bigl{|}(\mathrm{Ln}f)^{\prime\prime}(t+h)-(\mathrm{Ln}f)^{\prime\prime}(t)\bigr{|}$
$\displaystyle\leqslant$ $\displaystyle\int\limits_{|x|\leqslant
r}|e^{ihx}-1|(1+x^{2})\,d|G|(x)+\int\limits_{|x|>r}|e^{ihx}-1|(1+x^{2})\,d|G|(x)$
$\displaystyle\leqslant$ $\displaystyle\int\limits_{|x|\leqslant
r}|hx|(1+x^{2})\,d|G|(x)+\int\limits_{|x|>r}\bigl{(}|e^{ihx}|+1\bigr{)}(1+x^{2})\,d|G|(x)$
$\displaystyle\leqslant$ $\displaystyle|h|r\int\limits_{|x|\leqslant
r}(1+x^{2})\,d|G|(x)+2\int\limits_{|x|>r}(1+x^{2})\,d|G|(x).$

Let us fix arbitrary $\varepsilon>0$. Due to (10), we first chose
$r=r_{\varepsilon}$ such that

$\displaystyle\int\limits_{|x|>r_{\varepsilon}}(1+x^{2})\,d|G|(x)\leqslant\varepsilon,$

and after this we choose $h_{\varepsilon}>0$ such that

$\displaystyle h_{\varepsilon}r_{\varepsilon}\int\limits_{|x|\leqslant
r_{\varepsilon}}(1+x^{2})\,d|G|(x)\leqslant\varepsilon.$

Then for any $h\in\mathbb{R}$ satisfying $|h|\leqslant h_{\varepsilon}$ we
obtain

$\displaystyle\sup_{t\in\mathbb{R}}\bigl{|}(\mathrm{Ln}f)^{\prime\prime}(t+h)-(\mathrm{Ln}f)^{\prime\prime}(t)\bigr{|}\leqslant
h_{\varepsilon}r_{\varepsilon}\int\limits_{|x|\leqslant
r}(1+x^{2})\,d|G|(x)+2\int\limits_{|x|>r_{\varepsilon}}(1+x^{2})\,d|G|(x)\leqslant
3\varepsilon.$

This means that $(\mathrm{Ln}f)^{\prime\prime}$ is uniformly continuous on
$\mathbb{R}$.

Due to the proved continuity of $(\mathrm{Ln}f)^{\prime\prime}$, we can write

$\displaystyle(\mathrm{Ln}f)^{\prime}(s)=(\mathrm{Ln}f)^{\prime}(0)+\int\limits_{0}^{s}(\mathrm{Ln}f)^{\prime\prime}(u)\,du,\quad
s\in\mathbb{R}.$

Since $\mathrm{Ln}f(0)=0$ and $(\mathrm{Ln}f)^{\prime}$ must be continuous, we
have

$\displaystyle\mathrm{Ln}f(t)=\int\limits_{0}^{t}(\mathrm{Ln}f)^{\prime}(s)\,ds,\quad
t\in\mathbb{R}.$

Therefore

$\displaystyle\mathrm{Ln}f(t)=\int\limits_{0}^{t}\biggl{(}(\mathrm{Ln}f)^{\prime}(0)+\int\limits_{0}^{s}(\mathrm{Ln}f)^{\prime\prime}(u)\,du\biggr{)}\,ds=t\cdot(\mathrm{Ln}f)^{\prime}(0)+\int\limits_{0}^{t}\biggl{(}\int\limits_{0}^{s}(\mathrm{Ln}f)^{\prime\prime}(u)\,du\biggr{)}\,ds,\quad
t\in\mathbb{R}.$

Let us fix $t\in\mathbb{R}$ and $h>0$. We write

$\displaystyle\Delta^{2}_{h}\mathrm{Ln}f(t)$ $\displaystyle=$
$\displaystyle(t-h)\cdot(\mathrm{Ln}f)^{\prime}(0)+\int\limits_{0}^{t-h}\biggl{(}\int\limits_{0}^{s}(\mathrm{Ln}f)^{\prime\prime}(u)\,du\biggr{)}\,ds$
$\displaystyle{}+(t+h)\cdot(\mathrm{Ln}f)^{\prime}(0)+\int\limits_{0}^{t+h}\biggl{(}\int\limits_{0}^{s}(\mathrm{Ln}f)^{\prime\prime}(u)\,du\biggr{)}\,ds$
$\displaystyle{}-2t\cdot(\mathrm{Ln}f)^{\prime}(0)-2\int\limits_{0}^{t}\biggl{(}\int\limits_{0}^{s}(\mathrm{Ln}f)^{\prime\prime}(u)\,du\biggr{)}\,ds.$

Thus we get

$\displaystyle\Delta^{2}_{h}\mathrm{Ln}f(t)=\int\limits_{t}^{t+h}\biggl{(}\int\limits_{0}^{s}(\mathrm{Ln}f)^{\prime\prime}(u)\,du\biggr{)}\,ds+\int\limits_{t}^{t-h}\biggl{(}\int\limits_{0}^{s}(\mathrm{Ln}f)^{\prime\prime}(u)\,du\biggr{)}\,ds.$

On account of (11), we have

$\displaystyle\int\limits_{t}^{t\pm
h}\biggl{(}\int\limits_{0}^{s}(\mathrm{Ln}f)^{\prime\prime}(u)\,du\biggr{)}\,ds$
$\displaystyle=$ $\displaystyle-\int\limits_{t}^{t\pm
h}\biggl{(}\int\limits_{0}^{s}\biggl{(}\int\limits_{\mathbb{R}}e^{iux}(1+x^{2})\,dG(x)\biggr{)}\,du\biggr{)}\,ds$
$\displaystyle=$
$\displaystyle-\int\limits_{\mathbb{R}}\biggl{(}\int\limits_{t}^{t\pm
h}\biggl{(}\int\limits_{0}^{s}e^{iux}\,du\biggr{)}\,ds\biggr{)}(1+x^{2})\,dG(x).$

It is easily seen that the changing of the order of integration is valid due
to the boundedness of $e^{iux}$ for $u,x\in\mathbb{R}$. So we get

$\displaystyle\Delta^{2}_{h}\mathrm{Ln}f(t)=-\int\limits_{\mathbb{R}}\Biggl{[}\int\limits_{t}^{t+h}\biggl{(}\int\limits_{0}^{s}e^{iux}\,du\biggr{)}\,ds+\int\limits_{t}^{t-h}\biggl{(}\int\limits_{0}^{s}e^{iux}\,du\biggr{)}\,ds\Biggr{]}(1+x^{2})\,dG(x).$

Observe that

$\displaystyle\int\limits_{t}^{t\pm
h}\biggl{(}\int\limits_{0}^{s}e^{iux}\,du\biggr{)}\,ds=\int\limits_{t}^{t\pm
h}\biggl{(}\dfrac{e^{isx}-1}{ix}\biggr{)}\,ds=\dfrac{e^{i(t\pm
h)x}-e^{itx}}{(ix)^{2}}-\dfrac{\pm h}{ix}.$

Hence

$\displaystyle\Delta^{2}_{h}\mathrm{Ln}f(t)$ $\displaystyle=$
$\displaystyle-\int\limits_{\mathbb{R}}\biggl{(}\dfrac{e^{i(t+h)x}-e^{itx}}{(ix)^{2}}+\dfrac{e^{i(t-h)x}-e^{itx}}{(ix)^{2}}\biggr{)}(1+x^{2})\,dG(x)$
$\displaystyle=$
$\displaystyle-\int\limits_{\mathbb{R}}e^{itx}\biggl{(}\dfrac{e^{ihx}-1}{(ix)^{2}}+\dfrac{e^{-ihx}-1}{(ix)^{2}}\biggr{)}(1+x^{2})\,dG(x)$
$\displaystyle=$
$\displaystyle-\int\limits_{\mathbb{R}}e^{itx}\biggl{(}\dfrac{2\cos(hx)-2}{(ix)^{2}}\biggr{)}(1+x^{2})\,dG(x).$

Thus for any $t\in\mathbb{R}$ and $h>0$ we obtain

$\displaystyle\Delta^{2}_{h}\mathrm{Ln}f(t)=2\int\limits_{\mathbb{R}}e^{itx}\bigl{(}\cos(hx)-1\bigr{)}\tfrac{1+x^{2}}{x^{2}}\,dG(x).$

Using Corollary 1, we come to the required assertion. $\Box$

##  4 Acknowledgments

This research was supported by the Ministry of Science and Higher Education of
the Russian Federation, agreement 075-15-2022-289 date 06/04/2022.

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