00footnotetext:

# A digital CDS technique and the performance testing††thanks: Partially
supported by National Natural Science Foundation of China (10978002)

LIU Xiao-Yan1,2 LU Jing-Bin1 YANG Yan-Ji1,2 LU Bo2
WANG Yu-Sa2 XU Yu-Peng2 CUI Wei-Wei2 LI Wei2
LI Mao-Shun2 WANG Juan2 HAN Da-Wei2 CHEN Tian-Xiang2
HUO Jia2 HU Wei2 ZHANG Yi2 ZHU Yue2
ZHANG Zi-Liang2 YIN Guo-He2 Wang Yu2 Zhao Zhong-Yi2,3
Fu Yan-Hong2,3 Zhang Ya2,3 MA Ke-Yan1 CHEN Yong2;1<EMAIL_ADDRESS>1
College of Physics, Jilin University, No.2699, Qianjin Road, Changchun 130023,
China
2 Key Laboratory for Particle Astrophysics, Institute of High Energy Physics,
Chinese Academy of Sciences (CAS), 19B Yuquan Road, Beijing 100049, China
3 School of Physical Science and Technology, Yunnan University, Cuihu North
Road 2, Kunming, 650091, China

###### Abstract

Readout noise is a critical parameter for characterizing the performance of
charge-coupled devices (CCDs), which can be greatly reduced by the correlated
double sampling (CDS) circuit. However, conventional CDS circuit inevitably
introduces new noises since it consists of several active analog components
such as operational amplifiers. This paper proposes a digital CDS circuit
technique, which transforms the pre-amplified CCD signal into a train of
digital presentations by a high-speed data acquisition card directly without
the noisy CDS circuit first, then implement the digital CDS algorithm through
numerical method. The readout noise of 3.3 e- and the energy resolution of 121
<EMAIL_ADDRESS>can be achieved via the digital CDS technique.

###### keywords:

charge-coupled devices, readout noise, correlated double sampling

###### pacs:

2

9.30.Kv, 29.40.Wk, 29.85.-c

00footnotetext: $\scriptstyle\copyright$2013 Chinese Physical Society and the
Institute of High Energy Physics of the Chinese Academy of Sciences and the
Institute of Modern Physics of the Chinese Academy of Sciences and IOP
Publishing Ltd

##  1 Introduction

Owing to its advantages on smaller size, lower power dissipation, wider
response spectrum range, lower noise and higher resolution, Charge-coupled
devices (CCDs) have been widely applied in industrial inspection, night
vision, visible imaging, soft X-ray astronomical observations and so on. As a
critical parameter that has to be considered in designing and operating CCD,
the noises of CCD mainly come from two mechanisms, one comes from CCD itself,
including shot noise, dark current noise and the transfer noise[1], and the
other comes from the operation of CCD, such as output amplifier noise and the
reset noise.

It is common to suppress the reset noise with the correlated double sampling
(CDS) circuit. However, traditional CDS introduces new noises, and leaves some
useful information behind, e.g. the original waveform. The digital CDS, e.g.
Gach J L, 2003[2], is able to store all the initial information, which
provides the possibility for varies of the backend data process. In practice,
it shows a perfect performance on the reduction of the readout noise.

In this paper, a digital correlated double sampling circuit technique is
proposed. A PCI-9846H data acquisition card, which is manufactured by ADLINK
Inc., has been used to convert the pre-amplified CCD signal into the digital
representations. The digital CDS system can record a large amount of data and
the original waveform could be derived from these data for further analyzing
of the signal characteristics. So the data processing could be optimized to
get better performance such as a lower readout noise. The relationship between
the readout noise of the digital CDS system and the sample number has been
investigated. The results of the measured data show that the proposed digital
CDS system is better than the conventional analog CDS system, which has been
assembled on the low energy X-ray telescope (LE) of the Hard X-ray Modulation
Telescope (HXMT)[3], whatever the CCD operating temperature is below
$-110\,^{\circ}\mathrm{C}$, or the CCD is degraded after irradiation by
proton.

##  2 Reset noise and Correlated Double Sampling

Charge generation, collection, transfer and measurement, make all of the
operation of CCD. Taking two-phase CCD for example, the electronic schematic
of CCD output region is illustrated on the left part in Fig. 2, the right part
in Fig. 2 is the simple diagram of output signal. Reset noise is generated by
the periodic reset of the sense-node capacitance ($C_{\mathrm{n}}$) by the
reset switch. Reset noise, $n_{\mathrm{R}}(e^{-})$, can be expressed as

$\displaystyle n_{\mathrm{R}}(e^{-})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{kTC}}{q},$ (1)

where $k$ is the Boltzmann’s constant ($1.38\times 10^{-23}$ J/K), $T$ is the
absolute temperature (K), $C$ is the capacitance (F), and $q$ is the
electronic charge. Reset noise can also be called kTC noise [4]. The noise
voltage changes very quickly when reset switch is on, leaving an undefined
level on the $C_{\mathrm{n}}$ after the switch is off. After the reset switch
has turned off, the noise voltage does not change significantly over a pixel
period. Reset noise can be removed or reduced by using CDS circuit technique
which subtracts the sample taken from the signal level from the sample taken
from the reference level. However, conventional CDS circuit inevitably
introduces new noises since it consists of several active analog components
such as operational amplifiers. In this paper, we use digital CDS system
instead of the conventional analog CDS.

The schematic of CCD output region and output signal.

##  3 Brief introduction to the experiment equipment

###  3.1 The target CCD detector

The CCD detector discussed in this paper is CCD236. It is a two-phased swept
charge device (SCD) which is designed by the E2V Inc. for HXMT-LE,
collaborated with the Institute of High Energy Physics (IHEP). CCD236 achieves
a high transfer rate by abandoning the position information of the X-ray
photons. Charges are first transferred to the electrodes on the diagonal along
the way perpendicular of the electrodes, and then to the read-out section,
which locates in the center of the chips, along the diagonal. Besides the real
output, there is a dummy output the same as the real one. As the dummy output
only carries common-mode noise, the negative effect of common-mode noise can
be greatly suppressed by subtracting that noise from the real output
signal[5].

###  3.2 The data acquisition card

The PCI-9846H is a data acquisition card manufactured by the ADLINK inc.,
which is designed for digitizing high frequency and wide dynamic range
signals, taking advantages in the rapid read-out speed and lower system noise.
Its onboard 512 MB acquisition memory enables it to store the data of the
waveform for a considerable long time. It has a high linearity 16-bit A/D
converter, and has a sampling rate up to 40 million samples per second (MS/s).
In addition, the analog voltage range is $\pm$5V[6].

###  3.3 The readout system using the digital CDS

Digital CDS system block diagram is shown as Fig. 3.3, which is made up of
three parts: pre-amplifier, voltage buffer and data acquisition system. The
voltage gain of pre-amplifier is 11x. The voltage buffer ensures the accuracy
of the signal amplitude. The PCI-9846H digitizes the analog input at a certain
rate, and sends the conversion results to computer via PCI (peripheral
component interconnection) for backend data processing.

The digital CDS system block diagram.

##  4 The results of performance testing

###  4.1 Testing results at low CCD operating temperature

To investigate the performance of the digital CDS system, a test has been
carried out with the CCD operated below about $-110\,^{\circ}\mathrm{C}$, as
the effect of the dark current from the CCD can be ignored at such a low
temperature. The driving clock for the CCD is 83.3kHz, while the digital data
acquisition system is sampling at 40 MS/s (million samples per second). An
Fe-55 radioactive source is used to measure the energy resolution.

As shown in Fig. 4.1, the waveform is reconstructed from total 480 sample
points. Due to such non-idealities as clock feed-through, parasitic effects as
well as signal integrity problems, glitches are prominent at the edges of
driving clocks. However, only the two steady parts every cycle make sense,
since signals in these two parts are completely settled. A certain number of
sample points are selected as effective points for processing.

The typical signal waveform.

The flat section of the curve in Fig. 4.1 is selected on the basis of that the
difference of the voltage between two adjacent sampling points is no more than
40 mV. Taking the section of the signal level as an example, the initial point
(the 389th point) is exactly placed in the middle of the flat section, the
second one is located just left to the initial one (i.e. the 388th point), the
third on is situated just right to the initial one (i.e. the 390th point), and
so on. All the points right to the 477th point are ignored. The way to select
points on the reference level is just the same, and the initial point in this
case is the 150th point. During the following numerical calculation, the
number of effective points increases from 2 to 392 in order to find out the
optimal number. Eventually, the voltage amplitude of the incoming X-ray
photons can be obtained by the equation

$\displaystyle V_{\mathrm{ph}}$ $\displaystyle=$
$\displaystyle\overline{V_{\mathrm{Ref}}}-\overline{V_{\mathrm{Signal}}},$ (2)

where $\overline{V_{\mathrm{Signal}}}$ is the average voltage of the sampling
points on the flat section of the signal level, and
$\overline{V_{\mathrm{Ref}}}$ is the average voltage of the sampling points on
the flat section of the reference level.

Readout noise vs. the number of effective sampling points under 83.3kHz
operating frequency.

As shown in Fig. 4.1, below 350 samples, the readout noise decreases with the
effective points increasing. The function (Eq. 2) has been used to fit the
relationship between the readout noise and the number of effective sampling
points, which describes the error of the mean value for the independent
sampling data.

We try to use Eq. 3 to fit the data in Fig. 4.1.

$\displaystyle n$ $\displaystyle=$
$\displaystyle\frac{A_{\mathrm{0}}}{\sqrt{N}},$ (3)

where $n$ is readout noise ($\mathrm{e^{-}}$), and $N$ is the number of
effective sampling points. The fitting curve is shown in Fig. 4.1 as the solid
line, with the value of $A_{\mathrm{0}}$ $64.1\pm 0.6\mathrm{e^{-}}$. When
only one point are sampled on each level ($N=2$), the noise is about
$20\mathrm{e^{-}}$, which is significantly lower than the value of the reset
noise (53$\mathrm{e^{-}}$) calculated by Eq. 1, where $C$ is 0.032 pF for
CCD236 and $T$ is 160 K. However, the declination of the data in Fig. 4.1
indicates that there are also other sources of the noises or interferences
besides the reset noise.

Eq. 3 cannot fit the data well at both ends. In detail, the readout noise
increases again when the number of effective points exceeds 350, because the
data beyond the steady parts are used. The ranges of 350, 392 effective points
are shown in Fig. 4.1. Obviously, the optimal effective point number is around
350, so we select this number in all the test below in this paper. As the
sampling points are fewer, shown in left-hand side of Fig. 4.1, the declining
slope of the data is significantly shallower than that of the $1/\sqrt{N}$. It
is suggested that the sampling data are not independent in this situation.

We calculated the cross-correlation function of the voltage of each sampling
point and the initial sampling point (i.e. the 389th point). As shown in Fig.
4.1, the sampling points located faraway are weakly correlated, while the
adjacent sampling points show sharp oscillation, from strongly correlated to
anti-correlated. It indicates that the readout noise is probably not the white
noise.

The cross-correlation between the voltages of each sampling point and the
initial sampling point (the 389th point). The upper plot is the cross-
correlation data, and the lower plot is the mean value of the waveform for
reference.

For this purpose, the power spectrum of the readout noise (the part of the
signal level) are calculated by the fast Fourier transform (FFT) method. In
the power spectrum (Fig. 4.1), there are low-frequency noises substantially,
most likely $1/f^{\mathrm{\alpha}}$ noise with $\alpha$ from 1 to 2. The lower
frequency noise component may be responsible to the weak correlation in long
distance. On the other hand, it is found that there is an enhancement at about
5 MHz in the power spectrum, which can explain the strong correlation in short
distance. The lower frequency noise may come from the onchip field effect
transistor (FET) of the CCD readout circuit[7], and the $\mathrm{\sim}$5 MHz
noise is probably caused by the outside interference or the CCD itself.

The normalized power spectrum of the readout noise in the part of the signal
level, with a $1/f^{2}$ noise shown in dashed line for reference.

To determine the performance, especially the energy linearity, an X-ray tube
with the copper target is used. From the spectrum which is shown in Fig. 4.1,
four distinct peaks besides the noise peak are found. The integral
nonlinearity (INL) is satisfied with the value of $0.4\%$.

The spectrum from a copper target bombarded with X-rays generated by an X-ray
tube.

The integral nonlinearity of the digital CDS system.

To investigate the energy resolution of the system, the test has been
conducted with an Fe-55 radioactive source. The spectrum is shown in Fig. 4.1.
Fitting the noise peak and Mn Kα (5.9 keV) peak by Gaussian, the parameters as
the mean value and the standard deviation can be obtained, the energy
resolution of the CCD detector system is calculated to be
<EMAIL_ADDRESS>(FWHM), and the readout noise is 3.342$\pm$0.002e-.

An Fe-55 spectrum of CCD236 at $-113\,^{\circ}\mathrm{C}$.

Within the temperature range of $-116\,^{\circ}\mathrm{C}$ to
$-110\,^{\circ}\mathrm{C}$, the readout noise and energy resolution of the
digital CDS system and the conventional CDS readout system under the same
condition are compared in Table 4.1, which illustrates that the performance of
digital CDS system is better than that of the conventional CDS system.

The performance comparison between two systems.  CCD operational Energy
resolution(FWHM)/eV Readout noise/e- Digital CDS Conventional CDS Digital CDS
Conventional CDS temperature $/\,^{\circ}\mathrm{C}$ system system system
system -110.3 123.4$\pm$0.4 131.9$\pm$1.1 3.487$\pm$0.002 5.251$\pm$0.002
-113.0 121.0$\pm$0.4 131.0$\pm$1.1 3.342$\pm$0.002 5.428$\pm$0.002 -115.1
121.7$\pm$0.4 131.4$\pm$1.1 3.331$\pm$0.002 5.208$\pm$0.002 -116.2
121.3$\pm$0.4 129.7$\pm$1.0 3.329$\pm$0.002 5.416$\pm$0.002

###  4.2 The performance comparison between two systems after proton
irradiation

The readout noise and energy resolution of a proton-irradiated CCD236, with a
dose of $3\times 10^{8}\mathrm{p/cm}^{2}$, are measured with the temperature
ranging from $-60\,^{\circ}\mathrm{C}$ to $-10\,^{\circ}\mathrm{C}$[8].

As shown in Fig. 4.2, the readout noise of the digital CDS system is about 1
$\mathrm{e^{-}}$ lower than that of the conventional CDS readout system in the
temperature range from $-60\,^{\circ}\mathrm{C}$ to $-30\,^{\circ}\mathrm{C}$.
And the difference of readout noise gradually increases to 4 $\mathrm{e^{-}}$
when the temperature warms up to -10 . In addition, the energy resolution of
the digital CDS system also shows a better result similarly (Fig. 4.2). The
performance of digital CDS system is better than conventional CDS system after
proton irradiation. Therefore, it is a promising way to suppress the negative
effects of the proton-irradiation damage to the CCDs.

The comparison of the readout noise. The open circles, the solid circles and
the plus signs (‘+’) represent the data of the digital CDS system, the
conventional CDS system and the residuals, respectively.

The same as Fig. 4.2, but the comparison of the energy resolution.

##  5 Conclusions

In this paper, we have proposed a digital CDS system. Under the same
condition, the readout noise is reduced monotonously as the number of sampling
points per cycle increases to 350. And from the waveform obtained from the
digital CDS system, we have analysed the sources of the noise by the power
spectrum. Compared to the conventional CDS system, the digital CDS system has
a lower readout noise and a better energy resolution, not only under very low
CCD operating temperatures but also on the CCD after the proton-irradiation.
Additionally, the digital CDS system shows a satisfied result of INL. In
summary, the digital CDS is a promising technique to reduce the readout noise
of CCDs, to suppress the effect of the interferences, and to investigate the
properties of the noise.

## References

  * [1] ZHANG Jun-Min, XU Yu-Peng, CHEN Yong et al. Journal of Chinese Inertial Technology, 2009, 17(6): 666-669(in Chinese)
  * [2] Gach j L, Darson D, Guillaume C et al. The Astronomical Society of the Pacific, 2003, 115:1068-1071
  * [3] Li Ti-Pei. Nuclear Physics B Proceedings Supplements, 2007, 166:131-139
  * [4] Janesick J R. Scientific Charge-Coupled Devices, Bellingham, Washington: SPIE PRESS, 2001: 537-538
  * [5] Wang Yu-Sa, Chen Yong, Xu Yu-Peng, et al. Chinese Physics C, 2009, 33(X): 1-5
  * [6] http$\mathrm{:}$//www.adlinktech.com/PD/marketing/Datasheet/PCI-9816+9826+9846/PCI-9816+9826+9846 Datasheet en 1.pdf
  * [7] Centen P. $IEEETrans.ElectronDevices$, 1991, 38:1206-1216
  * [8] Yang Yan-Ji, Lu Jing-Bin, Wang Yu-Sa et al. Chinese Physics C, 2014, 38(8): 086004


# Operator relations characterizing higher-order differential operators

Włodzimierz Fechner, Eszter Gselmann and Aleksandra Świątczak

###### Abstract

Let $r$ be a positive integer, $N$ be a nonnegative integer and
$\Omega\subset\mathbb{R}^{r}$ be a domain. Further, for all multi-indices
$\alpha\in\mathbb{N}^{r}$, $|\alpha|\leq N$, let us consider the partial
differential operator $D^{\alpha}$ defined by

$D^{\alpha}=\frac{\partial^{|\alpha|}}{\partial
x_{1}^{\alpha_{1}}\cdots\partial x_{r}^{\alpha_{r}}},$

where $\alpha=(\alpha_{1},\ldots,\alpha_{r})$. Here by definition we mean
$D^{0}\equiv\mathrm{id}$. An easy computation shows that if
$f,g\in\mathscr{C}^{N}(\Omega)$ and $\alpha\in\mathbb{N}^{r},|\alpha|\leq N$,
then we have

$D^{\alpha}(f\cdot
g)=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}D^{\beta}(f)\cdot
D^{\alpha-\beta}(g).$

This paper is devoted to the study of identity $(\ast)$ in the space
$\mathscr{C}(\Omega)$. More precisely, if $r$ is a positive integer, $N$ is a
nonnegative integer and $\Omega\subset\mathbb{R}^{r}$ is a domain, then we
describe those mappings
$T_{\alpha}\colon\mathscr{C}(\Omega)\to\mathscr{C}(\Omega)$,
$\alpha\in\mathbb{N}^{r},|\alpha|\leq N$ that satisfy identity $(\ast)$ for
all possible multi-indices $\alpha\in\mathbb{N}^{r}$, $|\alpha|\leq N$. Our
main result says that if the domain is $\mathscr{C}(\Omega)$, then the
mappings $T_{\alpha}$ are of a rather special form. Related results in the
space $\mathscr{C}^{N}(\Omega)$ are also presented.

##  1 Introduction and preliminaries

In this paper the set of real numbers is denoted by $\mathbb{R}$, the set of
complex numbers by $\mathbb{C}$, and the set of nonnegative integers by
$\mathbb{N}$.

Let $r$ be a positive integer. Elements of $\mathbb{N}^{r}$ will be called
$r$-dimensional multi-indices. Sums and differences of multi-indices (of the
same dimension) are meant to be componentwise, i.e., if
$\alpha,\beta\in\mathbb{N}^{r}$, then

$\alpha\pm\beta=(\alpha_{1}\pm\beta_{1},\ldots,\alpha_{r}\pm\beta_{r})$

Further, if $\alpha,\beta\in\mathbb{N}^{r}$, then we write $\alpha\leq\beta$
if for all $i=1,\ldots,r$ we have $\alpha_{i}\leq\beta_{i}$, where
$\alpha=(\alpha_{1},\ldots,\alpha_{r})$ and
$\beta=(\beta_{1},\ldots,\beta_{r})$. If for the multi-indices
$\alpha,\beta\in\mathbb{N}^{r}$ we have $\alpha\leq\beta$ and
$\alpha\neq\beta$, we will write $\alpha<\beta$. By the height of a multi-
index $\alpha\in\mathbb{N}^{r}$ we understand
$|\alpha|=\sum_{i=1}^{r}\alpha_{i}$, where
$\alpha=(\alpha_{1},\ldots,\alpha_{r})$. Finally, we will also use the notion
of factorial and binomial coefficients in this multi-index setting. If
$\alpha,\beta\in\mathbb{N}^{r}$, then

$\alpha!=\alpha_{1}!\cdots\alpha_{r}!$

and

$\binom{\alpha}{\beta}=\binom{\alpha_{1}}{\beta_{1}}\cdots\binom{\alpha_{r}}{\beta_{r}}=\frac{\alpha!}{\beta!(\alpha-\beta)!},$

where $\alpha=(\alpha_{1},\ldots,\alpha_{r})$ and
$\beta=(\beta_{1},\ldots,\beta_{r})$.

Let $r$ be a positive integer, $N$ be a nonnegative integer and
$\Omega\subset\mathbb{R}^{r}$ be a domain (i.e. a nonempty, open and connected
set). For all multi-indices $\alpha\in\mathbb{N}^{r}$, $|\alpha|\leq N$, let
us consider the partial differential operator $D^{\alpha}$ defined by

$D^{\alpha}=\frac{\partial^{|\alpha|}}{\partial
x_{1}^{\alpha_{1}}\cdots\partial x_{r}^{\alpha_{r}}},$

where $\alpha=(\alpha_{1},\ldots,\alpha_{r})$. Here by definition we mean
$D^{0}=\mathrm{id}$. Let further

$\mathscr{C}^{N}(\Omega)=\left\\{f\colon\Omega\to\mathbb{R}\,|\,f\text{ is
}N\text{ times continuously differentiable}\right\\}.$

An easy computation shows that if $f,g\in\mathscr{C}^{N}(\Omega)$ and
$\alpha\in\mathbb{N}^{r},|\alpha|\leq N$, then we have

$D^{\alpha}(f\cdot
g)=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}D^{\beta}(f)\cdot
D^{\alpha-\beta}(g).$ (1)

The main aim of this paper will be about the converse in some sense. More
precisely, in this paper, we will study the solutions
$T_{\alpha}\colon\mathscr{C}^{N}(\Omega)\to\mathscr{C}(\Omega)$ of the
operator equation

$T_{\alpha}(f\cdot
g)=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}T_{\beta}(f)T_{\alpha-\beta}(g)\qquad\left(f,g\in\mathscr{C}^{N}(\Omega)\right)$

for all multi-indices $\alpha\in\mathbb{N}^{r}$, $|\alpha|\leq N$.

Equations analogous to (1) have an important role not only in connection to
characterization theorems related to differential operators but also in
harmonic and spectral analysis.

In the following, we will use the notations and the terminology of the
monographs Székelyhidi [7, 8] and while considering moment sequences of higher
rank, the terminology of [2].

Let $(G,\cdot)$ be an Abelian semigroup. A nonzero function $m\colon
G\to\mathbb{C}$ is called exponential, if

$m(x\cdot y)=m(x)m(y)$

holds for all $x,y$ in $G$. Let $N$ be a nonnegative integer. A function
$\varphi\colon G\to\mathbb{C}$ is termed to be a _moment function of order_
$N$, if there exist functions $\varphi_{k}\colon G\to\mathbb{C}$ such that
$\varphi_{0}=1$, $\varphi_{N}=\varphi$ and

$\varphi_{k}(x\cdot
y)=\sum_{j=0}^{k}\binom{k}{j}\varphi_{j}(x)\varphi_{k-j}(y)$ (2)

for all $x$ and $y$ in $G$ and $k=0,1,\dots,N$. If $G$ is a monoid with the
neutral element $1$, then this concept can be extended by relaxing the
assumption $\varphi_{0}\equiv 1$ to $\varphi_{0}(1)=1$. In this case,
$\varphi_{0}$ is an arbitrary exponential function and we say that
$\varphi_{0}$ generates the generalized moment sequence of order $N$ and the
function $\varphi_{k}$ is a generalized moment function of order $k$, or, if
we want to specify the exponential $\varphi_{0}$, then we say that
$\varphi_{k}$ is a generalized moment function of order $k$ associated with
the exponential $\varphi_{0}$.

######  Definition 1.

Let $G$ be an Abelian semigroup, $r$ a positive integer, and for each multi-
index $\alpha$ in $\mathbb{N}^{r}$ let $f_{\alpha}\colon G\to\mathbb{C}$ be a
function. We say that $(f_{\alpha})_{\alpha\in\mathbb{N}^{r}}$ is a
_generalized moment sequence of rank $r$_, if

$f_{\alpha}(x\cdot
y)=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}f_{\beta}(x)f_{\alpha-\beta}(y)$
(3)

holds whenever $x,y$ are in $G$. The function $f_{\mathbf{0}}$, where
$\mathbf{0}$ is the zero element in $\mathbb{N}^{r}$, is called the
_generating function_ of the sequence.

######  Remark 1.

For $r=1$, instead of multi-indices, we have nonnegative integer indices. Thus
generalized moment functions of rank $1$ are simply moment sequences.

######  Remark 2.

Assume now that $(G,\cdot)$ is an Abelian group (not only a semigroup). For
$\alpha=\mathbf{0}$ we have

$f_{\mathbf{0}}(x\cdot y)=f_{\mathbf{0}}(x)\cdot f_{\mathbf{0}}(y)$

for each $x,y$ in $G$, hence $f_{\mathbf{0}}=m$ is always an exponential, or
identically zero. It can be proved by induction on the height of the multi-
index $\alpha\in\mathbb{N}^{r}$ that if $f_{\mathbf{0}}$ is the identically
zero function, then for all multi-index $\alpha$, the mapping $f_{\alpha}$
must be identically zero, too.

In a rather natural way, the above notions can be extended from complex-valued
mappings to mappings whose range is a (commutative) ring. Indeed, if
$(G,\cdot)$ is an Abelian semigroup and $Q$ is a commutative ring, $r$ is a
positive integer, and $\alpha\in\mathbb{N}^{r}$ is a multi-index, then a
function $f\colon G\to Q$ is a generalized moment function of rank $r$ and of
order $N$, where $N=|\alpha|$, if for all multi-indices
$\beta\in\mathbb{N}^{r}$ with $|\beta|\leq N$, there exists a function
$f_{\beta}\colon G\to Q$ such that $f=f_{\alpha}$ and we have

$f_{\beta}(x\cdot
y)=\sum_{\gamma\leq\beta}\binom{\beta}{\gamma}f_{\gamma}(x)f_{\beta-\gamma}(y)$
(4)

holds whenever $x,y\in G$ and for all multi-indices $\beta\in\mathbb{N}^{r}$
with $|\beta|\leq N$.

######  Remark 3.

Using the above definition this means that if $N\geq 1$ and we consider
$\mathscr{C}^{N}(\Omega)$ with the pointwise product of functions, then it
will become an Abelian semigroup and we take $\mathscr{C}(\Omega)$ to be the
range, then the sequence of mappings $(D^{\alpha})_{|\alpha|\leq N}$ forms a
moment sequence of rank $r$.

##  2 Characterizations of higher order differential operators

The main aim of this paper is to investigate the following problem: Let $r$ be
a positive integer, $N$ be a nonnegative integer and
$\Omega\subset\mathbb{R}^{r}$ be a domain (i.e. a nonempty, open and connected
set). Determine the mappings
$T_{\alpha}\colon\mathscr{C}^{N}(\Omega)\to\mathscr{C}(\Omega)$,
$\alpha\in\mathbb{N}^{r},|\alpha|\leq N$ if they fulfill

$T_{\beta}(f\cdot
g)=\sum_{\gamma\leq\beta}\binom{\beta}{\gamma}T_{\gamma}(f)T_{\beta-\gamma}(g)$
(5)

for all $f,g\in\mathscr{C}^{N}(\Omega)$ and for all multi-indices
$\beta\in\mathbb{N}^{r}$, $|\beta|\leq N$.

Observe that if $\beta=\mathbf{0}=(0,\ldots,0)$, then the above identity
becomes

$T_{\mathbf{0}}(f\cdot g)=T_{\mathbf{0}}(f)\cdot
T_{\mathbf{0}}(g)\qquad\left(f,g\in\mathscr{C}^{N}(\Omega)\right).$

This means, that similarly to the group case, the first element of the
sequence, i.e. $T_{\mathbf{0}}$ is an ‘exponential’.

Recall again that if $(G,\cdot)$ is an Abelian _group_ , then a nonzero
function $m\colon G\to\mathbb{C}$ is an exponential, if

$m(x\cdot y)=m(x)m(y)$

holds for all $x,y$ in $G$. In the case of this concept, the fact that the
range of $m$ is the set of complex numbers plays a key role. Indeed, if $m$ is
an exponential on the Abelian group $G$, then either $m$ is identically zero,
or nowhere zero. At the same time, as we will see below, analogous statements
are not true for mappings
$T\colon\mathscr{C}^{N}(\Omega)\to\mathscr{C}(\Omega)$.

The study of multiplicative maps between function spaces has quite extensive
literature. Here we quote only two of them, but the interested reader can
consult e.g. Artstein-Avidan–Faifman–Milman [1], Milgram [4], Mrčun [5] and
Mrčun–Šemrl [6].

A result from [6] concerning _bijective_ multiplicative mappings between the
function spaces $\mathscr{C}(X)$ and $\mathscr{C}(Y)$ says that if we are
given compact Hausdorff spaces $X$ and $Y$, $\tau\colon Y\to X$ is a
homeomorphism and $p\in\mathscr{C}(Y)$ is a positive function, then the
mapping $\mathcal{M}\colon\mathscr{C}(X)\to\mathscr{C}(Y)$ defined by

$\mathcal{M}(f)(x)=\left|f(\tau(x))\right|^{p(x)}\cdot\mathrm{sgn}\left(f(\tau(x))\right)\qquad\left(x\in
Y,f\in\mathscr{C}(X)\right)$

is a bijective and multiplicative map, i.e. we have

$\mathcal{M}(f\cdot g)(x)=\mathcal{M}(f)(x)\cdot\mathcal{M}(g)(x)$

for all $x\in Y$ and $f,g\in\mathscr{C}(X)$.

In view of this, if $K\subset\mathbb{R}^{r}$ is a _compact_ set and
$\tau\colon K\to K$ is a homeomorphism, then the mapping
$\mathcal{M}\colon\mathscr{C}(K)\to\mathscr{C}(K)$ defined by

$\mathcal{M}(f)(x)=\left|f(\tau(x))\right|^{p(x)}\cdot\mathrm{sgn}\left(f(\tau(x))\right)\qquad\left(x\in
K,f\in\mathscr{C}(K)\right)$

is a bijective and multiplicative map. Firstly observe that this is only one
direction and not an ‘if and only if’ statement. Further, in general, we
intend to work on a _domain_ $\Omega\subset\mathbb{R}^{r}$ and we cannot a
priori assume that the mapping in question is _bijective_.

A corollary of a result from Mrčun [5] describes _bijective_ multiplicative
self-mappings of $\mathscr{C}^{N}(\Omega)$, where $N$ is a fixed positive
integer. Let $N,r$ be a positive integers and $\Omega\subset\mathbb{R}^{r}$ be
a $\mathscr{C}^{N}$-manifold. Then for any multiplicative bijection
$\mathcal{M}\colon\mathscr{C}^{N}(\Omega)\to\mathscr{C}^{N}(\Omega)$ there
exists a unique $\mathscr{C}^{N}$-diffeomorphism $\tau\colon\Omega\to\Omega$
such that

$\mathcal{M}(f)(x)=f(\tau(x))\qquad\left(x\in\Omega,f\in\mathscr{C}^{N}(\Omega)\right)$

holds.

In the cases we intend to study, unfortunately, the range of the mappings is
not $\mathscr{C}^{N}(\Omega)$, but the much larger function space
$\mathscr{C}(\Omega)$. In addition, in general, it cannot be guaranteed that
the mapping $T_{\mathbf{0}}$ is bijective. However, without the assumption of
bijectivity, we cannot expect to be able to describe the multiplicative
mappings in these spaces. Thus, in this paper, we will determine the moment
functions of the spaces in question in the case of some important
multiplicative mappings.

###  2.1 A non-bijective case

Let $r$ be a positive integer, $N$ be nonnegative a integer, and
$\Omega\subset\mathbb{R}^{r}$ be a domain. Then the mapping
$T_{\mathbf{0}}\colon\mathscr{C}^{N}(\Omega)\to\mathscr{C}(\Omega)$ defined by

$T_{\mathbf{0}}(f)(x)=1\qquad\left(x\in\Omega,f\in\mathscr{C}^{N}(\Omega)\right)$

is multiplicative (and non-bijective). Therefore, it can be suitable to
generate a moment sequence. As we will see, this mapping generates a fairly
trivial moment sequence.

######  Theorem 1.

Let $r$ be a positive integer, $N$ be nonnegative a integer, and
$\Omega\subset\mathbb{R}^{r}$ be a domain. Assume further that for all multi-
indices $\alpha\in\mathbb{N}^{r}$, $|\alpha|\leq N$, we are given a mapping
$T_{\alpha}\colon\mathscr{C}^{N}(\Omega)\to\mathscr{C}(\Omega)$ such that

$T_{\mathbf{0}}(f)(x)=1\qquad\left(x\in\Omega,f\in\mathscr{C}^{N}(\Omega)\right)$

and $(T_{\alpha})_{|\alpha|\leq N}$ forms a moment sequence of rank $r$ and of
order $N$. Then for all multi-indices $\alpha\in\mathbb{N}^{r}$ with
$\alpha\neq\mathbf{0}$ and $|\alpha|\leq N$ we have

$T_{\alpha}(f)(x)=0$

for all $x\in\Omega$ and $f\in\mathscr{C}^{N}(\Omega)$.

###### Proof.

We prove the statement on induction of the height of the multi-index
$\alpha\in\mathbb{N}^{r}$. Accordingly, let $\alpha\in\mathbb{N}^{r}$ be an
arbitrary multi-index with $|\alpha|=1$. Then

$T_{\alpha}(f\cdot
g)=T_{\mathbf{0}}(f)T_{\alpha}(g)+T_{\alpha}(f)T_{\mathbf{0}}(g)$

holds for all $f,g\in\mathscr{C}^{N}(\Omega)$. Since

$T_{\mathbf{0}}(f)(x)=1\qquad\left(x\in\Omega,f\in\mathscr{C}^{N}(\Omega)\right),$

this means that

$T_{\alpha}(f\cdot g)=T_{\alpha}(f)+T_{\alpha}(g)$

all $f,g\in\mathscr{C}^{N}(\Omega)$. Let $f$ and $g$ be the identically zero
functions on $\Omega$, we get that

$T_{\alpha}(0)=2T_{\alpha}(0),$

so $T_{\alpha}(0)=0$. This however yields that

$T_{\alpha}(f\cdot 0)=T_{\alpha}(f)+T_{\alpha}(0)$

for all $f\in\mathscr{C}^{N}(\Omega)$. Thus

$T_{\alpha}(f)(x)=0$

for all $f\in\mathscr{C}^{N}(\Omega)$ and $x\in\Omega$.

Let now $k\in\left\\{1,\ldots,N-1\right\\}$ be arbitrary, and suppose that

$T_{\beta}(f)(x)=0\qquad\left(f\in\mathscr{C}^{N}(\Omega),x\in\Omega\right)$

holds for all multi-indices $\beta$ with $|\beta|\leq k$. Let further
$\alpha\in\mathbb{N}^{r}$ be an arbitrary multi-index with $|\alpha|=k+1$.
Then

$\displaystyle T_{\alpha}(f\cdot g)$
$\displaystyle=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}T_{\beta}(f)\cdot
T_{\alpha-\beta}(g)$
$\displaystyle=T_{\mathbf{0}}(f)T_{\alpha}(g)+T_{\alpha}(f)T_{\mathbf{0}}(g)+\sum_{0<\beta<\alpha}\binom{\alpha}{\beta}T_{\beta}(f)\cdot
T_{\alpha-\beta}(g)$ $\displaystyle=T_{\alpha}(f)+T_{\alpha}(g)$

holds for all $f,g\in\mathscr{C}^{N}(\Omega)$. This is exactly the same
equation that we solved above. Thus

$T_{\alpha}(f)(x)=0$

for all $f\in\mathscr{C}^{N}(\Omega)$ and $x\in\Omega$. ∎

###  2.2 A bijective case

Let $r$ and $N$ be positive integers, $\Omega\subset\mathbb{R}^{r}$ be a
$\mathscr{C}^{N}$-manifold and $\tau\colon\Omega\to\Omega$ be a
$\mathscr{C}^{N}$-diffeomorphism. Define
$\tilde{T}_{\mathbf{0}}\colon\mathscr{C}^{N}(\Omega)\to\mathscr{C}(\Omega)$
through

$\tilde{T}_{\mathbf{0}}(f)(x)=f(\tau(x))\qquad\left(x\in\Omega,f\in\mathscr{C}^{N}(\Omega)\right).$

Then $\tilde{T}_{\mathbf{0}}$ is a multiplicative mapping. Thus it can be an
appropriate candidate to generate a moment sequence on
$\mathscr{C}^{N}(\Omega)$.

######  Lemma 1.

Let $r$ and $N$ be positive integers, $\Omega\subset\mathbb{R}^{r}$ be a
$\mathscr{C}^{N}$-manifold and $\tau\colon\Omega\to\Omega$ be a
$\mathscr{C}^{N}$-diffeomorphism. Further, let us consider the mappings
$T_{\mathbf{0}},\tilde{T}_{\mathbf{0}}\colon\mathscr{C}^{N}(\Omega)\to\mathscr{C}(\Omega)$
defined by

$T_{\mathbf{0}}(f)(x)=f(x)\qquad\text{and}\qquad\tilde{T}_{\mathbf{0}}(f)(x)=f(\tau(x))\qquad\left(x\in\Omega,f\in\mathscr{C}^{N}(\Omega)\right),$

respectively. Then the following statements are equivalent:

  1. (i)

the sequence of mappings
$T_{\alpha}\colon\mathscr{C}^{N}(\Omega)\to\mathscr{C}(\Omega)$,
$\alpha\in\mathbb{N}^{r},|\alpha|\leq N$ is a moment sequence generated by
$T_{\mathbf{0}}$

  2. (ii)

the sequence of mappings
$\tilde{T}_{\alpha}\colon\mathscr{C}^{N}(\Omega)\to\mathscr{C}(\Omega)$,
$\alpha\in\mathbb{N}^{r},|\alpha|\leq N$ is a moment sequence generated by
$\tilde{T}_{\mathbf{0}}$

###### Proof.

Let $r$ and $N$ be positive integers, $\Omega\subset\mathbb{R}^{r}$ be a
$\mathscr{C}^{N}$-manifold and $\tau\colon\Omega\to\Omega$ be a
$\mathscr{C}^{N}$-diffeomorphism. Further, let is consider the mappings
$T_{\mathbf{0}},\tilde{T}_{\mathbf{0}}\colon\mathscr{C}^{N}(\Omega)\to\mathscr{C}(\Omega)$
defined by

$T_{\mathbf{0}}(f)(x)=f(x)\qquad\text{and}\qquad\tilde{T}_{\mathbf{0}}(f)(x)=f(\tau(x))\qquad\left(x\in\Omega,f\in\mathscr{C}^{N}(\Omega)\right),$

respectively.

To prove the direction (i)$\Rightarrow$(ii), assume that the sequence of
mappings $T_{\alpha}\colon\mathscr{C}^{N}(\Omega)\to\mathscr{C}(\Omega)$,
$\alpha\in\mathbb{N}^{r},|\alpha|\leq N$ is a moment sequence generated by
$T_{\mathbf{0}}$. This means that for all $\alpha\in\mathbb{N}^{r}$,
$|\alpha|\leq N$ we have

$T_{\alpha}(f\cdot
g)(x)=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}T_{\beta}(f)(x)\cdot
T_{\alpha-\beta}(g)(x)$

for all $f,g\in\mathscr{C}^{N}(\Omega)$ and $x\in\Omega$. Thus we also have

$T_{\alpha}(f\cdot
g)(\tau(x))=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}T_{\beta}(f)(\tau(x))\cdot
T_{\alpha-\beta}(g)(\tau(x))\qquad\left(f,g\in\mathscr{C}^{N}(\Omega),x\in\Omega\right).$

For all multi-indices $\alpha\in\mathbb{N}^{r}$, $|\alpha|\leq N$, define the
mapping
$\tilde{T}_{\alpha}\colon\mathscr{C}^{N}(\Omega)\to\mathscr{C}(\Omega)$ by

$\tilde{T}_{\alpha}(f)(x)=T_{\alpha}(f)(\tau(x))\qquad\left(f\in\mathscr{C}^{N}(\Omega),x\in\Omega\right)$

to deduce that

$\tilde{T}_{\alpha}(f\cdot
g)(x)=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\tilde{T}_{\beta}(f)(x)\cdot\tilde{T}_{\alpha-\beta}(g)(x)$

for all $f,g\in\mathscr{C}^{N}(\Omega)$ and $x\in\Omega$. Thus the sequence of
mappings $(\tilde{T}_{\alpha})_{|\alpha|\leq N}$ is a moment sequence of rank
$r$ generated by $\tilde{T}_{0}$.

The proof of the implication (ii)$\Rightarrow$(i) is analogous. It is enough
to consider a point $x=\tau(y)$ with arbitrary $y\in\Omega$ and use the fact
that $\tau$ is a diffeomorphism. ∎

As we saw above, if $r$ and $N$ are positive integers,
$\Omega\subset\mathbb{R}^{r}$ is a $\mathscr{C}^{N}$-manifold and
$\tau\colon\Omega\to\Omega$ is a $\mathscr{C}^{N}$-diffeomorphism, then the
mapping
$\tilde{T}_{\mathbf{0}}\colon\mathscr{C}^{N}(\Omega)\to\mathscr{C}(\Omega)$
defined by

$\tilde{T}_{\mathbf{0}}(f)(x)=f(\tau(x))\qquad\left(x\in\Omega,f\in\mathscr{C}^{N}(\Omega)\right),$

is a multiplicative mapping. Thus it can be an appropriate candidate to
generate a moment sequence on $\mathscr{C}^{N}(\Omega)$. Nevertheless, the
previous lemma says that instead of multiplicative mappings of this form, it
suffices to consider the identity mapping. Accordingly, below we will describe
moment sequences generated by the identity mapping. Further, observe that
while describing the solutions of equation (5), not only the generator, i.e.,
the operator $T_{\mathbf{0}}$, but also the domain $\mathscr{C}^{N}(\Omega)$
can play a crucial role. In the second part of this section, we focus on the
largest possible domain, that is, we will work on $\mathscr{C}(\Omega)$.

During the proof of Theorem 2 we will use a corollary of [3, Theorem 3.5] and
also [3, Theorem 7.1] which are the following statements. Before stating these
results, however, we need two more notions from the theory of operator
relations.

######  Definition 2.

Let $k$ be a nonnegative integer, $r$ be a positive integer and
$\Omega\subset\mathbb{R}^{r}$ be an open set. An operator
$A\colon\mathscr{C}^{k}(\Omega)\to\mathscr{C}(\Omega)$ is _non-degenerate_ if
for each nonvoid open subset $U\subset\Omega$ and all $x\in U$, there exist
functions $g_{1},g_{2}\in\mathscr{C}^{k}(\Omega)$ with supports in $U$ such
that the vectors $(g_{i}(x),Ag_{i}(x))\in\mathbb{R}^{2}$, $i=1,2$ are linearly
independent in $\mathbb{R}^{2}$.

######  Definition 3.

Let $k$ and $r$ be positive integers with $k\geq 2$ and
$\Omega\subset\mathbb{R}^{r}$ be an open set. We say that the operator
$A\colon\mathscr{C}^{k}(\Omega)\to\mathscr{C}(\Omega)$ _depends non-trivially
on the derivative_ if there exists $x_{0}\in\Omega$ and there are functions
$f_{1},f_{2}\in\mathscr{C}^{k}(\Omega)$ such that

$f_{1}(x_{0})=f_{2}(x_{0})\quad\text{and}\quad Af_{1}(x_{0})\neq
Af_{2}(x_{0})$

holds.

######  Proposition 1.

Let $r$ be a positive integer and $\Omega\subset\mathbb{R}^{r}$ be a domain.
Suppose that the operator $T\colon\mathscr{C}(\Omega)\to\mathscr{C}(\Omega)$
satisfies the Leibniz rule, i.e.,

$T(f\cdot g)=f\cdot T(g)+T(f)\cdot
g\qquad\left(f,g\in\mathscr{C}(\Omega)\right).$

Then there exists a function $c\in\mathscr{C}(\Omega)$ such that for all
$f\in\mathscr{C}(\Omega)$ and $x\in\Omega$

$T(f)(x)=c(x)\cdot f(x)\cdot\ln\left(\left|f(x)\right|\right).$

Conversely, any such map $T$ satisfies the Leibniz rule.

######  Proposition 2.

Let $r$ be a positive integer, $k$ be a nonnegative integer and
$\Omega\subset\mathbb{R}^{r}$ be a domain. Assume that
$T,A\colon\mathscr{C}^{k}(\Omega)\to\mathscr{C}(\Omega)$ satisfy

$T(f\cdot g)=T(f)\cdot g+f\cdot T(g)+2A(f)\cdot
A(g)\qquad\left(f,g\in\mathscr{C}^{k}(\Omega)\right)$

and that in case $k\geq 2$ the mapping $A$ is non-degenerate and depends non-
trivially on the derivative. Then there are continuous functions
$a\colon\Omega\to\mathbb{R}$ and $b,c\colon\Omega\to\mathbb{R}^{r}$ such that
we have

$\begin{array}[]{rcl}T(f)(x)&=&\langle
f^{\prime\prime}(x)c(x),c(x)\rangle+R(f)(x)\\\\[5.69054pt] A(f)(x)&=&\langle
f^{\prime}(x),c(x)\rangle\end{array}\qquad\left(f\in\mathscr{C}^{k}(\Omega),x\in\Omega\right),$

where

$R(f)(x)=\langle
f^{\prime}(x),b(x)\rangle+a(x)f(x)\ln\left(|f(x)|\right)\qquad\left(f\in\mathscr{C}^{k}(\Omega)\right).$

If $k=1$, then necessarily $c\equiv 0$. Further, if $k=0$, then necessarily
$b\equiv 0$ and $c\equiv 0$.

Conversely, these operators satisfy the above second-order Leibniz rule.

Our main result for operators defined on $\mathscr{C}(\Omega)$ is the
following theorem.

######  Theorem 2.

Let $r$ and $N$ be positive integers, $\Omega\subset\mathbb{R}^{r}$ be a
domain and assume that for all multi-indices $\alpha\in\mathbb{N}^{r}$, with
$|\alpha|\leq N$ we are given a mapping
$T_{\alpha}\colon\mathscr{C}(\Omega)\to\mathscr{C}(\Omega)$ such that
$T_{\mathbf{0}}$ is the identity mapping and for all multi-indices
$\alpha\in\mathbb{N}^{r}$ with $0\neq|\alpha|\leq N$ we have

$T_{\alpha}(f\cdot
g)=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}T_{\beta}(f)\cdot
T_{\alpha-\beta}(g)$ (6)

for all $f,g\in\mathscr{C}(\Omega)$. Then there exist a family of functions
$\\{c_{\alpha}\in\mathscr{C}(\Omega):0\neq|\alpha|\leq N\\}$ such that

$\left[\sum_{\mathbf{0}<\beta<\alpha}\binom{\alpha}{\beta}c_{\beta}(x)\cdot
c_{\alpha-\beta}(x)\right]=0$ (7)

and

$T_{\alpha}(f)(x)=c_{\alpha}(x)f(x)\ln(|f(x)|)\qquad\left(x\in\Omega,f\in\mathscr{C}(\Omega),0\neq|\alpha|\leq
N\right).$ (8)

And also conversely, if $T_{\mathbf{0}}$ is the identity mapping on
$\mathscr{C}(\Omega)$, we are given a family of functions that satisfies (7)
and we define the mappings $T_{\alpha}$ on $\mathscr{C}(\Omega)$ by the
formula (8), then they satisfy equation (6) for all multi-indices $\alpha$
such that $0\neq|\alpha|\leq N$.

###### Proof.

Let $r$ and $N$ be positive integers, $\Omega\subset\mathbb{R}^{r}$ be a
domain and assume that for all multi-indices $\alpha\in\mathbb{N}^{r}$, with
$|\alpha|\leq N$ we are given a mapping
$T_{\alpha}\colon\mathscr{C}(\Omega)\to\mathscr{C}(\Omega)$ such that
$T_{\mathbf{0}}$ is the identity mapping and for all multi-indices
$\alpha\in\mathbb{N}^{r}$ with $0\neq|\alpha|\leq N$ we have

$T_{\alpha}(f\cdot
g)=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}T_{\beta}(f)\cdot
T_{\alpha-\beta}(g)$

for all $f,g\in\mathscr{C}(\Omega)$.

We prove the statement by induction on the multi-index $\alpha$.

Let $\alpha\in\mathbb{N}^{r}$ be an arbitrary multi-index for which
$|\alpha|=1$ holds. Then

$T_{\alpha}(f\cdot
g)=T_{\mathbf{0}}(f)T_{\alpha}(g)+T_{\alpha}(f)T_{\mathbf{0}}(g)=f\cdot
T_{\alpha}(g)+T_{\alpha}(f)\cdot
g\qquad\left(f,g\in\mathscr{C}(\Omega)\right),$

since $T_{\mathbf{0}}=\mathrm{id}$ was assumed. Using Proposition 1, we obtain
that there exists a continuous function $c_{\alpha}\in\mathscr{C}(\Omega)$
such that

$T_{\alpha}(f)(x)=c_{\alpha}(x)f(x)\ln(|f(x)|)\qquad\left(x\in\Omega,f\in\mathscr{C}(\Omega)\right).$

Let now $k\in\left\\{1,\ldots,N-1\right\\}$ be arbitrary and assume that the
statement of the theorem holds for all multi-indices $\beta\in\mathbb{N}^{r}$
for which we have $|\beta|\leq k$. Let further $\alpha\in\mathbb{N}^{r}$ be an
arbitrary multi-index for which $|\alpha|=k+1$. Then

$\displaystyle T_{\alpha}(f\cdot g)$
$\displaystyle=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}T_{\beta}(f)\cdot
T_{\alpha-\beta}(g)$ $\displaystyle=T_{\mathbf{0}}(f)\cdot
T_{\alpha}(g)+T_{\alpha}(f)\cdot
T_{\mathbf{0}}(g)+\sum_{\mathbf{0}<\beta<\alpha}\binom{\alpha}{\beta}T_{\beta}(f)\cdot
T_{\alpha-\beta}(g)$ $\displaystyle=f\cdot T_{\alpha}(g)+T_{\alpha}(f)\cdot
g+\sum_{\mathbf{0}<\beta<\alpha}\binom{\alpha}{\beta}c_{\beta}f\ln(|f|)\cdot
c_{\alpha-\beta}g\ln(|g|)$ $\displaystyle=f\cdot
T_{\alpha}(g)+T_{\alpha}(f)\cdot
g+\left[\sum_{\mathbf{0}<\beta<\alpha}\binom{\alpha}{\beta}c_{\beta}(x)\cdot
c_{\alpha-\beta}\right]\cdot f\ln(|f|)\cdot g(x)\ln(|g|)$

holds for all $f,g\in\mathscr{C}(\Omega)$. Using Proposition 2, taking into
account that $k=0$, we obtain that there exists a continuous function
$c_{\alpha}$ such that

$T_{\alpha}(f)(x)=c_{\alpha}(x)\cdot f(x)\cdot\ln(|f(x)|)$

is fulfilled for all $f\in\mathscr{C}(\Omega)$ and $x\in\Omega$. Further, the
family of functions $\\{c_{\alpha}\in\mathscr{C}(\Omega):0\neq|\alpha|\leq
N\\}$ necessarily satisfies (7).

The converse implication is an easy computation. ∎

As we saw in the previous theorem, the moment sequences are quite poor on the
$\mathscr{C}(\Omega)$ space. We note that if $N\geq 1$, then there are
substantially more diverse moment sequences in the space
$\mathscr{C}^{N}(\Omega)$, see Remark 3. However, this will be dealt with in
one of our future work.

######  Acknowledgment.

The research of Eszter Gselmann has been supported by project no. K134191 that
has been implemented with the support provided by the National Research,
Development and Innovation Fund of Hungary, financed under the K_20 funding
scheme.

The work of Aleksandra Świątczak is implemented under the project "Curriculum
for advanced doctoral education & training - CADET Academy of TUL" co-financed
by the STER Programme - Internationalization of doctoral schools.

This article has been completed while one of the authors (Aleksandra
Świątczak), was the Doctoral Candidate in the Interdisciplinary Doctoral
School at the Lodz University of Technology, Poland.

## References

  * [1] Shiri Artstein-Avidan, Dmitry Faifman, and Vitali Milman.  On multiplicative maps of continuous and smooth functions.  In Geometric aspects of functional analysis, volume 2050 of Lecture Notes in Math., pages 35–59. Springer, Heidelberg, 2012.
  * [2] Żywilla Fechner, Eszter Gselmann, and László Székelyhidi.  Moment functions on groups.  Results Math., 76(4):Paper No. 171, 16, 2021.
  * [3] Hermann König and Vitali Milman.  Operator relations characterizing derivatives.  Birkhäuser / Springer, Cham, 2018.
  * [4] Arthur N. Milgram.  Multiplicative semigroups of continuous functions.  Duke Math. J., 16:377–383, 1949.
  * [5] Janez Mrčun.  On isomorphisms of algebras of smooth functions.  Proc. Amer. Math. Soc., 133(10):3109–3113, 2005.
  * [6] Janez Mrčun and Peter Šemrl.  Multiplicative bijections between algebras of differentiable functions.  Ann. Acad. Sci. Fenn. Math., 32(2):471–480, 2007.
  * [7] László Székelyhidi.  Discrete spectral synthesis and its applications.  Springer Monographs in Mathematics. Springer, Dordrecht, 2006.
  * [8] László Székelyhidi.  Harmonic and spectral analysis.  World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014.


* [84] A. Torcianti and S. Matzka, “Explainable artificial intelligence for predictive maintenance applications using a local surrogate model,” in _2021 4th International Conference on Artificial Intelligence for Industries (AI4I)_. IEEE, 2021, pp. 86–88.
  * [85] Y. Remil, A. Bendimerad, M. Plantevit, C. Robardet, and M. Kaytoue, “Interpretable summaries of black box incident triaging with subgroup discovery,” in _2021 IEEE 8th International Conference on Data Science and Advanced Analytics (DSAA)_ , Oct. 2021, pp. 1–10.
  * [86] B. Ghasemkhani, O. Aktas, and D. Birant, “Balanced K-Star: An explainable machine learning method for Internet-of-Things-Enabled predictive maintenance in manufacturing,” _Machines_ , vol. 11, no. 3, p. 322, Feb. 2023.
  * [87] J. Liu, S. Zheng, and C. Wang, “Causal graph attention network with disentangled representations for complex systems fault detection,” _Reliability Engineering & System Safety_, vol. 235, p. 109232, 2023.
  * [88] A. Trilla, N. Mijatovic, and X. Vilasis-Cardona, “Unsupervised probabilistic anomaly detection over nominal subsystem events through a hierarchical variational autoencoder,” _International Journal of Prognostics and Health Management_ , vol. 14, no. 1, 2023.
  * [89] I. Errandonea, P. Ciáurriz, U. Alvarado, S. Beltrán, and S. Arrizabalaga, “Edge intelligence-based proposal for onboard catenary stagger amplitude diagnosis,” _Computers in Industry_ , vol. 144, p. 103781, 2023. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0166361522001774
  * [90] B. Steenwinckel, D. De Paepe, S. V. Hautte, P. Heyvaert, M. Bentefrit, P. Moens, A. Dimou, B. Van Den Bossche, F. De Turck, S. Van Hoecke _et al._ , “Flags: A methodology for adaptive anomaly detection and root cause analysis on sensor data streams by fusing expert knowledge with machine learning,” _Future Generation Computer Systems_ , vol. 116, pp. 30–48, 2021\.
  * [91] H. Li, D. Parikh, Q. He, B. Qian, Z. Li, D. Fang, and A. Hampapur, “Improving rail network velocity: A machine learning approach to predictive maintenance,” _Transportation Research Part C: Emerging Technologies_ , vol. 45, pp. 17–26, 2014.
  * [92] Z. Allah Bukhsh, A. Saeed, I. Stipanovic, and A. G. Doree, “Predictive maintenance using tree-based classification techniques: A case of railway switches,” _Transp. Res. Part C: Emerg. Technol._ , vol. 101, pp. 35–54, Apr. 2019.
  * [93] P. Cao, S. Zhang, and J. Tang, “Preprocessing-free gear fault diagnosis using small datasets with deep convolutional neural network-based transfer learning,” _IEEE Access_ , vol. 6, pp. 26 241–26 253, 2018.
  * [94] G. Hajgató, R. Wéber, B. Szilágyi, B. Tóthpál, B. Gyires-Tóth, and C. Hős, “PredMaX: Predictive maintenance with explainable deep convolutional autoencoders,” _Advanced Engineering Informatics_ , vol. 54, p. 101778, Oct. 2022.
  * [95] J. Jakubowski, P. Stanisz, S. Bobek, and G. J. Nalepa, “Explainable anomaly detection for hot-rolling industrial process,” in _2021 IEEE 8th International Conference on Data Science and Advanced Analytics (DSAA)_. IEEE, 2021, pp. 1–10.
  * [96] N. Mylonas, I. Mollas, N. Bassiliades, and G. Tsoumakas, “Local multi-label explanations for random forest,” in _Joint European Conference on Machine Learning and Knowledge Discovery in Databases_. Springer, 2022, pp. 369–384.
  * [97] J. Jakubowski, P. Stanisz, S. Bobek, and G. J. Nalepa, “Roll wear prediction in strip cold rolling with physics-informed autoencoder and counterfactual explanations,” in _2022 IEEE 9th International Conference on Data Science and Advanced Analytics (DSAA)_. IEEE, 2022, pp. 1–10.
  * [98] W. Xu, Z. Zhou, T. Li, C. Sun, X. Chen, and R. Yan, “Physics-constraint variational neural network for wear state assessment of external gear pump,” _IEEE Transactions on Neural Networks and Learning Systems_ , 2022.
  * [99] J. M. F. Salido and S. Murakami, “A comparison of two learning mechanisms for the automatic design of fuzzy diagnosis systems for rotating machinery,” _Applied Soft Computing_ , vol. 4, no. 4, pp. 413–422, 2004.
  * [100] R. Langone, A. Cuzzocrea, and N. Skantzos, “Interpretable anomaly prediction: Predicting anomalous behavior in industry 4.0 settings via regularized logistic regression tools,” _Data & Knowledge Engineering_, vol. 130, p. 101850, 2020.
  * [101] V. M. Janakiraman, “Explaining aviation safety incidents using deep temporal multiple instance learning,” in _Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, 2018, pp. 406–415.
  * [102] M. Berno, M. Canil, N. Chiarello, L. Piazzon, F. Berti, F. Ferrari, A. Zaupa, N. Ferro, M. Rossi, and G. A. Susto, “A machine learning-based approach for advanced monitoring of automated equipment for the entertainment industry,” in _2021 IEEE International Workshop on Metrology for Industry 4.0 & IoT (MetroInd4. 0&IoT)_. IEEE, 2021, pp. 386–391.
  * [103] E. Anello, C. Masiero, F. Ferro, F. Ferrari, B. Mukaj, A. Beghi, and G. A. Susto, “Anomaly detection for the industrial internet of things: an unsupervised approach for fast root cause analysis,” in _2022 IEEE Conference on Control Technology and Applications (CCTA)_. IEEE, 2022, pp. 1366–1371.
  * [104] D. Marcato, G. Arena, D. Bortolato, F. Gelain, V. Martinelli, E. Munaron, M. Roetta, G. Savarese, and G. A. Susto, “Machine learning-based anomaly detection for particle accelerators,” in _2021 IEEE Conference on Control Technology and Applications (CCTA)_. IEEE, 2021, pp. 240–246.
  * [105] L. Felsberger, A. Apollonio, T. Cartier-Michaud, A. Müller, B. Todd, and D. Kranzlmüller, “Explainable deep learning for fault prognostics in complex systems: A particle accelerator use-case,” in _Machine Learning and Knowledge Extraction: 4th IFIP TC 5, TC 12, WG 8.4, WG 8.9, WG 12.9 International Cross-Domain Conference, CD-MAKE 2020, Dublin, Ireland, August 25–28, 2020, Proceedings 4_. Springer, 2020, pp. 139–158.
  * [106] P. Bellini, D. Cenni, L. A. I. Palesi, P. Nesi, and G. Pantaleo, “A deep learning approach for short term prediction of industrial plant working status,” in _2021 IEEE Seventh International Conference on Big Data Computing Service and Applications (BigDataService)_. IEEE, 2021, pp. 9–16.
  * [107] H. Choi, D. Kim, J. Kim, J. Kim, and P. Kang, “Explainable anomaly detection framework for predictive maintenance in manufacturing systems,” _Applied Soft Computing_ , vol. 125, p. 109147, 2022.
  * [108] D. Kim, G. Antariksa, M. P. Handayani, S. Lee, and J. Lee, “Explainable anomaly detection framework for maritime main engine sensor data,” _Sensors_ , vol. 21, no. 15, p. 5200, 2021.
  * [109] K. Michałowska, S. Riemer-Sørensen, C. Sterud, and O. M. Hjellset, “Anomaly detection with unknown anomalies: Application to maritime machinery,” _IFAC-PapersOnLine_ , vol. 54, no. 16, pp. 105–111, 2021.
  * [110] A. Bakdi, N. B. Kristensen, and M. Stakkeland, “Multiple instance learning with random forest for event logs analysis and predictive maintenance in ship electric propulsion system,” _IEEE Trans. Ind. Inf._ , vol. 18, no. 11, pp. 7718–7728, Nov. 2022.
  * [111] M. McCann and A. Johnston, “SECOM,” UCI Machine Learning Repository, 2008, DOI: https://doi.org/10.24432/C54305.
  * [112] M. Gashi, B. Mutlu, and S. Thalmann, “Impact of interdependencies: Multi-component system perspective toward predictive maintenance based on machine learning and xai,” _Applied Sciences_ , vol. 13, no. 5, p. 3088, 2023\.
  * [113] Q. Cao, C. Zanni-Merk, A. Samet, F. d. B. de Beuvron, and C. Reich, “Using rule quality measures for rule base refinement in knowledge-based predictive maintenance systems,” _Cybernetics and Systems_ , vol. 51, no. 2, pp. 161–176, 2020.
  * [114] A. Klein, “Hard drive failure rates: A look at drive reliability,” Jul 2021. [Online]. Available: https://www.backblaze.com/blog/backblaze-hard-drive-stats-q1-2020/
  * [115] M. Amram, J. Dunn, J. J. Toledano, and Y. D. Zhuo, “Interpretable predictive maintenance for hard drives,” _Machine Learning with Applications_ , vol. 5, p. 100042, Sep. 2021.
  * [116] I. Katser, V. Kozitsin, V. Lobachev, and I. Maksimov, “Unsupervised offline changepoint detection ensembles,” _Applied Sciences_ , vol. 11, no. 9, p. 4280, 2021.
  * [117] D. Dua and C. Graff, “Uci machine learning repository,” 2017. [Online]. Available: http://archive.ics.uci.edu/ml
  * [118] K. Scott, D. Kakde, S. Peredriy, and A. Chaudhuri, “Computational enhancements to the mahalanobis-taguchi system to improve fault detection and diagnostics,” in _2023 Annual Reliability and Maintainability Symposium (RAMS)_. IEEE, 2023, pp. 1–7.
  * [119] K. S. Hansen, N. Vasiljevic, and S. A. Sørensen, “Wind farm measurements,” May 2021. [Online]. Available: https://data.dtu.dk/collections/Wind_Farm_measurements/5405418/3
  * [120] C. M. Roelofs, M.-A. Lutz, S. Faulstich, and S. Vogt, “Autoencoder-based anomaly root cause analysis for wind turbines,” _Energy and AI_ , vol. 4, p. 100065, 2021.
  * [121] M. Beretta, A. Julian, J. Sepulveda, J. Cusidó, and O. Porro, “An ensemble learning solution for predictive maintenance of wind turbines main bearing,” _Sensors_ , vol. 21, no. 4, Feb. 2021.
  * [122] M. Beretta, Y. Vidal, J. Sepulveda, O. Porro, and J. Cusidó, “Improved ensemble learning for wind turbine main bearing fault diagnosis,” _Applied Sciences_ , vol. 11, no. 16, 2021. [Online]. Available: https://www.mdpi.com/2076-3417/11/16/7523
  * [123] S. J. Upasane, H. Hagras, M. H. Anisi, S. Savill, I. Taylor, and K. Manousakis, “A big bang-big crunch type-2 fuzzy logic system for explainable predictive maintenance,” in _2021 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)_. IEEE, 2021, pp. 1–8.
  * [124] P. M. Attia, A. Grover, N. Jin, K. A. Severson, T. M. Markov, Y.-H. Liao, M. H. Chen, B. Cheong, N. Perkins, Z. Yang _et al._ , “Closed-loop optimization of fast-charging protocols for batteries with machine learning,” _Nature_ , vol. 578, no. 7795, pp. 397–402, 2020.
  * [125] R. Csalódi, Z. Bagyura, and J. Abonyi, “Mixture of survival analysis models-cluster-weighted weibull distributions,” _IEEE Access_ , vol. 9, pp. 152 288–152 299, 2021.
  * [126] F. Wang, Z. Zhao, Z. Zhai, Z. Shang, R. Yan, and X. Chen, “Explainability-driven model improvement for SOH estimation of lithium-ion battery,” _Reliab. Eng. Syst. Saf._ , vol. 232, p. 109046, Apr. 2023.
  * [127] B. Verkuil, C. E. Budde, and D. Bucur, “Automated fault tree learning from continuous-valued sensor data: a case study on domestic heaters,” _arXiv preprint arXiv:2203.07374_ , 2022.
  * [128] L. Lorenti, G. De Rossi, A. Annoni, S. Rigutto, and G. A. Susto, “Cuad-mo: Continuos unsupervised anomaly detection on machining operations,” in _2022 IEEE Conference on Control Technology and Applications (CCTA)_. IEEE, 2022, pp. 881–886.
  * [129] B. A. ugli Olimov, K. C. Veluvolu, A. Paul, and J. Kim, “Uzadl: Anomaly detection and localization using graph laplacian matrix-based unsupervised learning method,” _Computers & Industrial Engineering_, vol. 171, p. 108313, 2022.
  * [130] A. Lourenço, M. Fernandes, A. Canito, A. Almeida, and G. Marreiros, “Using an explainable machine learning approach to minimize opportunistic maintenance interventions,” in _International Conference on Practical Applications of Agents and Multi-Agent Systems_. Springer, 2022, pp. 41–54.
  * [131] O. Serradilla, E. Zugasti, J. Ramirez de Okariz, J. Rodriguez, and U. Zurutuza, “Adaptable and explainable predictive maintenance: Semi-supervised deep learning for anomaly detection and diagnosis in press machine data,” _Applied Sciences_ , vol. 11, no. 16, p. 7376, 2021.
  * [132] M. Hermansa, M. Kozielski, M. Michalak, K. Szczyrba, Ł. Wróbel, and M. Sikora, “Sensor-based predictive maintenance with reduction of false alarms—a case study in heavy industry,” _Sensors_ , vol. 22, no. 1, p. 226, 2021.
  * [133] D. Xu, W. Cheng, J. Ni, D. Luo, M. Natsumeda, D. Song, B. Zong, H. Chen, and X. Zhang, “Deep multi-instance contrastive learning with dual attention for anomaly precursor detection,” in _Proceedings of the 2021 SIAM International Conference on Data Mining (SDM)_. SIAM, 2021, pp. 91–99.
  * [134] B. Steurtewagen and D. Van den Poel, “Adding interpretability to predictive maintenance by machine learning on sensor data,” _Computers & Chemical Engineering_, vol. 152, p. 107381, 2021.
  * [135] A. T. Keleko, B. Kamsu-Foguem, R. H. Ngouna, and A. Tongne, “Health condition monitoring of a complex hydraulic system using deep neural network and deepshap explainable xai,” _Advances in Engineering Software_ , vol. 175, p. 103339, 2023.
  * [136] G. Chen, M. Liu, and Z. Kong, “Temporal-logic-based semantic fault diagnosis with time-series data from industrial internet of things,” _IEEE Transactions on Industrial Electronics_ , vol. 68, no. 5, pp. 4393–4403, 2020\.
  * [137] A. Schmetz, C. Vahl, Z. Zhen, D. Reibert, S. Mayer, D. Zontar, J. Garcke, and C. Brecher, “Decision support by interpretable machine learning in acoustic emission based cutting tool wear prediction,” in _2021 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM)_. ieeexplore.ieee.org, Dec. 2021, pp. 629–633.
  * [138] T. V. Addison Howard, Sohier Dane, “Vsb power line fault detection,” 2018. [Online]. Available: https://kaggle.com/competitions/vsb-power-line-fault-detection
  * [139] S. Simmons, L. Jarvis, D. Dempsey, and A. W. Kempa-Liehr, “Data mining on extremely long Time-Series,” in _2021 International Conference on Data Mining Workshops (ICDMW)_ , Dec. 2021, pp. 1057–1066.
  * [140] Y. Zhang, P. Wang, K. Liang, Y. He, and S. Ma, “An alarm and fault association rule extraction method for power equipment based on explainable decision tree,” in _2021 11th International Conference on Power and Energy Systems (ICPES)_ , Dec. 2021, pp. 442–446.
  * [141] S. J. Upasane, H. Hagras, M. H. Anisi, S. Savill, I. Taylor, and K. Manousakis, “A type-2 fuzzy based explainable AI system for predictive maintenance within the water pumping industry,” _IEEE Transactions on Artificial Intelligence_ , pp. 1–14, 2023.
  * [142] L. Xia, Y. Liang, J. Leng, and P. Zheng, “Maintenance planning recommendation of complex industrial equipment based on knowledge graph and graph neural network,” _Reliab. Eng. Syst. Saf._ , vol. 232, p. 109068, Apr. 2023.
  * [143] G. Tod, A. P. Ompusunggu, and E. Hostens, “An improved first-principle model of AC powered solenoid operated valves for maintenance applications,” _ISA Trans._ , vol. 135, pp. 551–566, Apr. 2023.
  * [144] M. Mahmoodian, F. Shahrivar, S. Setunge, and S. Mazaheri, “Development of digital twin for intelligent maintenance of civil infrastructure,” _Sustain. Sci. Pract. Policy_ , vol. 14, no. 14, p. 8664, Jul. 2022.
  * [145] I. Konovalenko and A. Ludwig, “Generating decision support for alarm processing in cold supply chains using a hybrid k-nn algorithm,” _Expert Systems with Applications_ , vol. 190, p. 116208, 2022.
  * [146] A. Dhaou, A. Bertoncello, S. Gourvénec, J. Garnier, and E. Le Pennec, “Causal and interpretable rules for time series analysis,” in _Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining_, ser. KDD ’21. New York, NY, USA: Association for Computing Machinery, Aug. 2021, pp. 2764–2772.
  * [147] [Online]. Available: https://catalogue.data.wa.gov.au/dataset/water-pipe-wcorp-002
  * [148] P. Castle, J. Ham, M. Hodkiewicz, and A. Polpo, “Interpretable survival models for predictive maintenance,” in _30th European Safety and Reliability Conference and 15th Probabilistic Safety Assessment and Management Conference_. research-repository.uwa.edu.au, 2020, pp. 3392–3399.
  * [149] V. Belle and I. Papantonis, “Principles and practice of explainable machine learning,” _Front Big Data_ , vol. 4, p. 688969, Jul. 2021.
  * [150] A. Saabas, “Interpreting random forests,” Oct 2014. [Online]. Available: http://blog.datadive.net/interpreting-random-forests/
  * [151] S. Bach, A. Binder, G. Montavon, F. Klauschen, K.-R. Müller, and W. Samek, “On Pixel-Wise explanations for Non-Linear classifier decisions by Layer-Wise relevance propagation,” _PLoS One_ , vol. 10, no. 7, p. e0130140, Jul. 2015\.
  * [152] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,” _Proceedings of the IEEE_ , vol. 86, no. 11, pp. 2278–2324, 1998.
  * [153] M. Sundararajan, A. Taly, and Q. Yan, “Axiomatic attribution for deep networks,” in _Proceedings of the 34th International Conference on Machine Learning_ , ser. Proceedings of Machine Learning Research, D. Precup and Y. W. Teh, Eds., vol. 70. PMLR, 2017, pp. 3319–3328.
  * [154] D. Janzing, D. Balduzzi, M. Grosse-Wentrup, and B. Schölkopf, “Quantifying causal influences,” _aos_ , vol. 41, no. 5, pp. 2324–2358, Oct. 2013\.
  * [155] D. Dandolo, C. Masiero, M. Carletti, D. Dalle Pezze, and G. A. Susto, “AcME—Accelerated model-agnostic explanations: Fast whitening of the machine-learning black box,” _Expert Syst. Appl._ , vol. 214, p. 119115, Mar. 2023.
  * [156] D. Smilkov, N. Thorat, B. Kim, F. Viégas, and M. Wattenberg, “Smoothgrad: removing noise by adding noise,” _arXiv preprint arXiv:1706.03825_ , 2017\.
  * [157] S. Wachter, B. Mittelstadt, and C. Russell, “Counterfactual explanations without opening the black box: Automated decisions and the GDPR,” _Harv. JL & Tech._, 2017.
  * [158] TeamHG-Memex, “Teamhg-memex/eli5: A library for debugging/inspecting machine learning classifiers and explaining their predictions.” [Online]. Available: https://github.com/TeamHG-Memex/eli5
  * [159] R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra, “Grad-CAM: Visual explanations from deep networks via gradient-based localization,” _Int. J. Comput. Vis._ , vol. 128, no. 2, pp. 336–359, Feb. 2020.
  * [160] M. Carletti, M. Terzi, and G. A. Susto, “Interpretable anomaly detection with DIFFI: Depth-based feature importance of isolation forest,” _Eng. Appl. Artif. Intell._ , vol. 119, p. 105730, Mar. 2023.
  * [161] I. Mollas, N. Bassiliades, and G. Tsoumakas, “Conclusive local interpretation rules for random forests,” _Data Min. Knowl. Discov._ , vol. 36, no. 4, pp. 1521–1574, Jul. 2022.
  * [162] “AI4I 2020 Predictive Maintenance Dataset,” UCI Machine Learning Repository, 2020, DOI: https://doi.org/10.24432/C5HS5C.
  * [163] K. Simonyan, A. Vedaldi, and A. Zisserman, “Deep inside convolutional networks: Visualising image classification models and saliency maps,” _arXiv preprint arXiv:1312.6034_ , 2013.
  * [164] Vaswani, Shazeer, Parmar, and others, “Attention is all you need,” _Adv. Neural Inf. Process. Syst._ , 2017.
  * [165] L. A. Zadeh, “Fuzzy logic,” _Computer_ , vol. 21, no. 4, pp. 83–93, Apr. 1988\.
  * [166] M. Ravanelli and Y. Bengio, “Speaker recognition from raw waveform with sincnet,” in _2018 IEEE spoken language technology workshop (SLT)_. IEEE, 2018, pp. 1021–1028.
  * [167] J. N. Morgan and J. A. Sonquist, “Problems in the analysis of survey data, and a proposal,” _Journal of the American statistical association_ , vol. 58, no. 302, pp. 415–434, 1963.
  * [168] D. Bertsimas and J. Dunn, “Optimal classification trees,” _Machine Learning_ , vol. 106, pp. 1039–1082, 2017.
  * [169] D. Bertsimas, J. Dunn, E. Gibson, and A. Orfanoudaki, “Optimal survival trees,” _Machine Learning_ , vol. 111, no. 8, pp. 2951–3023, 2022.
  * [170] G. Ke, Q. Meng, T. Finley, T. Wang, W. Chen, W. Ma, Q. Ye, and T.-Y. Liu, “Lightgbm: A highly efficient gradient boosting decision tree,” _Advances in neural information processing systems_ , vol. 30, 2017.
  * [171] H. A. Watson _et al._ , “Launch control safety study,” _Bell labs_ , 1961\.
  * [172] Student, “The probable error of a mean,” _Biometrika_ , vol. 6, no. 1, pp. 1–25, 1908.
  * [173] K. Pearson, “X. on the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling,” _The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science_ , vol. 50, no. 302, pp. 157–175, 1900.
  * [174] P. Veličković, G. Cucurull, A. Casanova, A. Romero, P. Lio, and Y. Bengio, “Graph attention networks,” _arXiv preprint arXiv:1710.10903_ , 2017.
  * [175] D. A. Reynolds _et al._ , “Gaussian mixture models.” _Encyclopedia of biometrics_ , vol. 741, no. 659-663, 2009.
  * [176] H. Nori, S. Jenkins, P. Koch, and R. Caruana, “Interpretml: A unified framework for machine learning interpretability,” _arXiv preprint arXiv:1909.09223_ , 2019.
  * [177] L. E. Baum and T. Petrie, “Statistical inference for probabilistic functions of finite state markov chains,” _The annals of mathematical statistics_ , vol. 37, no. 6, pp. 1554–1563, 1966.
  * [178] O. Maler and D. Nickovic, “Monitoring temporal properties of continuous signals,” in _International Symposium on Formal Techniques in Real-Time and Fault-Tolerant Systems_. Springer, 2004, pp. 152–166.
  * [179] M. Grieves and J. Vickers, “Digital twin: Mitigating unpredictable, undesirable emergent behavior in complex systems,” _Transdisciplinary perspectives on complex systems: New findings and approaches_ , pp. 85–113, 2017\.
  * [180] D. A. Augusto and H. J. Barbosa, “Symbolic regression via genetic programming,” in _Proceedings. Vol. 1. Sixth Brazilian symposium on neural networks_. IEEE, 2000, pp. 173–178.
  * [181] T. Hastie and R. Tibshirani, “Generalized additive models,” _Stat. Sci._ , vol. 1, no. 3, pp. 297–310, Aug. 1986.
  * [182] G. Taguchi, G. Taguchi, and R. Jugulum, _The Mahalanobis-Taguchi strategy: A pattern technology system_. John Wiley & Sons, 2002.
  * [183] E. Fix and J. L. Hodges, “Discriminatory analysis. nonparametric discrimination: Consistency properties,” _International Statistical Review/Revue Internationale de Statistique_ , vol. 57, no. 3, pp. 238–247, 1989\.
  * [184] A. Barredo Arrieta, N. Díaz-Rodríguez, J. Del Ser, A. Bennetot, S. Tabik, A. Barbado, S. Garcia, S. Gil-Lopez, D. Molina, R. Benjamins, R. Chatila, and F. Herrera, “Explainable artificial intelligence (XAI): Concepts, taxonomies, opportunities and challenges toward responsible AI,” _Inf. Fusion_ , vol. 58, pp. 82–115, Jun. 2020.
  * [185] T. Miller, “Explanation in artificial intelligence: Insights from the social sciences,” _Artif. Intell._ , vol. 267, pp. 1–38, Feb. 2019.
  * [186] L. Coroama and A. Groza, “Evaluation metrics in explainable artificial intelligence (xai),” in _International Conference on Advanced Research in Technologies, Information, Innovation and Sustainability_. Springer, 2022, pp. 401–413.
  * [187] M. Sisk, M. Majlis, C. Page, and A. Yazdinejad, “Analyzing xai metrics: Summary of the literature review,” _TechRxiv preprint techrxiv.21262041.v1_ , 2022.
  * [188] M. A. Kadir, A. Mosavi, and D. Sonntag, “Assessing xai: Unveiling evaluation metrics for local explanation, taxonomies, key concepts, and practical applications,” _engrXiv preprint_ , 2023.
  * [189] R. R. Hoffman, S. T. Mueller, G. Klein, and J. Litman, “Metrics for explainable ai: Challenges and prospects,” _arXiv preprint arXiv:1812.04608_ , 2018.

|  Logan Cummins (Member, IEEE) received their B.S. degree in Computer Science
and Engineering from Mississippi State University. They are currently pursuing
a Ph.D degree in Computer Science at Mississippi State University with a minor
in Cognitive Science. They are a Graduate Research Assistant with the
Predictive Analytics and Technology Integration (PATENT) Laboratory in
collaboration with the Institute for Systems Engineering Research.
Additionally, they perform research with the Social Therapeutic and Robotic
Systems (STaRS) research lab. Their research interests include explainable
artificial intelligence and its applications, cognitive science, and human-
computer interactions as applied to human-agent teamming. They are a member of
ACM and IEEE at Mississippi State University.
---|---
|  Alexander Sommers (Member, IEEE) received his B.S. in Computer Science from
Saint Vincent Collage and his M.S. in Computer Science from Southern Illinois
University. He is pursuing A Ph.D. in Computer Science at Mississippi State
University with a concentration in machine learning. He is a Graduate Research
Assistant in the Predictive Analytics and Technology Integration Laboratory
(PATENT Lab), in collaboration with the Institute for Systems Engineering
Research. His work concerns synthetic time-series generation and remaining-
useful-life prediction. His interests are the application of machine learning
to reliability engineering and lacuna discovery respectively. He is a member
of IEEE and ACM.
---|---
|  Somayeh Bakhtiari Ramezani (Member, IEEE) received the B.S. degree in
computer engineering and the M.S. degree in information technology engineering
from the Iran University of Science and Technology, in 2004 and 2008,
respectively. She is currently pursuing the Ph.D. degree in computer science
with Mississippi State University. She is a Graduate Research Assistant with
the Predictive Analytics and Technology Integration (PATENT) Laboratory in
collaboration with the Institute for Systems Engineering Research. Prior to
joining Mississippi State University, in 2019, she was with several companies
in the energy and healthcare sectors as an HPC programmer and a data
scientist. She is a 2021 SIGHPC Computational and Data Science Fellow. Her
research interests include probabilistic modeling and optimization of dynamic
systems, the application of ML, quantum computation, and time-series
segmentation in the healthcare sector. She is a member of an ACM, the
President of the ACM-W Student Chapter at Mississippi State University, and
the Chair of the IEEE-WIE AG Mississippi section.
---|---
|  Sudip Mittal (Member, IEEE) is an Assistant Professor in the Department of
Computer Science & Engineering at the Mississippi State University. He
graduated with a Ph.D. in Computer Science from the University of Maryland
Baltimore County in 2019. His primary research interests are cybersecurity and
artificial intelligence. Mittal’s goal is to develop the next generation of
cyber defense systems that help protect various organizations and people. At
Mississippi State, he leads the Secure and Trustworthy Cyberspace (SECRETS)
Lab and has published over 80 journals and conference papers in leading
cybersecurity and AI venues. Mittal has received funding from the NSF, USAF,
USACE, and various other Department of Defense programs. He also serves as a
Program Committee member or Program Chair of leading AI and cybersecurity
conferences and workshops. Mittal’s work has been cited in the LA times,
Business Insider, WIRED, the Cyberwire, and other venues. He is a member of
the ACM and IEEE.
---|---
|  Joseph Jabour received his B. S. of Computer Science from the University of
Mississippi in 2019, and is currently pursuing an M. S. of Computer Science
from Mississippi State University. He is a Computer Scientist at the
Information Technology Lab (ITL) of the Engineering Research and Development
Center (ERDC) in Vicksburg, MS. He began working at the ERDC in 2019, and he
has pursued research in the field of Artificial Intelligence and Machine
Learning. Additionally, he has performed a significant amount of work in the
fields of Data Visualization, Digital Twins, and many other forms of research.
He has since presented at several nationally recognized conferences, has held
a vice chair position in the ERDC Association of Computing Machinery, and is
currently a facilitator of the ERDC ITL Field Training Exercise based off of
leadership principles from the Echelon Front. He has received awards for his
research and development including but not limited to the ERDC Award for
Outstanding Innovation in Research and Development. He seeks to push past the
forefront of technological development and innovation and endeavors to
identify and implement solutions to our nation’s leading causes of concern.
---|---
|  Maria Seale received the B.S. degree in computer science from the
University of Southern Mississippi, in 1987, and the M.S. and Ph.D. degrees in
computer science from Tulane University, in 1992 and 1995, respectively. Prior
to joining the Information Technology Laboratory, U.S. Army Engineer Research
and Development Center (ERDC), in 2016, she held positions with the Institute
for Naval Oceanography, the U.S. Naval Research Laboratory, and various
private companies, as well as a tenured Associate Professorship with the
University of Southern Mississippi. At ERDC, she has been involved with
research in making scalable machine learning algorithms available on high-
performance computing platforms and expanding the center’s capabilities to
manage and analyze very large data sets. Her research interests include
natural language processing, machine learning, natural computing, high-
performance data analytics, and prognostics and health management for
engineered systems. She is a member of the Prognostics and Health Management
Society, the American Society of Mechanical Engineers, and the Association of
Computing Machinery.
---|---
|  Shahram Rahimi (Member, IEEE) is currently a Professor and the Head of the
Department of Computer Science and Engineering, Mississippi State University.
Prior to that, he led the Department of Computer Science, Southern Illinois
University, for five years. He is also a recognized leader in the area of
artificial and computational intelligence, with over 220 peer-reviewed
publications and a few patents or pending patents in this area. He is a member
of the IEEE New Standards Committee in Computational Intelligence. He provides
advice to staff and administration at the federal government on predictive
analytics for foreign policy. He was a recipient of the 2016 Illinois Rising
Star Award from ISBA, selected among 100s of highly qualified candidates. His
intelligent algorithm for patient flow optimization and hospital staffing is
currently used in over 1000 emergency departments across the nation. He was
named one of the top ten AI technology for healthcare, in 2018, by HealthTech
Magazine. He has secured over $20M of federal and industry funding as a PI or
a co-PI in the last 20 years. He has also organized 15 conferences and
workshops in the areas of computational intelligence and multi-agent systems
over the past two decades. He has served as the Editor-in-Chief for two
leading computational intelligence journals and is on the editorial board of
several other journals.
---|---