# The sample of eight LLAGNs:
X-ray properties.

Nadiia G. Pulatova1 Anatoliy V. Tugay2 Lidiia V. Zadorozhna2 1 Main
Astronomical Observatory of the National Academy of Sciences of Ukraine,
Akademika Zabolotnoho str. 27, Kyiv, Ukraine, 03143
email<EMAIL_ADDRESS>
2Taras Shevchenko National University of Kyiv, Physical faculty,
ave. Glushkova 2, building 1, Kyiv, Ukraine, 03680
emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
(2021)

###### Abstract

LLAGN are very important objects for studying as they are found in a large
fraction of all massive galaxies. Nevertheless this topic needs more
investigation as fraction of LLAGN in all AGN are much more higher than
fraction of researches dedicated to LLAGN among all AGN studies. The goal of
our work is checking out X-ray properties of LLAGN. For this purpose we
created a sample of LLAGN by selecting most prominent LLAGN from literature
and analyzed their X-ray spectral properties. As a result, we obtained 12
LLAGN and for 8 of them XMM X-ray observations are available. The spectra from
one XMM camera, PN, were fitted with power law + absorption of neutral
hydrogen. In the current report we present the previous results of this study.
We plan to increase numbers of objects in our future studies.

###### keywords:

LLAGNs, X-ray spectra, XMM Newton.

††volume: xxx††journal: IAU Symposium 367/VIRTUAL Education and Heritage in
the Era of Big Data in Astronomy††editors: A.C. Editor, B.D. Editor & C.E.
Editor, eds.

##  1 Introduction

For nowadays numerous observations in all electromagnetic spectrum are
available for public using and searching for a new ideas with a purpose to
clarify the nature of AGN (active galactic nuclei). Mostly all scientific
attention is paid to bright AGNs: Sy1 and Sy2 types. In our current study we
will investigate X-ray properties of Low-Luminosity AGN (LLAGN). It is
generally accepted that most time of its “life” galaxy spend in a quiescent
mode. Therefore all studies connected with LLAGN plays very important role in
understanding of the nature and physical mechanisms that take place in a
galactic nuclei. The intrinsic faintness, i.e. $L_{Bol}<10^{44}erg\cdot
s^{-1}$ , and the low level of activity are the distinctive characteristics of
LLAGNs, which mainly consists of local Seyfert galaxies and Low-Ionization
Nuclear Emission-line Regions (LINERs).

Table 1: The sample of selected LLAGNs 1.

Object | RA | DEC | Optical | z | Morph. | Flux, | X-Ray Flux, | XMM
---|---|---|---|---|---|---|---|---
name |  |  | classif. |  | type | V mag | erg/cm2/s | obs.
NGC 404 | 01 09 27.1 | +35 43 05 | LINER | 0.000107 | dE/S0 | 11.73 |  $6.71*10-14$, 3 | no
NGC 1052 | 02 41 04.8 | -08 15 20.8 | LINER h | 0.004930 | E | 10.47 |  $2.7*10^{-11}$, 2 | 5
NGC 3368 | 10 46 45.7 | +11 49 11.7 | LINER | 0.003012 | SBa | 9.25 |  $2.1*10^{-13}$, 4 | no
NGC 3486 | 11 00 23.9 | +28 58 29.3 | LINER/Sy 2 | 0.002272 | SABc | 10.53 |  $1.1*10^{-13}$, 5 | 1
NGC 3642 | 11 22 18.0 | +59 04 27.2 | LINER b | 0.00529 | SAbc | 14.04 | - | no
NGC 3998 | 11 57 56.1 | +55 27 12.9 | LINER 1.9 | 0.003646 | SA0 | 12.10 |  $1.4*10^{-11}$, 2 | 2
NGC 4203 | 12 15 05.0 | +33 11 50.4 | LINER b | 0.00362 | SA0 | 11.99 |  $31.42*10^{-13}$, 7 | no
NGC 4552 | 12 35 39.8 | +12 33 22.8 | LINER/Sy 2/HII | 0.000914 | S0 | 9.75 |  $9.60*10^{-13}$, 6 | 1
NGC 4579 | 12 37 43.5 | +11 49 05.1 | LINER 1.9/Sy 1.9 | 0.00562 | Sab | 9.66 |  $8.9*10^{-12}$, 2 | 2
NGC 4594 | 12 39 59.4 | -11 37 22.9 | Sy 2/Sy 1.9 | 0.003416 | Sa | 8.00 |  $29.24*10^{-13}$, 6 | 1
M 87 | 12 30 49.4 | +12 23 28.0 | LINER | 0.00420 | E-E/S0 | 8.63 |  $1.05*10^{-10}$ , 8 | 3
M 81 | 09 55 33.1 | +69 03 55.0 | LINER/Sy1.8/ QSO | -0.000140 | SAab | 6.94 |  $3.10*10^{-11}$, 5 | 5

Notes:
1 Parameters for every galaxy were taken from NED
(http://ned.ipac.caltech.edu/), SIMBAD (http://simbad.u-strasbg.fr/) and
Hyperleda (http://leda.univ-lyon1.fr/) databases.
2 15-150keV, 3 0.3-8keV, 4 0.7-7keV, 5 2-10 keV, 6 0.2-4 keV, 7 0.2-4 keV, 8
0.1-2.4 keV

Table 2: The sample of selected LLAGNs 1.

Object | Obs ID | Obs Date | Dur., s | XSPEC | Pho | $N_{H},10^{22}$ | kT , | kT , | $\chi^{2}/d.o.f.$
---|---|---|---|---|---|---|---|---|---
name |  |  |  | Model | Index |  | (ap/mek) keV | (zgauss) keV |
NGC 1052 | 0093630101 | 2001-08-15 | 16308 | apec+zgauss+ | 1.33 | 2.28 | 0.81 | 5.96 | 55/56
|  |  |  | po+ phabs·po | $\pm 0.15$ | $\pm 0.97$ |  |  |
| 0306230101 | 2006-01-12 | 54909 | apec+zgauss+ | 1.03 | 6.42 | 0.74 | 5.65 | 336/296
|  |  |  | po+ phabs·po | $\pm 0.16$ | $\pm 1.08$ |  |  |
| 0553300301 | 2009-01-14 | 52319 | apec+zgauss+ | 1.48 | 9.26 | 0.75 | 6.41 | 282/287
|  |  |  | po+ phabs·po | $\pm 0.074$ | $\pm 0.56$ |  |  |
| 0553300401 | 2009-08-12 | 59018 | apec+zgauss+ | 1.13 | 9.95 | 0.75 | 6.37 | 356/324
|  |  |  | po+ phabs·po | $\pm 0.067$ | $\pm 0.51$ |  |  |
| 0790980101 | 2017-01-17 | 70500 | apec+zgauss+ | 1.06 | 9.59 | 0.75 | 6.37 | 293/264
|  |  |  | po+ phabs·po | $\pm 3.62$ | $\pm 5.29$ |  |  |
NGC 3486 | 0112550101 | 2001-05-09 | 15172 | powerlaw | 2.9 |  |  |  | 3/3
|  |  |  |  | $\pm 0.50$ |  |  |  |
NGC 3998 | 0790840101 | 2016-10-26 | 25000 | phabs·po | 1.86 |  2.13·$10^{-2}$ |  |  | 158/163
|  |  |  |  | $\pm 0.03$ | $\pm 0.0061$ |  |  |
| 0090020101 | 2001-05-09 | 13222 | phabs·po | 1.85 |  2.31·$10^{-2}$ |  |  | 403/373
|  |  |  |  | $\pm 0.015$ | $\pm 0.0029$ |  |  |
NGC 4552 | 0141570101 | 2003-07-10 | 44838 | phabs·po+apec | 2.1 |  7.59·$10^{-2}$ | 0.72 |  | 96/120
|  |  |  |  | $\pm 0.085$ | $\pm 0.016$ |  |  |
NGC 4579 | 0790840201 | 2016-12-06 | 23002 | po·phabs + mekal | 1.8 |  3.82·$10^{-2}$ | 0.39 |  | 510/ 452
|  |  |  |  | $\pm 0.013$ | $\pm 0.0018$ |  |  |
| 0112840101 | 2003-06-12 | 23669 | po·phabs + mekal | 1.8 |  2.62·$10^{-2}$ | 0.62 |  | 396/ 309
|  |  |  |  | $\pm 0.024$ | $\pm 0.0043$ |  |  |
NGC 4594 | 0084030101 | 2001-12-28 | 43456 | po·phabs | 1.9 | 0.16 |  |  | 116/111
|  |  |  |  | $\pm 0.038$ | $\pm 0.0098$ |  |  |
M 81 | 0112521101 | 2002-04-16 | 11518 | phabs·po+apec | 1.87 | 0.2 | 0.93 |  | 187/ 187
|  |  |  |  | $\pm 0.030$ | $\pm 0.0078$ |  |  |
| 0200980101 | 2004-09-26 | 119166 | phabs·po+apec+ | 2.55 | 0.25 | 1.01 | 2.15·10-3 | 818/712
|  |  |  | zgauss | $\pm 0.16$ | $\pm 0.021$ |  |  |
| 0693851101 | 2012-11-16 | 13316 | phabs·po+apec+ | 1.74 | 0.23 | 1.23 | 1.55 | 178/164
|  |  |  | zgauss | $\pm 0.069$ | $\pm 0.014$ |  |  |
M 87 | 0114120101 | 2000-06-19 | 60109 | phabs·po+apec | 2.45 | 8.85·10-2 | 1.32 |  | 922/508
|  |  |  |  | $\pm 0.032$ | $\pm 0.0041$ |  |  |

Notes:
1 Parameters for every galaxy were taken from NED
(http://ned.ipac.caltech.edu/), SIMBAD (http://simbad.u-strasbg.fr/) and
Hyperleda (http://leda.univ-lyon1.fr/) databases.
2 15-150keV, 3 0.3-8keV, 4 0.7-7keV, 5 2-10 keV, 6 0.2-4 keV, 7 0.2-4 keV, 8
0.1-2.4 keV

##  2 Overview

The common properties of LLAGNs were studied on the base of different samples
of LLAGNs. Very good review of LINER‘s properties, the most common form of
LLAGNs, was made by [Dan Maoz (2008), Dan Maoz (2008)]. In this paper author
argues that there is actually more similarity than contrast between the
observed spectral properties of different AGNs, with only a mild and
continuous change in the properties as a function of decreasing accretion
rate, going from quasars to the faint LINERs in nearby galaxies. A sample of
seven nearby, early type spiral galaxies was studied in[Roberts at all (2001),
Roberts at all (2001)]. The authors used ROSAT HRI spatial data and ASCA
spectral measurements to address the question of whether a LLAGN is present in
galaxies that have a LINER 2 classification. All seven galaxies have X-ray
spectra consistent with a two-component, soft thermal plus hard power-law,
spectral form. In the paper an idea is claimed, that in many, perhaps the
majority, of LINER 2 galaxies, the nuclear X-ray luminosity does not derive
directly from the presence of a LLAGN. [Minfeng Gu & Xinwu Cao (2009), Minfeng
Gu & Xinwu Cao (2009)] found an interesting difference between Low-Ionization
Nuclear Emission-line Regions (LINERs) and luminous active galactic nuclei
(AGNs). A significant anti-correlation was revealed between the hard X-ray
photon index and the Eddington ratio$L_{Bol}/L_{Edd}$ for a sample of LINERs
and local Seyfert galaxies compiled from literature with Chandra or XMM-Newton
observations. To the most brightest LLAGNs were dedicated numerous studies by
many scientific groups from different countries. For example, M87 that is
included to our sample of LLAGNs. In [Jolley, E.J.D., Kuncic Z. (2007),
Jolley, E.J.D., Kuncic Z. (2007)] to the well known LLAGN M87 was applied an
accretion model that attributes the low radiative output to a low mass
accretion rate, rather than a low radiative efficiency. Authors calculated the
combined disk-jet steady-state broadband spectrum. A comparison between
predicted and observed spectra indicates that M87 may be a maximally spinning
black hole accreting at a rate of $\sim 10^{-3}M_{sun}yr^{-1}$ In the resent
report made by [Younes et al (2019), Younes et al (2019)] is presented the
analysis of simultaneous XMM-Newton+NuSTAR observations of two low-luminosity
Active Galactic Nuclei (LLAGN), NGC 3998 and NGC 4579. In the paper were
calculated parameters of X-ray emmision of these LLAGNs, and were discussed
these results in the context of hard X-ray emission from bright AGN, other
LLAGN, and hot accretion flow models.

The goal of our study was to obtain XMM spectra for LLAGNs and to search for
the common behavior of their spectra. Therefore we performed facial analysis
of XMM observations for every source. To find the best spectral fit to the
data, several models were tested in this work. For the beginning we fitted
every obtained spectra with a simple phabs·powerlaw model that shows that
X-ray emission is driven by non-thermal processes. We found that X -ray
spectra of only 3 sources from our sample can be fitted with simple
phabs·powerlaw model. Another 5 LLAGNs have very complicated spectra and
fitting with phabs·powerlaw model doesn‘t provide good $\chi^{2}$ statistic.

The data extraction was performed with the Science Analysis System (SAS)
version 16.0.0, The PN data are selected using event patterns 0–4 and 0–12,
respectively. We extracted source events for all observations from a circle
that should hold 90% of X-ray emission of the source and the center should be
equidistant from the ring.

We used archived XMM-Newton observations for 8 from 12 selected LLAGNs. (see
Table 1) NGC3642 is not bright in X-ray range and for 3 LLAGNs, namely NGC404
NGC3368 and NGC4203 there is no XMM observations. For sources with more than 5
XMM observations we chose observations distributed in time with the reason
that we can observe some changes in their spectral properties, if they exist
(see Table 2 ).

Table 1.

##  3 Implications

Conclusions.We created the sample of 12 LLAGNs for investigation of their
X-ray properties. Redshift z in our sample varies from -0.000140 in M81 to
0.00562 in NGC 4579. It means that all objects are relatively close to us
galaxies. From Table 1 we can see that 75% (9 from 12) of the host galaxies of
selected LLAGNs are spiral and only 25% (3 from 12) are elliptical. We
obtained XMM spectra for LLAGNs and tried to find common behaviors in their
spectra. Nevertheless their relatively low brightness (if compare with AGNs),
we found that X-ray spectra of only 3 sources from our sample can be fitted
with simple phabs·powerlaw model. X-ray spectra of another 5 X-ray sources
required models with multiple components and shows complicated features.
Obtained model’s parameters are: Photon Index varies from 1.2 in NGC 1052 to
2.90 in NGC 3486; absorption by neutral hydrogen $N_{H}=2.22\cdot
10^{20}cm^{-2}$ in NGC 3998 to $N_{H}=7.5\cdot 10^{22}cm^{-2}$.

## References

  * [Dan Maoz (2008)] Dan Maoz 2008,  J. Phys.: Conf. Ser. 131, 012036
  * [Jolley, E.J.D., Kuncic Z. (2007)] Jolley, E.J.D., Kuncic, Z. 2007, Ap&SS 311, 257
  * [Minfeng Gu & Xinwu Cao (2009)] Minfeng Gu, Xinwu Cao 2009, MNRAS, 399, 349
  * [ Roberts at all (2001)] Roberts T. P., N. J. Schurch, R. S. Warwick 2001, MNRAS, 324, 737
  * [Younes et al (2019)] Younes, George; Ptak, Andrew; Ho, Luis C.; Xie, Fu-Guo; Terasima, Yuichi; Yuan, Feng; Huppenkothen, Daniela; Yukita, Mihoko 2019, ApJ, 870, 73


# Linear canonical wavelet transform and the associated uncertainty principles

Bivek Guptaa, Amit K. Vermab,
a,b Department of Mathematics, IIT Patna, Bihta, Patna 801103.
Carlo Cattanic
cEngineering School (DEIM), University of Tuscia,
Largo dell’Università, 01100 Viterbo, Italy.
<EMAIL_ADDRESS>author email:
akverma<EMAIL_ADDRESS>
###### Abstract

We define a novel time-frequency analyzing tool, namely linear canonical
wavelet transform (LCWT) and study some of its important properties like inner
product relation, reconstruction formula and also characterize its range. We
obtain Donoho-Stark’s and Lieb’s uncertainty principle for the LCWT and give a
lower bound for the measure of its essential support. We also give Shapiro’s
mean dispersion theorem for the proposed LCWT.

Keywords: Linear canonical transform; Linear canonical wavelet transform;
Uncertainty principle; Shapiro’s theorem
AMS Subject Classification 2020: 42B10, 42C40, 43A32

##  1 Introduction

We first mention below some important abbreviations that will be used
throughout this paper.

List of Abbreviations
FT - Fourier transform
FrFT - Fractional Fourier transform
LCT - Linear canonical transform
WT - Wavelet transform
FrWT - Fractional wavelet transform
WLCT - Windowed linear canonical transform
LCWT - Linear canonical wavelet transform
ONS - Orthonormal sequence
RKHS - Reproducing kernel Hilbert Space
UP - Uncertainty principle

As a generalization of FT [1] and FrFT [2, 3, 4], LCT is a four parameter
family of linear integral transform proposed by Mohinsky and Quesne [5] and is
considered as the important tool for non-stationary signal processing. Because
of the extra degrees of freedom, as compared to the FT and FrFT, its
application can be found in number of fields including signal separation [6],
signal reconstruction [7], filter designing [8] and many more. Recently in [9]
the authos studied octonion linear canonical transform. For more detail on LCT
and its application we refer the reader to [10].

Even though the wavelet transform (WT)[11] is a potential tool for the
analysis of non-stationary signals, it is incompetent for analyzing the
signals with not well concentrated energy in the time-frequency plane, for
example, the chirp like signal which is ubiquitous in nature [12]. On the
other hand, for the signal whose energy in the frequency domain is not well
concentrated, LCT is an appropriate tool. However, because of its global
kernel it is not capable of indicating the time localization of the LCT
spectral components, and thus LCT is not suitable for processing the non-
stationary signal whose LCT spectral characteristics changes with time. The
WLCT [13] is thus proposed to overcome this drawback. In this case, the
original signal is first segmented with time localization window, followed by
performing the LCT spectral analysis for these segment. WLCT is capable of
offering a joint signal representation in both time and LCT domain, but its
fixed window width limits the practical application, it is impossible to
provide good time resolution and spectral resolution simultaneously.

Thus to circumvent these limitations of LCT, WT and WLCT we propose a novel
LCWT. In fact, Wei et al. [14] and Guo et al. [15] generalized the FrWT,
studied in [16], to the LCWT. Wei et al. [14] studied its resolution in time
and linear canonical domains and Guo et al. [15] studied its properties on
Sobolev spaces. Dai et al. [12] gave a new definition of the FrWT(also see
[17]), which we generalize in the context of the LCT and study the associated
UP.

In Harmonic analysis, the UP is a relation between a function and its FT which
says that a function (non-zero) and its FT cannot be very well localize
simultaneously. This general fact is interpreted in several different ways,
for this we refer the reader to a survey paper by Folland and Sitaram [18].
Shapiro, in [19], studied the localization for an ONS and proved that if an
ONS $\\{\phi_{k}\\}$ in $L^{2}(\mathbb{R})$ and the sequence of their FT
$\\{\hat{\phi}_{k}\\}$ are such that their means and dispersions are uniformly
bounded, then $\\{\phi_{k}\\}$ is finite. Jaming and Powell [20] proved a
quantitative version of Shapiro’s theorem which says that for an ONS
$\\{\phi_{k}\\}$ in $L^{2}(\mathbb{R})$ and $N\in\mathbb{N},$

$\sum_{k=1}^{N}\left(\|t\phi_{k}\|_{L^{2}(\mathbb{R})}^{2}+\|\xi\hat{\phi}_{k}\|_{L^{2}(\mathbb{R})}^{2}\right)\geq\frac{(N+1)^{2}}{2\pi}.$

A multivariable quantitative version of Shapiro’s theorem for generalized
dispersion was proved by Malinnikova [21]. It states that if $\\{\phi_{k}\\}$
be an ONS in $L^{2}(\mathbb{R}^{d}),$ $N\in\mathbb{N}$ and $p>0$ then
$\exists$ $C_{p,d}$ for which

$\sum_{k=1}^{N}\left(\||t|^{\frac{p}{2}}\phi_{k}\|_{L^{2}(\mathbb{R}^{d})}^{2}+\||\xi|^{\frac{p}{2}}\hat{\phi}_{k}\|_{L^{2}(\mathbb{R}^{d})}^{2}\right)\geq
C_{p,d}N^{1+\frac{p}{2d}}.$

Recently, in this direction Shapiro’s mean dispersion theorem has been proved
for many integral transforms like short-time FT [22], WT [23], Hankel WT [24],
Hankel Stockwell transform [25], shearlet transform [26], etc.

The main objectives of this paper are (i) to define a new time-frequency
analysing tool, namely LCWT, which generalizes the FrWT studied in [12] in the
context of LCT, and study some of its basic properties along with the inner
product relation, reconstruction formula and also characterize its range (ii)
to study the time-LCT frequency analysis and the associated constant
$Q-$factor (iii) to establish an UP for the LCWT for a finite energy signal.
The UP for the LCWT can be derived from the UP of the LCT following the
strategy adopted by Wilczok [27] and Verma et al. [28] while deriving the UP
for the WT and the FrWT respectively. Similar UP has been introduced for
several integral transform like fractional WT [29], non-isotropic angular
Stockwell transform [30], etc. However, we are interested in proposing an
uncertainty principle directly for the LCWT without using the UP associated
with LCT. In this regard we establish the Donoho-Stark’s and the Lieb’s UP for
the LCWT, which in turn provide a lower bound for the measure of essential
support of the LCWT. See also [13],[31], for similar results in case of other
integral transforms. (iv) to study the Shapiro’s mean dispersion theorem for
the LCWT.

The paper is arranged as follows. In section 2, we recall the definition of
LCT and some of its properties. In section 3, we define LCWT and study some of
its basic properties including inner product relation, reconstruction formula
and also characterize the range of the transform. Donoho-Stark’s and Lieb’s UP
for the proposed LCWT are studied in section 4. Section 5 is devoted to
Shapiro’s mean dispersion theorem for LCWT. Finally, in section 6, we conclude
our paper.

##  2 Preliminaries

We briefly recall the definition of LCT and its important properties that we
will be using in the sequel.

###  2.1 LCT

######  Definition 2.1.

The LCT of $f\in L^{2}(\mathbb{R})$, with respect to a matrix parameter

$M=\begin{bmatrix}A&B\\\ C&D\end{bmatrix},AD-BC=1$

is defined as

$(\mathcal{L}^{M}f)(\xi)=\begin{cases}\displaystyle\int_{\mathbb{R}}f(t)K_{M}(t,\xi)dt,B\neq
0,\\\ \sqrt{D}e^{\frac{i}{2}CD\xi^{2}}f(D\xi),~{}B=0,\end{cases}$

where $K_{M}(t,\xi)$ is a kernel given by

$K_{M}(t,\xi)=\frac{1}{\sqrt{2\pi
iB}}e^{\frac{i}{2}\left(\frac{A}{B}t^{2}-\frac{2}{B}\xi
t+\frac{D}{B}\xi^{2}\right)},~{}\xi\in\mathbb{R}.$ (1)

Among several important properties of the LCT, the important among them, that
will be used in the sequel, is the Parseval’s formula

$\int_{\mathbb{R}}f(t)\overline{g(t)}dt=\int_{\mathbb{R}}(\mathcal{L}^{M}f)(\xi)\overline{(\mathcal{L}^{M}g)(\xi)}d\xi,~{}\mbox{where}~{}f,~{}g\in
L^{2}(\mathbb{R}).$ (2)

Particularly, if $f=g,$ then we have the Plancherel’s formula

$\|f\|_{L^{2}(\mathbb{R})}=\|\mathcal{L}^{M}f\|_{L^{2}(\mathbb{R})}.$ (3)

The LCTs satisfies the additive property, i.e.,

$\mathcal{L}^{M}\mathcal{L}^{N}f=\mathcal{L}^{MN}f,~{}\mbox{where}~{}f\in
L^{2}(\mathbb{R}),$ (4)

and the inversion property

$\mathcal{L}^{M^{-1}}\left(\mathcal{L}^{M}f\right)=f,$ (5)

where, $M^{-1}$ denotes the inverse of $M.$ For convenient, we now denote the
matrix $M$ by $(A,B;C,D).$

##  3 LCWT

We propose a new integral transform namely the LCWT. This definition is mainly
motivated from the definition of FrWT defined by Dai et al. [12]. We shall
discuss some of its basic properties along with the inner product relation,
reconstruction formula and also prove that its range is a RKHS.

Motivated by the definition of the admissible wavelet pair in [32], we first
define it in the setting of LCT domain.

######  Definition 3.1.

A pair $\\{\psi,\phi\\}$ of functions in $L^{2}(\mathbb{R})$ is said to be an
admissible linear canonical wavelet pair (ALCWP) if they satisfy the following
admissibility condition

$C_{\psi,\phi,M}:=\int_{\mathbb{R^{+}}}\overline{(\mathcal{L}^{M}\psi)\left(\frac{\xi}{a}\right)}(\mathcal{L}^{M}\phi)\left(\frac{\xi}{a}\right)\frac{da}{a}$
(6)

is a non-zero complex constant independent of $\xi\in\mathbb{R}$ satisfying
$|\xi|=1.$ In case $\psi=\phi,$ we denote $C_{\psi,\psi,M}$ by $C_{\psi,M}$
and the required admissibility condition reduces to

$C_{\psi,M}:=\int_{\mathbb{R^{+}}}\left|(\mathcal{L}^{M}\psi)\left(\frac{\xi}{a}\right)\right|^{2}\frac{da}{a}$
(7)

is a positive constant independent of $\xi$ satisfying $|\xi|=1.$ We call
$\psi\in L^{2}(\mathbb{R}),$ satisfying equation (7), the admissible linear
canonical wavelet (ALCW).

We now give the definition of the novel LCWT.

######  Definition 3.2.

Let $f\in L^{2}(\mathbb{R}),$ $M=(A,B;C,D)$ be a matrix with $AD-
BC=1~{}\mbox{and}~{}B\neq 0$ then we define the LCWT of $f$ with respect to
$M$ and an ALCW $\psi$ by

$(W^{M}_{\psi}f)(a,b)=e^{-\frac{iA}{2B}b^{2}}\left\\{f(t)e^{\frac{iA}{2B}t^{2}}\star\overline{\sqrt{a}\psi(-at)e^{\frac{iA}{2B}(at)^{2}}}\right\\}(b),~{}a\in\mathbb{R^{+}},b\in\mathbb{R},$

where $\star$ denotes the convolution given by

$(f\star g)(\nu)=\int_{\mathbb{R}}f(x)g(\nu-x)dx,~{}\nu\in\mathbb{R}.$

Equivalently,

$\displaystyle\left(W^{M}_{\psi}f\right)(a,b)$ $\displaystyle=$ $\displaystyle
e^{-\frac{iA}{2B}b^{2}}\int_{\mathbb{R}}f(t)e^{\frac{iA}{2B}t^{2}}\overline{\sqrt{a}\psi(-a(b-t))e^{\frac{iA}{2B}(a(t-b))^{2}}}dt$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}f(t)\overline{e^{-\frac{iA}{2B}\\{(t^{2}-b^{2})-(a(t-b))^{2}\\}}\sqrt{a}\psi(a(t-b))}dt$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}f(t)\overline{\psi^{M}_{a,b}(t)}dt,$

where, with $\psi_{a,b}(t)=\sqrt{a}\psi(a(t-b))$

$\psi^{M}_{a,b}(t)=e^{-\frac{iA}{2B}\\{(t^{2}-b^{2})-(a(t-b))^{2}\\}}\psi_{a,b}(t).$
(8)

Thus, we have an equivalent definition of the LCWT as

$(W^{M}_{\psi}f)(a,b)=\langle f,\psi^{M}_{a,b}\rangle_{L^{2}(\mathbb{R})}.$
(9)

It is to be noted that depending on the different choice of the matrix $M,$ we
have different integral transforms:

  * •

For $M=(\cos\alpha,\sin\alpha;-\sin\alpha,\cos\alpha),\alpha\neq n\pi,$ we
obtain the FrWT as discussed in [12].

  * •

For $M=(0,1;-1,0)$ we obtain the traditional WT [32].

We now establish a fundamental relation between LCWT and the LCT. This
relation will be useful in obtaining the resolution of time and linear
canonical spectrum in the time-LCT-frequency plane and inner product relation
associated with the LCWT.

######  Proposition 3.1.

If $W_{\psi}^{M}f$ and $\mathcal{L}^{M}f$ are respectively the LCWT and the
LCT of $f\in L^{2}(\mathbb{R}).$ Then,

$\mathcal{L}^{M}\left((W_{\psi}^{M}f)(a,\cdot)\right)(\xi)=\frac{\sqrt{-2\pi
iB}}{\sqrt{a}}e^{\frac{iD}{2B}(\frac{\xi}{a})^{2}}(\mathcal{L}^{M}f)(\xi)\overline{(\mathcal{L}^{M}\psi)\left(\frac{\xi}{a}\right)}.$
(10)

###### Proof.

Form the definition of the LCT and $\psi_{a,b}^{M},$ it follows that

$\displaystyle\left(\mathcal{L}^{M}\psi_{a,b}^{M}\right)(\xi)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\sqrt{a}\psi(a(t-b))\sqrt{\frac{1}{2\pi
iB}}e^{\frac{i}{2}\left\\{\frac{Ab^{2}}{B}+\frac{A}{B}(a(t-b))^{2}-\frac{2}{B}\xi
t+\frac{D}{B}\xi^{2}\right\\}}dt$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\sqrt{a}\psi(at)\sqrt{\frac{1}{2\pi
iB}}e^{\frac{i}{2}\left\\{\frac{Ab^{2}}{B}+\frac{A}{B}(at)^{2}-\frac{2}{Ba}(at+ab)\xi+\frac{D}{B}\xi^{2}\right\\}}dt$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{a}}\int_{\mathbb{R}}\psi(t)\sqrt{\frac{1}{2\pi
iB}}e^{\frac{i}{2}\left(\frac{Ab^{2}}{B}-\frac{2}{B}\xi
b+\frac{D}{B}\xi^{2}\right)}e^{\frac{i}{2}\left(\frac{Ab^{2}}{B}-\frac{2}{B}t\left(\frac{\xi}{a}\right)+\frac{D}{B}\left(\frac{\xi}{a}\right)^{2}\right)}e^{\frac{-iD}{2B}\left(\frac{\xi}{a}\right)^{2}}dt$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{a}}e^{\frac{-iD}{2B}\left(\frac{\xi}{a}\right)^{2}}\sqrt{2\pi
iB}\int_{\mathbb{R}}\psi(t)K_{M}(b,\xi)K_{M}\left(t,\frac{\xi}{a}\right)dt.$

Therefore, we have

$\left(\mathcal{L}^{M}\psi_{a,b}^{M}\right)(\xi)=\frac{\sqrt{2\pi
iB}}{\sqrt{a}}K_{M}(b,\xi)\left(\mathcal{L}^{M}\psi\right)\left(\frac{\xi}{a}\right).$
(11)

Using (2) in (9), we get

$(W_{\psi}^{M}f)(a,b)=\langle\mathcal{L}^{M}f,\mathcal{L}^{M}\left(\psi_{a,b}^{M}\right)\rangle_{L^{2}(\mathbb{R})}.$

Using equation (11), we have

$(W_{\psi}^{M}f)(a,b)=\frac{\sqrt{-2\pi
iB}}{\sqrt{a}}\int_{\mathbb{R}}e^{\frac{-iD}{2B}\left(\frac{\xi}{a}\right)^{2}}\left(\mathcal{L}^{M}f\right)(\xi)\overline{\left(\mathcal{L}^{M}\psi\right)\left(\frac{\xi}{a}\right)}K_{M^{-1}}(b,\xi)d\xi.$
(12)

Therefore, it follows that

$\mathcal{L}^{M}\left((W_{\psi}^{M}f)(a,\cdot)\right)(\xi)=\frac{\sqrt{-2\pi
iB}}{\sqrt{a}}e^{\frac{iD}{2B}(\frac{\xi}{a})^{2}}(\mathcal{L}^{M}f)(\xi)\overline{(\mathcal{L}^{M}\psi)\left(\frac{\xi}{a}\right)}.$

This completes the proof. ∎

###  3.1 Time-LCT frequency analysis

From equation (9) it follows that if $\psi^{M}_{a,b}$ is supported in the time
domain, then so is $(W^{M}_{\psi}f)(a,b).$ Also, from equation (12), it
follows that the LCWT can provide the local property of $f(t)$ in the linear
canonical domain. Thus the LCWT is capable of producing simultaneously the
time-LCT frequency information and represent the signal in the time-LCT
frequency domain. More precisely, if $\psi$ and $\mathcal{L}^{M}\psi$ are
window functions in time and linear canonical domain respectively with
$E_{\psi}$ and $E_{\mathcal{L}^{M}\psi}$ as centres and $\Delta_{\psi}$ and
$\Delta_{\mathcal{L}^{M}\psi}$ are radii respectively. Then the centre and
radius of $\psi^{M}_{a,b}$ are given respectively by

$E[\psi^{M}_{a,b}]=\frac{1}{a}E_{\psi}+b,$

and

$\Delta[\psi^{M}_{a,b}]=\frac{1}{a}\Delta_{\psi}.$

Similarly, the centre and radius of window function
$(\mathcal{L}^{M}\psi)\left(\frac{\xi}{a}\right)$ are given by

$E\left[(\mathcal{L}^{M}\psi)\left(\frac{\xi}{a}\right)\right]=aE_{\mathcal{L^{M}\psi}},$

and

$\Delta\left[(\mathcal{L}^{M}\psi)\left(\frac{\xi}{a}\right)\right]=a\Delta_{\mathcal{L^{M}\psi}}.$

Thus, the $Q$-factor of the window function of the linear canonical transform
domain is

$Q=\frac{\Delta_{\mathcal{L^{M}\psi}}}{E_{\mathcal{L^{M}\psi}}},$

which is independent of the scaling parameter $a$ for a given parameter $M.$
This is called the constant $Q-$property of the LCWT.

###  3.2 Time-LCT frequency resolution

The LCWT $(W_{\psi}^{M}f)(a,b)$ localizes the signal $f$ in the time window

$\left[\frac{1}{a}E_{\psi}+b-\frac{1}{a}\Delta_{\psi},\frac{1}{a}E_{\psi}+b+\frac{1}{a}\Delta_{\psi}\right].$

Similarly, we get that the LCWT gives linear canonical spectrum content of $f$
in the window

$\left[aE_{\mathcal{L}^{M}\psi}-a\Delta_{\mathcal{L}^{M}\psi},aE_{\mathcal{L}^{M}\psi}+a\Delta_{\mathcal{L}^{M}\psi}\right].$

Thus, the joint resolution of the LCWT in the time and linear canonical domain
is given by the window

$\left[\frac{1}{a}E_{\psi}+b-\frac{1}{a}\Delta_{\psi},\frac{1}{a}E_{\psi}+b+\frac{1}{a}\Delta_{\psi}\right]\times\left[aE_{\mathcal{L}^{M}\psi}-a\Delta_{\mathcal{L}^{M}\psi},aE_{\mathcal{L}^{M}\psi}+a\Delta_{\mathcal{L}^{M}\psi}\right],$

with constant area $4\Delta_{\psi}\Delta_{\mathcal{L}^{M}\psi}$ in the time-
LCT-frequency plane. Thus it follows that for a given parameter $M,$ the
window area depends on the linear canonical admissible wavelets and is
independent of the parameters $a$ and $b.$ But it is to be noted that the the
window gets narrower for large value of $a$ and wider for small value of $a.$
Thus the window given by the transform is flexible and hence, it is capable of
simultaneously providing the time linear canonical domain information. This
flexibility of the window makes the proposed LCWT more advantageous then the
WLCT as in this case the window is rigid.

Some basic properties of LCWT is given below.

######  Theorem 3.1.

Let $g,h\in L^{2}(\mathbb{R}),$ $\psi$ and $\phi$ are ALCWs,
$\alpha,\beta\in\mathbb{C}$, $\lambda>0$ and $y\in\mathbb{R}.$ Then

  1. 1.

$W^{M}_{\psi}(\alpha g+\beta h)=\alpha(W^{M}_{\psi}g)+\beta(W^{M}_{\psi}h)$

  2. 2.

$W^{M}_{\alpha\psi+\beta\phi}(g)=\bar{\alpha}(W^{M}_{\psi}g)+\bar{\beta}(W^{M}_{\phi}g)$

  3. 3.

$(W^{M}_{\psi}\delta_{\lambda}g)(a,b)=(W^{\tilde{M}}_{\psi}g)(\frac{a}{\lambda},b\lambda),~{}\mbox{where}~{}(\delta_{\lambda}g)(t)=\sqrt{\lambda}g(\lambda
t)$ and $\tilde{M}=(A,{\lambda}^{2}B;\frac{C}{{\lambda}^{2}},D)$

  4. 4.

$(W^{M}_{\psi}\tau_{y}g)(a,b)=e^{\frac{iA}{B}y(y-b)}(W^{M}_{\psi}e^{\frac{iA}{B}yt}g)(a,b-y),~{}\mbox{where}~{}(\tau_{y}g)(t)=g(t-y)$.

###### Proof.

The proofs are immediate and can be omitted. ∎

If $\\{\psi,\phi\\}$ is admissible linear canonical wavelet pair such that
each $\phi$ and $\psi$ are ALCWs and $f,g\in L^{2}(\mathbb{R})$ and is such
that they are orthogonal then $W_{\psi}^{M}(f)$ and $W_{\phi}^{M}(g)$ are
orthogonal in $L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb).$ This result is
justified by the following theorem, which further gives the resolution of
identity for the LCWT.

######  Theorem 3.2.

(Inner product relation for LCWT). Let $\\{\psi,\phi\\}$ be an ALCWP such that
$\psi$ and $\phi$ are ALCWs and $f,g\in L^{2}(\mathbb{R}),$ then

$\langle
W^{M}_{\psi}f,W^{M}_{\phi}g\rangle_{L^{2}(\mathbb{R^{+}}\times\mathbb{R})}=2\pi|B|C_{\psi,\phi,M}\langle
f,g\rangle_{L^{2}(\mathbb{R})},$ (13)

where $C_{\psi,\phi,M}$ is provided in (6).

###### Proof.

Using equation (10), we get

$\displaystyle\langle
W^{M}_{\psi}f,W^{M}_{\phi}g\rangle_{L^{2}(\mathbb{R^{+}}\times\mathbb{R})}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R^{+}}\times\mathbb{R}}\left(W^{M}_{\psi}f\right)(a,b)\overline{\left(W^{M}_{\phi}g\right)(a,b)}dadb$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R^{+}}\times\mathbb{R}}\left(\mathcal{L}^{M}\left(W^{M}_{\psi}f\right)(a,\cdot)\right)(\xi)\overline{\left(\mathcal{L}^{M}\left(W^{M}_{\phi}g\right)(a,\cdot)\right)(\xi)}d\xi
da$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R^{+}}\times\mathbb{R}}\frac{2\pi|B|}{a}\left(\mathcal{L}^{M}f\right)(\xi)\overline{\left(\mathcal{L}^{M}\psi\right)\left(\frac{\xi}{a}\right)}\overline{\left(\mathcal{L}^{M}g\right)(\xi)}\left(\mathcal{L}^{M}\phi\right)\left(\frac{\xi}{a}\right)d\xi
da$ $\displaystyle=$ $\displaystyle
2\pi|B|\int_{\mathbb{R}}\left(\mathcal{L}^{M}f\right)(\xi)\overline{\left(\mathcal{L}^{M}g\right)(\xi)}\left\\{\int_{\mathbb{R}^{+}}\overline{\left(\mathcal{L}^{M}\psi\right)\left(\frac{\xi}{a}\right)}\left(\mathcal{L}^{M}\phi\right)\left(\frac{\xi}{a}\right)\frac{da}{a}\right\\}d\xi$
$\displaystyle=$ $\displaystyle
2\pi|B|C_{\psi,\phi,M}\left\langle\mathcal{L}^{M}f,\mathcal{L}^{M}g\right\rangle_{L^{2}(\mathbb{R})}$
$\displaystyle=$ $\displaystyle 2\pi|B|C_{\psi,\phi,M}\langle f,g\rangle.$

∎

######  Remark 3.1.

Replacing $\psi=\phi$ in equation (13), we have

$\langle
W^{M}_{\psi}f,W^{M}_{\psi}g\rangle_{L^{2}(\mathbb{R^{+}}\times\mathbb{R})}=2\pi|B|C_{\psi,M}\langle
f,g\rangle_{L^{2}(\mathbb{R})},$ (14)

where $C_{\psi,M}$ is provided in (7).

######  Remark 3.2.

(Plancherel’s theorem for $W^{M}_{\psi}$) Replacing $f=g$ and $\phi=\psi$ in
equation (13) we have the Plancherel’s theorem for $W_{\psi}^{M}$ given by

$\|W^{M}_{\psi}f\|_{L^{2}(\mathbb{R^{+}}\times\mathbb{R})}=\left(2\pi|B|C_{\psi,M}\right)^{\frac{1}{2}}\|f\|_{L^{2}(\mathbb{R})}.$
(15)

Thus, from equation (15), it follows that LCWT from $L^{2}(\mathbb{R})$ into
$L^{2}(\mathbb{R^{+}\times\mathbb{R}})$ is a continuous linear operator. If
further ALCW $\psi$ is such that $C_{\psi,M}=\frac{1}{2\pi|B|},$ then the
operator is an isometry.

######  Theorem 3.3.

(Reconstruction formula). Let $\\{\psi,\phi\\}$ be an ALCWP such that $\psi$
and $\phi$ are ALCWs and $f\in L^{2}(\mathbb{R})$, then $f$ can be given by
the formula

$f(t)=\frac{1}{2\pi|B|C_{\psi,\phi,M}}\int_{\mathbb{R^{+}}\times\mathbb{R}}(W^{M}_{\psi}f)(a,b)\phi^{M}_{a,b}(t)dadb~{}~{}\mbox{a.e.}~{}t\in\mathbb{R}.$
(16)

###### Proof.

From equation (13), we get

$\displaystyle 2\pi|B|C_{\psi,\phi,M}\langle f,g\rangle_{L^{2}(\mathbb{R})}$
$\displaystyle=$ $\displaystyle\langle
W^{M}_{\psi}f,W^{M}_{\phi}g\rangle_{L^{2}(\mathbb{R^{+}}\times\mathbb{R})}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{+}\times\mathbb{R}}\left(W_{\psi}^{M}f\right)(a,b)\overline{\left(\int_{\mathbb{R}}g(t)\overline{\phi_{a,b}^{M}(t)}dt\right)}dadb$
$\displaystyle=$
$\displaystyle\left\langle\int_{\mathbb{R}^{+}\times\mathbb{R}}\left(W_{\psi}^{M}f\right)(a,b)\phi_{a,b}^{M}(t)dadb,g(t)\right\rangle_{L^{2}(\mathbb{R})}.$

Since $g\in L^{2}(\mathbb{R})$ is arbitrary, we have

$f(t)=\frac{1}{2\pi|B|C_{\psi,\phi,M}}\int_{\mathbb{R}^{+}\times\mathbb{R}}\left(W_{\psi}^{M}f\right)(a,b)\phi_{a,b}^{M}(t)dadb~{}\mbox{a.e.}$

The proof is complete. ∎

In particular, if $\psi=\phi$ then we have the following reconstruction
formula

$f(t)=\frac{1}{2\pi|B|C_{\psi,M}}\int_{\mathbb{R}^{+}\times\mathbb{R}}\left(W_{\psi}^{M}f\right)(a,b)\psi_{a,b}^{M}(t)dadb~{}\mbox{a.e.}~{}t\in\mathbb{R}$

The following theorem characterizes the range of the LCWT and proves that the
range is a RKHS. It also gives the explicit expression for the reproducing
kernel.

######  Theorem 3.4.

For $\psi$ being ALCW, $W^{M}_{\psi}(L^{2}(\mathbb{R}))$ is a RKHS with the
kernel

$K^{M}_{\psi}(x,y;a,b)=\frac{1}{2\pi|B|C_{\psi,M}}\langle\psi^{M}_{a,b},\psi^{M}_{x,y}\rangle_{L^{2}(\mathbb{R})},(x,y),(a,b)\in\mathbb{R^{+}\times\mathbb{R}}.$

Moreover, the kernel is such that
$|K^{M}_{\psi}(x,y;a,b)|\leq\frac{1}{2\pi|B|C_{\psi,M}}\|\psi\|^{2}_{L^{2}(\mathbb{R})}.$

###### Proof.

For $(a,b)\in\mathbb{R}^{+}\times\mathbb{R},$ we see that

$K^{M}_{\psi}(x,y;a,b)=\frac{1}{2\pi|B|C_{\psi,M}}\left(W_{\psi}^{M}\psi_{a,b}^{M}\right)(x,y)~{}\mbox{for
all}~{}(x,y)\in\mathbb{R}^{+}\times\mathbb{R}.$

Now,

$\displaystyle\|K^{M}_{\psi}(\cdot,\cdot;a,b)\|^{2}_{L^{2}(\mathbb{R}^{+}\times\mathbb{R})}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi|B|C_{\psi,M}}\|W_{\psi}^{M}\psi_{a,b}^{M}\|_{L^{2}(\mathbb{R}^{+}\times\mathbb{R})}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi|B|C_{\psi,M}}\|\psi\|^{2}_{L^{2}(\mathbb{R})}.$

Therefore, for $(a,b)\in\mathbb{R}^{+}\times\mathbb{R},$
$K^{M}_{\psi}(x,y;a,b)\in L^{2}(\mathbb{R}^{+}\times\mathbb{R}).$ Now, let
$f\in L^{2}(\mathbb{R})$

$\displaystyle\left(W_{\psi}^{M}f\right)(a,b)$ $\displaystyle=$
$\displaystyle\langle f,\psi_{a,b}^{M}\rangle_{L^{2}(\mathbb{R})}$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi|B|C_{\psi,M}}\langle
W_{\psi}^{M}f,2\pi|B|C_{\psi,M}K_{\psi}^{M}(\cdot,\cdot;a,b)\rangle_{L^{2}(\mathbb{R}^{+}\times\mathbb{R})}$
$\displaystyle=$ $\displaystyle\langle
W_{\psi}^{M}f,K_{\psi}^{M}(\cdot,\cdot;a,b)\rangle_{L^{2}(\mathbb{R}^{+}\times\mathbb{R})}.$

Thus, it follows that

$K^{M}_{\psi}(x,y;a,b)=\frac{1}{2\pi|B|C_{\psi,M}}\langle\psi^{M}_{a,b},\psi^{M}_{x,y}\rangle_{L^{2}(\mathbb{R})},$

is the reproducing kernel of $W_{\psi}^{M}(L^{2}(\mathbb{R}).$

Again,

$\displaystyle|K^{M}_{\psi}(x,y;a,b)|$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi|B|C_{\psi,M}}|\langle\psi^{M}_{a,b},\psi^{M}_{x,y}\rangle_{L^{2}(\mathbb{R})}|$
$\displaystyle\leq$
$\displaystyle\frac{1}{2\pi|B|C_{\psi,M}}\|\psi_{a,b}^{M}\|_{L^{2}(\mathbb{R})}\|\|\psi_{x,y}^{M}\|_{L^{2}(\mathbb{R})}$
$\displaystyle=$
$\displaystyle\frac{\|\psi\|^{2}_{L^{2}(\mathbb{R})}}{2\pi|B|C_{\psi,M}}.$

This completes the proof. ∎

##  4 Uncertainty principle

We prove some UPs that limits the concentration of the LCWT in some subset in
$\mathbb{R}^{+}\times\mathbb{R}$ of small measure. For related results in case
of Fourier transform and windowed Fourier transform we refer the reader to
[33],[34]. Kou et al. [35] studied the same for the WLCT.

######  Definition 4.1.

Let $0\leq\epsilon<1,$ $f\in L^{2}(\mathbb{R})$ and $E\subset\mathbb{R}$ be
measurable, then $f$ is $\epsilon-$concentrated on $E$ if

$\left(\int_{E^{c}}|f(x)|^{2}dx\right)^{\frac{1}{2}}\leq\epsilon\|f\|_{L^{2}(\mathbb{R})}.$

If $0\leq\epsilon\leq\frac{1}{2},$ then we say that most of the energy of $f$
is concentrated on $E$ and $E$ is called the essential support of $f$. If
$\epsilon=0,$ then support of $f$ is contained in $E.$

######  Lemma 4.1.

If $\psi$ is an ALCW and $f\in L^{2}(\mathbb{R}).$ Then
$W^{M}_{\psi}f\in{L^{p}(\mathbb{R^{+}}\times\mathbb{R})},$ for all
$p\in[2,\infty].$ Moreover,

$\|W^{M}_{\psi}f\|_{L^{p}(\mathbb{R^{+}}\times\mathbb{R})}\leq(2\pi|B|)^{\frac{1}{p}}C_{\psi,M}^{\frac{1}{p}}\|f\|_{L^{2}(\mathbb{R})}\|\psi\|^{1-\frac{2}{p}}_{L^{2}(\mathbb{R})},~{}p\in[2,\infty)$
(17)
$\|W^{M}_{\psi}f\|_{L^{\infty}(\mathbb{R^{+}}\times\mathbb{R})}\leq\|\psi\|_{L^{2}(\mathbb{R})}\|f\|_{L^{2}(\mathbb{R})}.$
(18)

###### Proof.

Since $\psi$ is an ALCW, it follows that $W^{M}_{\psi}f\in
L^{2}(\mathbb{R^{+}}\times\mathbb{R}).$ Again

$\displaystyle\left|\left(W^{M}_{\psi}f\right)(a,b)\right|$
$\displaystyle\leq$
$\displaystyle\|\psi\|_{L^{2}(\mathbb{R})}\|f\|_{L^{2}(\mathbb{R})}.$

Thus, $W^{M}_{\psi}f\in L^{\infty}(\mathbb{R^{+}}\times\mathbb{R}).$ Also,
since $W^{M}_{\psi}f\in L^{2}(\mathbb{R^{+}}\times\mathbb{R}),$ we have
$W^{M}_{\psi}f\in L^{p}(\mathbb{R^{+}}\times\mathbb{R}),~{}p\in[2,\infty).$
Moreover,

$\displaystyle\|W^{M}_{\psi}f\|_{L^{p}(\mathbb{R^{+}}\times\mathbb{R})}$
$\displaystyle\leq$
$\displaystyle\|W^{M}_{\psi}f\|^{\frac{2}{p}}_{L^{2}(\mathbb{R^{+}}\times\mathbb{R})}\|W^{M}_{\psi}f\|_{L^{\infty}(\mathbb{R^{+}}\times\mathbb{R})}^{1-\frac{2}{p}}$
$\displaystyle\leq$
$\displaystyle(2\pi|B|C_{\psi,M})^{\frac{1}{p}}\|f\|^{\frac{2}{p}}_{L^{2}(\mathbb{R})}\|f\|^{1-\frac{2}{p}}_{L^{2}(\mathbb{R})}\|\psi\|^{1-\frac{2}{p}}_{L^{2}(\mathbb{R})}.$

This proves the lemma. ∎

######  Definition 4.2.

Let $0\leq\epsilon<1,$ $F\in L^{2}(\mathbb{R^{+}}\times\mathbb{R})$ and
$\Omega\subset\mathbb{R^{+}}\times\mathbb{R}$ be measurable, then $F$ is
$\epsilon-$concentrated on $\Omega$ if

$\left(\int_{\Omega^{c}}|F(x,y)|^{2}dxdy\right)^{\frac{1}{2}}\leq\epsilon\|F\|_{L^{2}(\mathbb{R^{+}}\times\mathbb{R})}.$

If $0\leq\epsilon\leq\frac{1}{2},$ then we say that most of the energy of $F$
is concentrated on $\Omega$ and $\Omega$ is called the essential support of
$F.$ If $\epsilon=0,$ then support of $F$ is contained in $\Omega.$

We now prove the Donoho-Stark’s UP for the propose LCWT.

######  Theorem 4.1.

Let $0\leq\epsilon<1,$ $\psi$ is an ALCW and a non-zero $f\in
L^{2}(\mathbb{R}).$ Also let $W^{M}_{\psi}f$ is $\epsilon-$concentrated on
$\Omega\subset\mathbb{R^{+}}\times\mathbb{R}$ then

$|\Omega|\|\psi\|^{2}_{L^{2}(\mathbb{R})}\geq
2\pi|B|C_{\psi,M}(1-\epsilon^{2}),$ (19)

where $|\Omega|$ denoted the measure of $\Omega.$

###### Proof.

In equation (15), we have

$\|W_{\psi}^{M}f\|^{2}_{L^{2}(\mathbb{R^{+}}\times\mathbb{R})}=2\pi|B|C_{\psi,M}\|f\|^{2}_{L^{2}(\mathbb{R})}.$

Now,

$\int_{\mathbb{R^{+}}\times\mathbb{R}}|\left(W^{M}_{\psi}f\right)(a,b)|^{2}dadb\leq\int_{\mathbb{R^{+}}\times\mathbb{R}}\chi_{\Omega}(a,b)|\left(W^{M}_{\psi}f\right)(a,b)|^{2}dadb+\epsilon^{2}\|W_{\psi}^{M}f\|_{L^{2}(\mathbb{R^{+}}\times\mathbb{R})}.$

This gives

$(1-\epsilon^{2})\|W^{M}_{\psi}f\|_{L^{2}(\mathbb{R^{+}}\times\mathbb{R})}\leq|\Omega|\|W^{M}_{\psi}f\|^{2}_{L^{\infty}(\mathbb{R^{+}}\times\mathbb{R})}.$

Thus, using (18), we get

$2\pi|B|C_{\psi,M}\|f\|^{2}_{L^{2}(\mathbb{R})}\leq|\Omega|\|f\|_{L^{2}(\mathbb{R})}\|\psi\|_{L^{2}(\mathbb{R})}.$

The result follows, since $f\neq 0.$ ∎

######  Corollary 4.1.

If $f\in L^{2}(\mathbb{R})\cap L^{4}(\mathbb{R}),$ in
$L^{2}(\mathbb{R})-$norm, is $\epsilon_{E}-$concentrated on
$E\subset\mathbb{R}$ and $W^{M}_{\psi}f$ is $\epsilon_{\Omega}-$concentrated
on $\Omega\subset\mathbb{R^{+}}\times\mathbb{R},$ then

$|\Omega|m(E)\|\psi\|^{2}_{L^{2}(\mathbb{R})}\|f\|^{4}_{L^{4}(\mathbb{R})}\geq
2\pi|B|C_{\psi,M}(1-\epsilon_{\Omega}^{2})(1-\epsilon_{E}^{2})^{2}\|f\|^{4}_{L^{2}(\mathbb{R})},$

where, $m(E)$ denotes the measure of $E.$

###### Proof.

Since, $W^{M}_{\psi}f$ is $\epsilon_{\Omega}-$concentrated on
$\Omega\subset\mathbb{R^{+}}\times\mathbb{R}$ in
$L^{2}(\mathbb{R^{+}}\times\mathbb{R})-$norm, so we have
$|\Omega|\|\psi\|^{2}_{L^{2}(\mathbb{R})}\geq
2\pi|B|C_{\psi,M}(1-\epsilon_{\Omega}^{2}).$ Again, since $f$ is
$\epsilon_{E}-$concentrated, we have

$\left(\int_{E^{c}}|f(x)|^{2}dx\right)^{\frac{1}{2}}\leq\epsilon_{E}\|f\|_{L^{2}(\mathbb{R})},$

which further implies that

$\|f\|_{L^{2}(\mathbb{R})}(1-\epsilon_{E}^{2})\leq\int_{\mathbb{R}}\chi_{E}(x)|f(x)|^{2}dx.$

We have by Holder’s inequality

$\int_{\mathbb{R}}\chi_{E}(x)|f(x)|^{2}dx\leq\left(\int_{\mathbb{R}}|\chi_{E}(x)|^{2}dx\right)^{\frac{1}{2}}\|f\|^{2}_{L^{4}(\mathbb{R})}.$

Thus

$(1-\epsilon_{E}^{2})\|f\|^{2}_{L^{2}(\mathbb{R})}\leq(m(E))^{\frac{1}{2}}\|f\|_{L^{4}(\mathbb{R})}.$
(20)

Therefore,

$\displaystyle|\Omega|m(E)\|\psi\|^{2}_{L^{2}(\mathbb{R})}\|f\|^{4}_{L^{4}(\mathbb{R})}\geq
2\pi|B|C_{\psi,M}(1-\epsilon_{\Omega}^{2})(1-\epsilon_{E}^{2})^{2}\|f\|^{4}_{L^{2}(\mathbb{R})}.$

The proof is complete. ∎

######  Corollary 4.2.

If $f\in L^{2}(\mathbb{R})\cap L^{\infty}(\mathbb{R}),$ in
$L^{2}(\mathbb{R})-$norm, is $\epsilon_{E}-$concentrated on
$E\subset\mathbb{R}$ and $W^{M}_{\psi}f$ is $\epsilon_{\Omega}-$concentrated
on $\Omega\subset\mathbb{R^{+}}\times\mathbb{R},$ then

$|\Omega|m(E)\|\psi\|^{2}_{L^{2}(\mathbb{R})}\|f\|^{2}_{L^{\infty}(\mathbb{R})}\geq
2\pi|B|C_{\psi,M}(1-\epsilon_{\Omega}^{2})(1-\epsilon_{E}^{2})\|f\|^{2}_{L^{2}(\mathbb{R})}.$

###### Proof.

Since, $W^{M}_{\psi}f$ is $\epsilon_{\Omega}-$concentrated on
$\Omega\subset\mathbb{R^{+}}\times\mathbb{R}$ in
$L^{2}(\mathbb{R^{+}}\times\mathbb{R})-$norm, so we have
$|\Omega|\|\psi\|^{2}_{L^{2}(\mathbb{R})}\geq
2\pi|B|C_{\psi,M}(1-\epsilon_{\Omega}^{2}).$ Again, since $f$ is
$\epsilon_{E}-$concentrated, we have

$\left(\int_{E^{c}}|f(x)|^{2}dx\right)^{\frac{1}{2}}\leq\epsilon_{E}\|f\|_{L^{2}(\mathbb{R})},$

which further implies that

$\|f\|_{L^{2}(\mathbb{R})}(1-\epsilon_{E}^{2})\leq\int_{\mathbb{R}}|f(x)|^{2}\chi_{E}(x)dx.$

Since $f\in L^{\infty}(\mathbb{R}),$ so

$\displaystyle\int_{\mathbb{R}}\chi_{E}(x)|f(x)|^{2}dx$ $\displaystyle\leq$
$\displaystyle m(E)\|f\|^{2}_{L^{\infty}(\mathbb{R})}.$

Thus

$\|f\|^{2}_{L^{\infty}(\mathbb{R})}m(E)\geq(1-\epsilon_{E}^{2})\|f\|^{2}_{L^{2}{(\mathbb{R}})}.$
(21)

Therefore,

$|\Omega|m(E)\|\psi\|^{2}_{L^{2}(\mathbb{R})}\|f\|^{2}_{L^{\infty}(\mathbb{R})}\geq
2\pi|B|C_{\psi,M}(1-\epsilon_{\Omega}^{2})(1-\epsilon_{E}^{2})\|f\|^{2}_{L^{2}(\mathbb{R})}.$

The proof is complete. ∎

######  Theorem 4.2.

(Lieb’s uncertainty principle). Let $0\leq\epsilon<1,$ $\psi$ is an ALCW and a
non-zero$f\in L^{2}(\mathbb{R}).$ Also let $W^{M}_{\psi}f$ is
$\epsilon-$concentrated on $\Omega\subset\mathbb{R^{+}}\times\mathbb{R},$ then

$|\Omega|\|\psi\|^{2}_{L^{2}(\mathbb{R})}\geq
2\pi|B|C_{\psi,M}(1-\epsilon^{2})^{\frac{p}{p-2}},~{}p>2.$ (22)

###### Proof.

Since $W_{\psi}^{M}f$ is $\epsilon-$concentrated on $\Omega,$ we have

$\|\chi_{{\Omega}^{c}}W_{\psi}^{M}f\|_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}\leq\epsilon^{2}\|W_{\psi}^{M}f\|_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}.$

Now,

$\displaystyle\|\chi_{{\Omega}^{c}}W_{\psi}^{M}f\|_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}$
$\displaystyle\leq$
$\displaystyle\|\chi_{\Omega}W_{\psi}^{M}f\|_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}+\epsilon^{2}\|W_{\psi}^{M}f\|_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}.$

This implies,

$2\pi
C_{\psi,M}\|f\|_{L^{2}(\mathbb{R})}(1-\epsilon^{2})\leq\int_{\mathbb{R}^{+}\times\mathbb{R}}\chi_{\Omega}(a,b)|(W_{\psi}^{M}f)(a,b)|^{2}dadb.$

By Holder’s inequality, we get

$\displaystyle 2\pi C_{\psi,M}\|f\|_{L^{2}(\mathbb{R})}(1-\epsilon^{2})$
$\displaystyle\leq$
$\displaystyle\left(\int_{\mathbb{R}^{+}\times\mathbb{R}}(\chi_{\Omega}(a,b))^{\frac{p}{p-2}}dadb\right)^{\frac{p-2}{p}}\left(\int_{\mathbb{R}^{+}\times\mathbb{R}}|(W_{\psi}^{M}f)(a,b)|^{p}dadb\right)^{\frac{2}{p}}$
$\displaystyle\leq$
$\displaystyle|\Omega|^{\frac{p-2}{p}}\|W^{M}_{\psi}f\|^{2}_{L^{p}(\mathbb{R}^{+}\times\mathbb{R},dadb)}.$

Using equation (17), we get

$(2\pi|B|)^{1-\frac{2}{p}}C_{\psi,M}^{1-\frac{2}{p}}(1-\epsilon^{2})\leq|\Omega|^{\frac{p-2}{p}}\|\psi\|^{2-\frac{4}{p}}_{L^{2}(\mathbb{R})}.$

Therefore, we get

$|\Omega|\|\psi\|^{2}_{L^{2}(\mathbb{R})}\geq
2\pi|B|C_{\psi,M}(1-\epsilon^{2})^{\frac{p}{p-2}}.$

The proof is complete. ∎

######  Corollary 4.3.

If $f\in L^{2}(\mathbb{R})\cap L^{4}(\mathbb{R}),$ in
$L^{2}(\mathbb{R})-$norm, is $\epsilon_{E}-$ concentrated on
$E\subset\mathbb{R}$ and $W^{M}_{\psi}f$ is $\epsilon_{\Omega}-$concentrated
on $\Omega\subset\mathbb{R^{+}}\times\mathbb{R},$ then

$|\Omega|m(E)\|\psi\|^{2}_{L^{2}(\mathbb{R})}\|f\|^{4}_{L^{4}(\mathbb{R})}\geq
2\pi|B|C_{\psi,M}(1-\epsilon_{\Omega}^{2})^{\frac{p}{p-2}}(1-\epsilon_{E}^{2})^{2}\|f\|^{4}_{L^{2}(\mathbb{R})},~{}p>2.$

###### Proof.

The proof follows similarly as theorem 4.1, we use Lieb’s UP instead of the
Donoho-Stark’s UP. ∎

######  Corollary 4.4.

If$f\in L^{2}(\mathbb{R})\cap L^{\infty}(\mathbb{R}),$ in
$L^{2}(\mathbb{R})-$norm, is $\epsilon_{E}-$ concentrated on
$E\subset\mathbb{R}$ and $W^{M}_{\psi}f$ is $\epsilon_{\Omega}-$concentrated
on $\Omega\subset\mathbb{R^{+}}\times\mathbb{R},$ then

$|\Omega|m(E)\|\psi\|^{2}_{L^{2}(\mathbb{R})}\|f\|^{2}_{L^{\infty}(\mathbb{R})}\geq
2\pi|B|C_{\psi,M}(1-\epsilon_{\Omega}^{2})^{\frac{p}{p-2}}(1-\epsilon_{E}^{2})\|f\|^{2}_{L^{2}(\mathbb{R})},~{}p>2.$

###### Proof.

The proof follows similarly as theorem 4.2, we use Lieb’s UP instead of the
Donoho-Stark’s UP. ∎

##  5 Orthonormal sequences and uncertainty principle

We now express the UP in term of the generalized dispersion of $W_{\psi}^{M},$
which is defined by

$\rho_{p}\left(W_{\psi}^{M}f\right)=\left(\int_{\mathbb{R}^{+}\times\mathbb{R}}|(a,b)|^{p}|(W_{\psi}^{M}f)(a,b)|^{2}dadb\right)^{\frac{1}{p}},$
(23)

where $|(a,b)|=\sqrt{a^{2}+b^{2}},~{}p>0.$

######  Definition 5.1.

Let $T$ be a bounded linear operator on a Hilbert space $\mathbb{H}$ over the
field $\mathbb{F}$ (where $\mathbb{F}$ is $\mathbb{R}$ or $\mathbb{C}$) and
$\\{u_{n}\\}_{n\in\mathbb{N}}$ be an orthonormal basis of $\mathbb{H},$ then
$T$ is called a Hilbert-Schmidt operator if

$\|T\|_{HS}=\left(\sum_{n=1}^{\infty}\|Tu_{n}\|^{2}\right)^{\frac{1}{2}}<\infty.$

Before discussing the main result of this section, we estimate the Hilbert-
Schmidt norm of the product of some orthogonal projection operators and use it
to estimate the concentration of $W_{\psi}^{M}$ on subset of
$\mathbb{R}^{+}\times\mathbb{R}.$ Similar results were first studied by
Wilczok [27] in the case of windowed FT and WT.

######  Theorem 5.1.

Let $f\in L^{2}(\mathbb{R}),$ $\psi$ is an ALCW and
$\Omega\subset\mathbb{R}^{+}\times\mathbb{R}$ such that
$|\Omega|<\frac{2\pi|B|C_{\psi,M}}{\|\psi\|^{2}_{L^{2}(\mathbb{R})}}.$ Then

$\|\chi_{\Omega^{c}}W_{\psi}^{M}f\|_{L^{2}(\mathbb{R}^{+}\times\mathbb{R})}\geq\sqrt{2\pi|B|C_{\psi,M}-|\Omega|\|\psi\|^{2}_{L^{2}(\mathbb{R})}}\|f\|_{L^{2}(\mathbb{R})}.$

###### Proof.

We consider the orthogonal projections $P_{\psi}$ from
$L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)$ on the RKHS
$W_{\psi}^{M}(L^{2}(\mathbb{R}))$ and $P_{\Omega}$ on
$L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)$ defined by
$P_{\Omega}F=\chi_{\Omega}F,~{}\mbox{for all}~{}F\in
L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb).$ According to Saitoh [36], for
every $(a,b)\in\mathbb{R}^{+}\times\mathbb{R}$ and $F\in
L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb),$ we get

$\displaystyle\left(P_{\Omega}P_{\psi}F\right)(a,b)$ $\displaystyle=$
$\displaystyle\chi_{\Omega}(a,b)\left\langle
F,K_{\psi}^{M}\left(\cdot,\cdot;a,b\right)\right\rangle_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{+}\times\mathbb{R}}\chi_{\Omega}(a,b)F(x,y)\overline{K_{\psi}^{M}(x,y;a,b)}dxdy.$

Thus the integral operator $P_{\Omega}P_{\psi}$ has the kernel
$\mathcal{N}_{\psi,\Omega}^{M}$ defined on
$(\mathbb{R}^{+}\times\mathbb{R})^{2}$ by

$\mathcal{N}_{\psi,\Omega}^{M}(x,y;a,b)=F(x,y)\overline{K_{\psi}^{M}(x,y;a,b)}$

such that

$\displaystyle\int_{\mathbb{R}^{+}\times\mathbb{R}}\int_{\mathbb{R}^{+}\times\mathbb{R}}$
$\displaystyle|\mathcal{N}_{\psi,\Omega}^{M}(x,y;a,b)|^{2}dxdydadb$
$\displaystyle=\int_{\mathbb{R}^{+}\times\mathbb{R}}\left(\int_{\mathbb{R}^{+}\times\mathbb{R}}|K_{\psi}^{M}(x,y;a,b)|^{2}dxdy\right)|\chi_{\Omega}(a,b)|^{2}dadb$
$\displaystyle=\int_{\mathbb{R}^{+}\times\mathbb{R}}\chi_{\Omega}(a,b)\|K_{\psi}^{M}(\cdot,\cdot;a,b)\|^{2}_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}dadb$
$\displaystyle=\frac{|\Omega|\|\psi\|^{2}_{L^{2}(\mathbb{R})}}{2\pi|B|C_{\psi,M}}.$

Now,

$\chi_{\Omega}W_{\psi}^{M}f=P_{\Omega}P_{\psi}(W_{\psi}^{M}f)$

implies

$\|\chi_{\Omega}W_{\psi}^{M}f\|^{2}_{L^{2}(\mathbb{R}^{+}\times\mathbb{R})}\leq\|P_{\Omega}P_{\psi}\|^{2}\|W_{\psi}^{M}f\|^{2}_{L^{2}(\mathbb{R}^{+}\times\mathbb{R})}.$

Therefore

$\|W_{\psi}^{M}f\|^{2}_{L^{2}(\mathbb{R}^{+}\times\mathbb{R})}\leq\|P_{\Omega}P_{\psi}\|^{2}\|W_{\psi}^{M}f\|^{2}_{L^{2}(\mathbb{R}^{+}\times\mathbb{R})}+\|\chi_{\Omega^{c}}W_{\psi}^{M}f\|^{2}_{L^{2}(\mathbb{R}^{+}\times\mathbb{R})}$
$\mbox{i.e.,}~{}\|\chi_{\Omega^{c}}W_{\psi}^{M}f\|^{2}_{L^{2}(\mathbb{R}^{+}\times\mathbb{R})}\geq\left(1-\|P_{\Omega}P_{\psi}\|^{2}\right)2\pi|B|C_{\psi,M}\|f\|^{2}_{L^{2}(\mathbb{R})}.$

Now using the fact that
$\|P_{\Omega}P_{\psi}\|\leq\|P_{\Omega}P_{\psi}\|_{HS},$ where $\|\cdot\|$
denotes the operator norm, we obtain

$\displaystyle\|\chi_{\Omega^{c}}W_{\psi}^{M}f\|^{2}_{L^{2}(\mathbb{R}^{+}\times\mathbb{R})}\geq\left(1-\|P_{\Omega}P_{\psi}\|_{HS}^{2}\right)2\pi|B|C_{\psi,M}\|f\|^{2}_{L^{2}(\mathbb{R})}.$

Hence, we obtain

$\|\chi_{\Omega^{c}}W_{\psi}^{M}f\|_{L^{2}(\mathbb{R}^{+}\times\mathbb{R})}\geq\sqrt{2\pi|B|C_{\psi,M}-|\Omega|\|\psi\|^{2}_{L^{2}(\mathbb{R})}}\|f\|_{L^{2}(\mathbb{R})}.$

This proves the theorem. ∎

######  Theorem 5.2.

If $\psi$ is an ALCW, $\\{\phi_{n}\\}_{n\in\mathbb{N}}\subset
L^{2}(\mathbb{R})$ be an ONS and $\Omega\subset\mathbb{R}^{+}\times\mathbb{R}$
be such that its measure $|\Omega|<\infty,$ then for any non-empty
$\wedge\subset\mathbb{N},$

$\sum_{n\in\wedge}\left(1-\left\|\chi_{\Omega^{c}}W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right\|_{L^{2}(\mathbb{R^{+}}\times\mathbb{R},dadb)}\right)\leq\frac{|\Omega|\|\psi\|_{L^{2}(\mathbb{R})}^{2}}{2\pi|B|C_{\psi,M}}.$
(24)

###### Proof.

Consider the orthonormal basis $\\{h_{n}\\}_{n\in\mathbb{N}}$ of
$L^{2}(\mathbb{R^{+}}\times\mathbb{R},dadb).$ It has been proved in the above
theorem that $P_{\Omega}P_{\psi}$ is a Hilbert-Schmidt operator such that
$\|P_{\Omega}P_{\psi}\|^{2}_{HS}=\frac{|\Omega|\|\psi\|_{L^{2}(\mathbb{R})}^{2}}{2\pi|B|C_{\psi,M}}.$
Since $P_{\Omega}^{2}=P_{\Omega}$ and both $P_{\psi},~{}P_{\Omega}$ are self-
adjoint, the operator
$T=(P_{\Omega}P_{\psi})^{\star}(P_{\Omega}P_{\psi})=P_{\psi}P_{\Omega}P_{\psi}$
is positive and is such that

$\displaystyle\sum_{n\in\mathbb{N}}\langle
Th_{n},h_{n}\rangle_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}$
$\displaystyle=$ $\displaystyle\sum_{n\in\mathbb{N}}\langle
P_{\Omega}P_{\psi}h_{n},P_{\Omega}P_{\psi}h_{n}\rangle_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}$
$\displaystyle=$
$\displaystyle\sum_{n\in\mathbb{N}}\|P_{\Omega}P_{\psi}h_{n}\|^{2}_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}$
$\displaystyle=$ $\displaystyle\|P_{\Omega}P_{\psi}\|^{2}_{HS}$
$\displaystyle=$
$\displaystyle\frac{|\Omega|\|\psi\|_{L^{2}(\mathbb{R})}^{2}}{2\pi|B|C_{\psi,M}}<\infty.$

Therefore, $T$ is a trace class operator with
$Tr(T)=\frac{|\Omega|\|\psi\|_{L^{2}(\mathbb{R})}^{2}}{2\pi|B|C_{\psi,M}}.$
Now as $\\{\phi_{n}\\}_{n\in\mathbb{N}}$ is an ONS, from equation (14), it
follows that
$\left\\{W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right\\}_{n\in\mathbb{N}}$
is an ONS in $L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb).$
Hence, we have

$\displaystyle\sum_{n\in\wedge}$ $\displaystyle\left\langle
P_{\Omega}W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right),W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right\rangle_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}$
$\displaystyle=\sum_{n\in\wedge}\left\langle
P_{\psi}P_{\Omega}P_{\psi}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right),W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right\rangle_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}$
$\displaystyle=\sum_{n\in\wedge}\left\langle
T\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right),W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right\rangle_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}$
$\displaystyle\leq\sum_{n\in\mathbb{N}}\left\langle
T\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right),W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right\rangle_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}$
$\displaystyle=Tr(T)=\frac{|\Omega|\|\psi\|_{L^{2}(\mathbb{R})}^{2}}{2\pi|B|C_{\psi,M}}.$

For each $n\in\wedge,$ we have

$\displaystyle\left\langle
P_{\Omega}W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right),W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right\rangle_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}$
$\displaystyle=\left\langle\chi_{\Omega}W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right),W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right\rangle_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}$
$\displaystyle=1-\left\langle\chi_{\Omega^{c}}W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right),W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right\rangle_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}$
$\displaystyle\geq
1-\left\|\chi_{\Omega^{c}}W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right\|_{L^{2}(\mathbb{R}^{+}\times\mathbb{R},dadb)}.$

Thus, we have

$\sum_{n\in\wedge}\left(1-\left\|\chi_{\Omega^{c}}W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right\|_{L^{2}(\mathbb{R^{+}}\times\mathbb{R},dadb)}\right)\leq\frac{|\Omega|\|\psi\|_{L^{2}(\mathbb{R})}^{2}}{2\pi|B|C_{\psi,M}}.$

This proves the theorem . ∎

The theorem below shows that, if the LCWT of each member of an ONS are
$\epsilon-$concentrated in a set of finite measure then the sequence is
necessarily finite. The theorem also gives an upper bound of the cardinality
of the so proved finite sequence.

######  Theorem 5.3.

Let $s,\epsilon>0$ such that $\epsilon<1.$ Let
$G_{s}=\\{(a,b)\in\mathbb{R}^{+}\times\mathbb{R}:a^{2}+b^{2}\leq s^{2}\\}$ and
$\psi$ is an ALCW. Also let $\wedge\subset\mathbb{N}$ be non-empty and
$\\{\phi_{n}\\}_{n\in\wedge}\subset L^{2}(\mathbb{R})$ be an ONS. If
$W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)$ is
$\epsilon-$concentrated in $G_{s}$ for all $n\in\wedge,$ then $\wedge$ is
finite and

$Card(\wedge)\leq\frac{s^{2}\|\psi\|^{2}_{L^{2}(\mathbb{R})}}{4|B|C_{\psi,M}(1-\epsilon)},$
(25)

where $Card(\wedge)$ denotes the cardinality of $\wedge.$

###### Proof.

Applying above theorem, we have

$\sum_{n\in\wedge}\left(1-\left\|\chi_{G_{s}^{c}}W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right\|_{L^{2}(\mathbb{R^{+}}\times\mathbb{R},dadb)}\right)\leq\frac{|G_{s}|\|\psi\|_{L^{2}(\mathbb{R})}^{2}}{2\pi|B|C_{\psi,M}}.$

Again, since for each
$W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)$ is
$\epsilon-$concentrated in $G_{s},$ we have

$\left\|\chi_{G_{s}^{c}}W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right\|_{L^{2}(\mathbb{R}^{+}\times\mathbb{R})}\leq\epsilon.$

Therefore, it follows that

$\sum_{n\in\wedge}(1-\epsilon)\leq\frac{|G_{s}|\|\psi\|_{L^{2}(\mathbb{R})}^{2}}{2\pi|B|C_{\psi,M}}$
$\mbox{i.e.,}~{}Card(\wedge)(1-\epsilon)\leq\frac{|G_{s}|\|\psi\|_{L^{2}(\mathbb{R})}^{2}}{2\pi|B|C_{\psi,M}}.$

Thus $Card(\wedge)$ is finite and using $|G_{s}|=\frac{\pi s^{2}}{2},$ we
obtain

$Card(\wedge)\leq\frac{s^{2}\|\psi\|_{L^{2}(\mathbb{R})}^{2}}{4|B|(1-\epsilon)C_{\psi,M}}.$

The proof is complete. ∎

######  Corollary 5.1.

Let $p>0,$ $R>0$ and $\psi$ is an ALCW. Also let $\wedge\subset\mathbb{N},$ be
non-empty and $\\{\phi_{n}\\}_{n\in\wedge}\subset L^{2}(\mathbb{R})$ be an
ONS. Then $\wedge$ is finite if
$\left\\{\rho_{p}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)\right\\}_{n\in\wedge}$
is uniformly bounded. Moreover, if it is uniformly bounded by $R,$ then

$Card(\wedge)\leq\frac{2^{\frac{4}{p}+1}R^{2}\|\psi\|^{2}_{L^{2}(\mathbb{R})}}{4|B|C_{\psi,M}}.$

###### Proof.

Since for each $n\in\wedge,$
$\rho_{p}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)\leq
R,$ and thus

$\displaystyle\int_{|(a,b)|\geq
R2^{\frac{2}{p}}}\left|\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)(a,b)\right|^{2}dadb$
$\displaystyle=\int_{|(a,b)|\geq
R2^{\frac{2}{p}}}|(a,b)|^{-p}|(a,b)|^{p}\left|\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)(a,b)\right|^{2}dadb$
$\displaystyle\leq\frac{1}{\left(R2^{\frac{2}{p}}\right)^{p}}\int_{\mathbb{R}^{+}\times\mathbb{R}}|(a,b)|^{p}\left|\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)(a,b)\right|^{2}dadb$
$\displaystyle\leq\frac{1}{4}.$

Thus it follows that, for each $n\in\wedge,$
$W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)$ is
$\frac{1}{2}-$concentrated in

$G_{R2^{\frac{2}{p}}}=\left\\{(a,b)\in\mathbb{R}^{+}\times\mathbb{R}:|(a,b)|<R2^{\frac{2}{p}}\right\\}.$

Thus, from theorem 5.3, it follows that $\wedge$ is finite and

$Card(\wedge)\leq\frac{\left(R2^{\frac{2}{p}}\right)^{2}\|\psi\|^{2}_{L^{2}(\mathbb{R})}}{4|B|\left(1-\frac{1}{2}\right)C_{\psi,M}},$
$\mbox{i.e.,}~{}Card(\wedge)\leq\frac{2^{\frac{4}{p}+1}R^{2}\|\psi\|^{2}_{L^{2}(\mathbb{R})}}{4|B|C_{\psi,M}}.$

Thus the proof is complete. ∎

######  Lemma 5.1.

Let $p>0,$ $\psi$ is an ALCW and $\\{\phi_{n}\\}_{n\in\mathbb{N}}\subset
L^{2}(\mathbb{R})$ be an ONS, then $\exists$ $m_{0}\in\mathbb{Z}$ for which

$\rho_{p}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)\geq
2^{m_{0}},~{}\forall~{}n\in\mathbb{N}.$

###### Proof.

Define
$P_{m}=\left\\{n\in\mathbb{N}:\rho_{p}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)\in[2^{m-1},2^{m})\right\\},$
for each $m\in\mathbb{Z}.$
Then for each $n\in P_{m},$ we get

$\int_{\mathbb{R}^{+}\times\mathbb{R}}|(a,b)|^{p}\left|\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)(a,b)\right|^{2}dadb<2^{mp}.$

Now,

$\displaystyle\int_{|(a,b)|\geq
2^{m+\frac{2}{p}}}\left|\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)(a,b)\right|^{2}dadb$
$\displaystyle\leq\frac{1}{2^{mp+2}}\int_{\mathbb{R}^{+}\times\mathbb{R}}|(a,b)|^{p}\left|\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)(a,b)\right|^{2}dadb$
$\displaystyle\leq\frac{1}{2^{mp+2}}\left\\{\rho_{p}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)\right\\}^{p}.$

This gives

$\int_{|(a,b)|\geq
2^{\frac{2}{p}+m}}\left|\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)(a,b)\right|^{2}dadb\leq\frac{1}{4}.$

Thus it follows that, for each $n\in P_{m},$
$W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)$ is
$\frac{1}{2}-$concentrated on the set

$G_{2^{m+\frac{2}{p}}}=\left\\{(a,b)\in\mathbb{R}^{+}\times\mathbb{R}:|(a,b)|<2^{m+\frac{2}{p}}\right\\}.$

Therefore, $P_{m}$ is finite and

$Card(P_{m})\leq\frac{2^{2m+\frac{4}{p}+1}\|\psi\|^{2}_{L^{2}(\mathbb{R})}}{4|B|C_{\psi,M}},~{}\mbox{for
all}~{}m\in\mathbb{Z}.$

Letting $m\to-\infty,$ we get

$\lim_{m\to-\infty}Card(P_{m})=0.$

Hence $\exists$ $m_{0}\in\mathbb{Z}$ such that for all $m<m_{0},$ $P_{m}$ are
empty sets. Therefore,
$\rho_{p}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)\geq
2^{m_{0}},~{}\mbox{for all}~{}n\in\mathbb{N}.$ ∎

######  Theorem 5.4.

(Shapiro’s Dispersion theorem). Let $\psi$ be an ALCW and
$\\{\phi_{n}\\}_{n\in\mathbb{N}}\subset L^{2}(\mathbb{R})$ be an ONS, then for
every $p>0$ and non-empty finite $\wedge\subset\mathbb{N},$

$\sum_{n\in\wedge}\left\\{\rho_{p}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)\right\\}^{p}\geq\frac{(Card(\wedge))^{\frac{p}{2}+1}}{2^{p+1}}\left(\frac{3|B|C_{\psi,M}}{2^{\frac{4}{p}+2}\|\psi\|^{2}_{L^{2}(\mathbb{R})}}\right)^{\frac{p}{2}}.$

###### Proof.

Let $m_{0}$ be an integer defined in the above lemma. Let $k\in\mathbb{Z}$
such that $k\geq m_{0}.$ Define $Q_{k}=\bigcup\limits_{m=m_{0}}^{k}P_{m}.$
Then we have

$\displaystyle Card(Q_{k})=\sum_{m=m_{0}}^{k}Card(P_{m})$ $\displaystyle\leq$
$\displaystyle\sum_{m=m_{0}}^{k}\frac{2^{2m+\frac{4}{p}+1}\|\psi\|^{2}_{L^{2}(\mathbb{R})}}{4|B|C_{\psi,M}}$
$\displaystyle=$
$\displaystyle\frac{2^{\frac{4}{p}+1}\|\psi\|^{2}_{L^{2}(\mathbb{R})}}{4|B|C_{\psi,M}}\sum_{m=m_{0}}^{k}2^{2m}$
$\displaystyle\leq$
$\displaystyle\frac{2^{\frac{4}{p}+1}\|\psi\|^{2}_{L^{2}(\mathbb{R})}}{4|B|C_{\psi,M}}\frac{2^{2k+2}}{3}$
$\mbox{i.e.,}~{}Card(Q_{k})\leq\frac{2^{\frac{4}{p}+1}\|\psi\|^{2}_{L^{2}(\mathbb{R})}}{3|B|C_{\psi,M}}2^{2k}.$

Let
$C=\frac{2^{\frac{4}{p}+2}\|\psi\|^{2}_{L^{2}(\mathbb{R})}}{3|B|C_{\psi,M}}.$
Then $Card(Q_{k})\leq\frac{C}{2}2^{2k}.$ If $Card(\wedge)>2^{2(m_{0}+1)},$
then $\frac{1}{2\log{2}}\log\left(\frac{Card(\wedge)}{C}\right)>m_{0}+1.$
Let us choose an integer $k>m_{0}+1$ such that

$k-1<\frac{1}{2\log{2}}\log(\frac{Card(\wedge)}{C})\leq k.$

Then it results in

$C2^{2(k-1)}<Card(\wedge)\leq C2^{2k}.$

Thus, we have

$Card(Q_{k-1})=\frac{C}{2}2^{2(k-1)}<\frac{Card(\wedge)}{2}.$

Therefore,

$\displaystyle\sum_{n\in\wedge}\left\\{\rho_{p}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)\right\\}^{p}$
$\displaystyle\geq$ $\displaystyle\sum_{n\not\in
Q_{k-1}}\left\\{\rho_{p}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)\right\\}^{p}$
$\displaystyle\geq$ $\displaystyle\frac{Card(\wedge)}{2}2^{(k-1)p}$
$\displaystyle=$ $\displaystyle\frac{Card(\wedge)}{2.2^{p}}2^{kp}.$

Since, $Card(\wedge)\leq C2^{2k},$ we have
$\left(\frac{Card(\wedge)}{C}\right)^{\frac{p}{2}}\leq 2^{kp}.$
Therefore,

$\sum_{n\in\wedge}\left\\{\rho_{p}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)\right\\}^{p}\geq\frac{(Card(\wedge))^{\frac{p}{2}+1}}{2^{p+1}}\left(\frac{1}{C}\right)^{\frac{p}{2}}.$

Again, if $Card(\wedge)\leq 2^{2(m_{0}+1)},$ then

$\sum_{n\in\wedge}\left\\{\rho_{p}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)\right\\}^{p}\geq
Card(\wedge)2^{m_{0}p},~{}\mbox{(using lemma \ref{P3lemma5.2})}.$

Now, $Card(\wedge)\leq C2^{2(m_{0}+1)}$ implies
$\frac{1}{2^{p}}\left(\frac{Card(\wedge}{C})\right)^{\frac{p}{2}}\leq
2^{m_{0}p}.$ Thus we have

$\sum_{n\in\wedge}\left\\{\rho_{p}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)\right\\}^{p}\geq\frac{(Card(\wedge))^{\frac{p}{2}+1}}{2^{p}}\left(\frac{1}{C}\right)^{\frac{p}{2}}.$

Hence, for any non-empty finite $\wedge\subset\mathbb{N},$ we have

$\sum_{n\in\wedge}\left\\{\rho_{p}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)\right\\}^{p}\geq\frac{(Card(\wedge))^{\frac{p}{2}+1}}{2^{p+1}}\left(\frac{1}{C}\right)^{\frac{p}{2}}.$

Therefore, putting the value of $C$ we get

$\sum_{n\in\wedge}\left\\{\rho_{p}\left(W_{\psi}^{M}\left(\frac{\phi_{n}}{\sqrt{2\pi|B|C_{\psi,M}}}\right)\right)\right\\}^{p}\geq\frac{(Card(\wedge))^{\frac{p}{2}+1}}{2^{p+1}}\left(\frac{3|B|C_{\psi,M}}{2^{\frac{4}{p}+2}\|\psi\|^{2}_{L^{2}(\mathbb{R})}}\right)^{\frac{p}{2}}.$

This completes the proof. ∎

##  6 Conclusions

We have proposed a novel time-frequency analyzing tool, namely LCWT, which
combines the advantages of the LCT and the WT and offers time and linear
canonical domain spectral information simultaneously in the time-LCT-frequency
plane. We have studied its properties like inner product relation,
reconstruction formula and also characterized its range. We also gave a lower
bound of the measure of essential support of the LCWT via UP of Donoho-Stark
and Lieb. Finally, we have studied the Shapiro’s mean dispersion theorem
associated with the LCWT.

##  7 Acknowledgement

This work is partially supported by UGC File No. 16-9(June
2017)/2018(NET/CSIR), New Delhi, India.

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CANDIDATE TRIPLE-STAR SYSTEM WITH ELLIPSOIDAL COMPONENTS

DETECTED IN VULPECULA THROUGH TESS PHOTOMETRY

G. Conzo1; M. Moriconi1; I. Peretto2

${1})$ Gruppo Astrofili Palidoro, Fiumicino, Italy
<EMAIL_ADDRESS>${2})$ MarSEC Marana Space Explorer Center,
Crespadoro, Italy<EMAIL_ADDRESS>

Abstract: MaGiV-1 is a candidate triple ellipsoidal star system in Vulpecula
at coordinates RA(J2000) $19:52:19.13$ DEC(J2000) $+23:29:59.7$ classified as
ELL+ELL, number 2344411 in AAVSO VSX database. Through photometry from the
TESS Space Telescope, two significant periods describing the orbital times of
the components were identified using the Fourier transform. The analysis led
to determining $P_{A-BC}=(4.269\pm 0.213)d$ the orbital period of A-BC pair,
the primary component with the secondary component described by another pair,
and $P_{BC}=(0.610\pm 0.031)d$ the orbital period of B-C pair, the inner
ellipsoidal system. However, it cannot be completely ruled out that the
shorter period can be explained by pulsations of one of the two components
(e.g. by the GDOR type).
---

##  1 Introduction

Rotating ellipsoidal variable star systems have two stars that brightness
variation shows up in photometric analysis from mutual deformation of the
components by gravitational interaction. Therefore it is not generated from
eclipses, but it is generated by cross-sectional area changes of the stars and
it is represented by rather homogeneous light curves with an almost sinusoidal
pattern and its variations are less than a tenth of a magnitude. It is very
difficult to classify this type of variable star because the light curve
profile does not contain the amount of information that light plots of eclipse
binaries do (Morris, 1985).

This paper is about a very complex variability closely related to the ELL type
(Ellipsoidal Variable Star), as it is a candidate system consisting of three
ellipsoidal components describing a variation of the classical ellipsoidal
binary star system. MaGiV-1 has been defined as a “double rotating ellipsoidal
variable” (ELL+ELL) star, a triple hierarchical type system, therefore, three
stars orbiting each other simultaneously, and specifically two of these
objects form a narrow binary, called inner binary, then the third companion is
orbiting at a distance that far exceeds the separation length of the inner
binary (Meiron, Kocsis, and Loeb, 2017). Again, the brightness variation is
represented by a light curve with variability less than a tenth of a
magnitude, but, from the Fourier analysis for finding the period, two
significant values emerge instead of one, thus returning two distinct light
curves (one child of each other). Both resulting periods represent the orbital
period for each pair, just as in binary ellipsoidal variables (Jerzykiewicz
_et al._ , 2011).

##  2 Photometric observations

For detection of MaGiV-1 system, TESS photometry (Ricker, G. R.; et al., 2014)
was used, showing millesimal brightness changes. TESS observes the sky in
sectors measuring $24^{\circ}\times 96^{\circ}$. Each sector is observed for
two orbits of the satellite around the Earth, or about 27 days on average. In
our case, sector 41 and sector 54 were used, then photometrc observations
derive from cameras as shown in Figure 1 and Figure 2:

Figure 1: Map of TESS camera fields-of-view in celestial coordinates (Ricker
_et al._ , 2014). Sector 41. Figure 2: Map of TESS camera fields-of-view in
celestial coordinates (Ricker _et al._ , 2014). Sector 54.

Using the MAST portal (Conti _et al._ , 2011), at coordinates RA(J2000)
$19:52:19.13$ DEC(J2000) $+23:29:59.7$, raw data was obtained for star
identified as TIC 300480214. In addition, from the TESS observation FITS, the
CCD camera pixels with which MaGiV-1 was observed were obtained, and the color
indicates the amount of flux in each pixel111Electrons per second, as shown in
Figure 4 and Figure 4 (Ricker, G. R.; et al., 2014).

Figure 3: TESS Sector 41 pixels on the CCD camera obtained in Python with
Lightkurve (Lightkurve Collaboration _et al._ , 2018) and Matplotlib (Hunter,
2007) packages.

Figure 4: TESS Sector 54 pixels on the CCD camera obtained in Python with
Lightkurve (Lightkurve Collaboration _et al._ , 2018) and Matplotlib (Hunter,
2007) packages.

Before the light curve analysis, the offset 2457000 was added to the BJD
values from the fits files. (Eastman, Siverd, and Gaudi, 2010).

In addition, brightness was converted from flux to magnitudes using the
following law (Bessell, 2000):

$Mag=-2.5log(Flux)$ (1)

Finally, a normalized magnitude around zero was used, obtained by subtracting
each measure from the average of the entire data set, highlighting the very
small brightness variation. It can be shown in Figure 5 and Figure 6 on 24 Jul
2021 and 9 Jul 2022 observations data respectively.

Figure 5: Full TESS Plot for MaGiV-1 (Observation on July 24, 2021 in HJD time
domain). Plot obtained with VStar software (Benn, D. , 2012). Figure 6: Full
TESS Plot for MaGiV-1 (Observation on July 9, 2022 in HJD time domain). Plot
obtained with VStar software (Benn, D. , 2012).

The data, thus organized, were processed with Peranso software (Paunzen and
Vanmunster, 2016), obtaining the identification of two (or more) significant
periods. The data, thus organized, were processed with the method ANOVA of the
Peranso software (Schwarzenberg-Czerny, 1996).

Two signific periods were obtained to describe and characterize MaGiV-1
candidate triple-star system in the phase domain on two observation dates (24
Jul 2021 and 9 Jul 2022). First light curve in Figure 7 characterized by a
period $P_{A-BC}=(4.269\pm 0.213)d$, shows two minima within an entire cycle
(0 to 1) with a typical pattern of rotating ellipsoidal binary systems (ELL),
but including obvious and repeated oscillations along the entire plot, with
0.004 delta magnitudes. These oscillations make sense when analyzing the
second light curve in Figure 8 characterized by a period $P_{BC}=(0.610\pm
0.031)d$, showing a common plot of ELL star systems with two minima within an
entire cycle (0 to 1) and 0.003 delta magnitudes.

Figure 7: Phase Plot for MaGiV-1 (1st ELL: Primary star ‘A’ with second couple
consisting of ‘B’ and ‘C’ components). Plot obtained with VStar software
(Benn, D. , 2012). Figure 8: Phase Plot for MaGiV-1 (2nd ELL: Couple
consisting of ‘B’ and ‘C’ components). Plot obtained with VStar software
(Benn, D. , 2012).

Since the depths of the two minima barely differ, the true period could also
be half of $P_{BC}$ value, then $0.305d$.

Analyzing the periodogram, four significant values corresponding to the
fundamental peaks were evaluated.

Figure 9: Periodogram for MaGiV-1. Plot obtained with VStar software (Benn, D.
, 2012).

The periodogram first shows two significant periods that could be interpreted
as half of the orbital periods of the A-BC pair $P_{A-BC}=(4.269\pm 0.213)d$
and the BC pair $P_{BC}=(0.610\pm 0.031)d$. In addition, two periods below
$<0.2d$ are present $P_{puls_{1}}=(0.195\pm 0.050)d$ and
$P_{puls_{2}}=(0.165\pm 0.050)d$, which could represent Delta Scuti type
pulsations of one of the components. Since some Delta Scuti stars also show
pulsations in the GDOR region (Skarka _et al._ , 2022), it cannot be
completely ruled out that the presumed period of the BC pair is a pulsation
frequency, even if this seems unlikely.

##  3 Results

MaGiV-1 variable star was identified through data-mining from the MAST
website, where it was possible to view TESS photometric observations lasting
26.594 days, therefore reference series
TESS_TIC_00000300480214_Sector_41_PDCSAP and
TESS_TIC_00000300480214_Sector_54_PDCSAP were used. It were extracted BJD
(Barycentric Julian Date) time domain and the detrend flow PDCSAP_FLUX
(Toonen, Hamers, and Portegies Zwart, 2016). In Download Data Products (DDP)
only infrared photometry (Ic) was extracted, using FITS Viewer software, a
graphical program for viewing and editing any FITS format image or table (W.
Pence, P. Chai , 2020), and choosing the TIME and PDCSAP_FLUX columns (Conti
_et al._ , 2011). The data thus obtained were appropriately converted and
processed using Peranso software (Paunzen and Vanmunster, 2016) and analysing
it with Fourier Transform derived by ANOVA method, using periodic orthogonal
polynomials to fit the observations and statistics of variance analysis,
defining their quality. This greatly improved the sensitivity of spike
detection by dampening aliasing periods (Schwarzenberg-Czerny, 1996). It was
identified a first period $P_{A-BC}=(4.269\pm 0.213)d$ with 0.007 mag
amplitude and a second period $P_{BC}=(0.610\pm 0.031)d$ with 0.003 mag
amplitude. From obtained light curves and about variable stars guidelines
classification in Variable Stars Index (VSX) (Otero, 2020), it was possible to
classify the MaGiV-1 as a candidate triple star system ELL+ELL type candidate,
then a triple-star system with ellipsoidal components whose luminosity
variation changes with the orbital period.

Evaluating corresponding delta-magnitude for the light curves of ELL pairs
respectively, it can be used the Period-Luminosity (P-L) relationship in low
amplitude conditions (Pawlak _et al._ , 2014). This consideration makes it
possible to define the overall brightness of two binary systems as a whole
through a linear extrapolation that takes into account the orbital period.
Using the relationship

$W=-2.78(\pm 0.08)logP+19.30(\pm 0.18)$ (2)

it was obtained following results

$W_{bc}=(19.90\pm 0.11)mag$ (3)

$W_{a-bc}=(17.55\pm 0.16)mag$ (4)

Where $W_{bc}$ is the overall brightness of the inner binary system and
$W_{a-bc}$ is the overall brightness of the outer binary system. With these
results, it was defined the ‘giant-giant’ type components, in according by
linear extrapolation model for low-amplitude proposed.

Finally, MaGiV-1 distance was obtained through the relationship linking it
with the magnitudes resulting from the photometric analysis

$d=10^{(m_{v}-M_{v}+5-A_{v})/5}$ (5)

where $m_{v}$ and $M_{v}$ are relative and absolute magnitudes respectively.

The information provided by GAIA regarding parallax was used
$p=0.835079181mas$ and some features related to field such as extinction
$Av=(1.40\pm 0.56)mag$ and magnitude of source $m=(10.144\pm 0.557)mag$, it
was obtained (Hughes, 2006).

Distance evaluated is $d=(629\pm 245)pc$ and comparing it to the GAIA distance
$d_{GAIA}=1018pc$, shows an underestimation, because GAIA distance derived
from parallax, brings with it a more significant detection uncertainty, in
fact, the index $RUWE=2.387$ (Gaia Collaboration, 2020) is much more than 1
(value of high affidability).

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