# Multidimensional Fractional Wavelet Transforms and Uncertainty Principles

Navneet Kaura, Bivek Guptab, Amit K. Vermac,
a,b,cDepartment of Mathematics, IIT Patna, Bihta, Patna, 801106.
<EMAIL_ADDRESS>author email<EMAIL_ADDRESS>
###### Abstract

In this paper, we have given a new definition of continuous fractional wavelet
transform in $\mathbb{R}^{N},$ namely the multidimensional fractional wavelet
transform (MFrWT) and studied some of the basic properties along with the
inner product relation and the reconstruction formula. We have also shown that
the range of the proposed transform is a reproducing kernel Hilbert space and
obtain the associated kernel. We have obtained the uncertainty principle like
Heisenberg’s uncertainty principle, logarithmic uncertainty principle and
local uncertainty principle of the multidimensional fractional Fourier
transform (MFrFT). Based on these uncertainty principles of the MFrFT we have
obtained the corresponding uncertainty principles i.e., Heisenberg’s,
logarithmic and local uncertainty principles for the proposed MFrWT.

Keywords: Multidimensional Fractional Fourier Transform; Multidimensional
Fractional Wavelet Transform; Heisenberg’s Uncertainty Principle; Logarithmic
Uncertainty Principle; Local Uncertainty Principle
AMS Subject Classification 2020: 42C40, 46E30, 47G10

##  1 Introduction

In 1980 Namias introduced the fractional Fourier transform (FrFT), also known
as essentially equivalent transforms, as a generalization of the traditional
Fourier transform (FT)([18]) . This theory was later refined and studied in
[1], [16]. FrFT has been used as a substantial tool for analysis of the
signal, sensor data, transmission signals such as radio signals and many
others, for example [1],[4],[14],[22],[20]. Because of the scarcity of
localization information, It is not suitable for processing signals with
varying fractional frequencies over time. The Fourier transform, on the other
hand, has been found to be unsuitable for characterizing some practical uses
or dealing with their inherent mathematical issues. As a repercussion, some
Fourier transforms have been developed to compensate for the FT’s imprecision,
such as the FrFT, wavelet transform ([7]) and the windowed Fourier transform
([9]). Mendlovich et al. ([17]) devised the Fractional Wavelet Transform to
analyze with optical signals in 1997. The proposal was to use the wavelet
transform of the signal’s fractional spectrum, which was obtained using FrFT.
Since then, FrFT has produced fractional frequencies that last for the entire
duration of the signal rather than a specific time, preventing the signal from
communicating local information. As a result, the fractional wavelet
transformation investigated in [17] fails to capture the signal’s local
properties. As a result, Shi et al. ([23]) proposed a new fractional wavelet
transform that combines aspects of both the classical and FrFT wavelet
transforms and returns the signal’s local information. In [6], Dai et al.
studied a new FrWT, which is more general than the transform studied in [23],
[21]. They also studied the multiresolution analysis associated with it.
Luchko et al. ([15]) gave a novel definition of FrFT and the corresponding
fractional wavelet transform have been studied in [25],[27]. For more
information on the fractional Fourier transform introduced in [15], we refer
the reader to [13],[26].

In recent times, Zayed [31] proposed 2-dimensional FrFT

$\mathfrak{F}_{\alpha,\beta}(u,v)=\displaystyle\int_{\mathbb{R}^{2}}k(x,y,u,v;\alpha,\beta)f(x,y)dxdy,$

where $k(x,y,u,v;\alpha,\beta)$ is given by equation $(3.13)$ in [31], which
is not a tensor product of two 1-dimensional FrFT ([1]). He verified its
properties with the convolution theorem, the inverse theorem, and the Poisson
summation formula. Kamalakkanan et al. [12] proved the inverse formula for
MFRFT ([19]) which is defined as the tensor product of N-copies of a
1-dimensional FrFT. He also studied the related convolution theorem and
product theorem, as well as presenting a generalised fractional convolution
that was more general than the one in [11]. Verma et al. ([28]) extended the
continuous fractional wavelet transforms in $\mathbb{R}^{N}$ with the dilation
parameter in $\mathbb{R}^{N}$ and studied the associated uncertainty
principles along with its boundedness on Morrey space. As per taking the
context on multidimensional fractional Fourier transform and one-dimensional
fractional wavelet transforms, in this paper, we provide definition of
multidimensional fractional wavelet transforms with ${N}-$dimensional
parameter $\bm{\alpha}$ in a more precise way, including properties of
continuous fractional wavelet transforms.

The signal’s frequency and time at any point in the time-frequency plane are
unknown. To put it another way, we have no way of knowing which spectral
components are present at any given time. All we can do is look at which
spectrum components are present at any given time frame. This issue is called
the uncertainty principle. Heisenberg discovered and formulated the
uncertainty principle, which states that a moving particle’s momentum and
state cannot be determined simultaneously. The principles of logarithmic,
Heisenberg, and local uncertainty help us to perceive the interrelationships
between different transformed domains better than the unrelated appearance
[6], [30]. Because they are signal processing elements, the three uncertainty
principles are well suited for potential later applications ([10]). The
detailed description and history of these three inequalities are given in [8].

The purpose of this paper is to define novel multidimensional fractional
wavelet transform (MFrWT) with parameter
$(\alpha_{1},\alpha_{2},\cdots,\alpha_{N})$ that is broader in scope than the
transforms defined earlier. We give some basic properties of the suggested
transform and obtain the inner product relation,
reconstruction/retransformation formula and also characterize its range. We
derive the Heisenberg’s uncertainty inequality, local uncertainty inequality
and logarithmic uncertainty inequality for the MFrFT. Based on the properties
of the MFrWT and the uncertainty inequalities associated with the MFrFT we
derive the same for the MFrWT. The rest of the paper is organized as follows.
In section 2, we review some fundamental definitions. In section 3, The
MFrWT’s theoretical framework, including its definition, properties, and
inverse transformation, has been established. In addition, we defined the
range of transformations and demonstrated that the range is the reproducing
kernel Hilbert space. In section 4, we have obtained several uncertainty
principles for the MFrWT. Lastly, in section 5, we conclude our paper.

##  2 Preliminaries

Let $\|\cdot\|$ denote the Euclidean norm and $\mathbb{R}^{N}$ be the
$N$-dimensional Euclidean space that is, for
$\mathbf{x}=(x_{1},x_{2},\cdots,x_{N})\in\mathbb{R}^{N}$

$\displaystyle\|\mathbf{x}\|=\sqrt{\sum_{i=1}^{N}x_{i}^{2}}.$

We specify $|\mathbf{x}|_{m}=|x_{1}x_{2}x_{3}\cdots x_{N}|$ and
$\mathbb{R}_{0}^{N}=\\{\mathbf{x}\in\mathbb{R}^{N}:|\mathbf{x}|_{m}\neq 0\\}$.
For
$\mathbf{x}=(x_{1},x_{2},\cdots,x_{N}),~{}\mathbf{y}=(y_{1},y_{2},\cdots,y_{N})\in\mathbb{R}^{N},~{}\mathbf{x}+\mathbf{y}=(x_{1}+y_{1},x_{2}+y_{2},\cdots,x_{N}+y_{N}),~{}\textbf{xy}=(x_{1}y_{1},x_{2}y_{2},\cdots,x_{N}y_{N}).$
Moreover $y\in\mathbb{R}_{0}^{N}$ then,
$\frac{\mathbf{x}}{\mathbf{y}}=\left(\frac{x_{1}}{y_{1}},\frac{x_{2}}{y_{2}},\cdots,\frac{x_{N}}{y_{N}}\right)$
and if $\bm{\alpha}=(\alpha_{1},\alpha_{2},\cdots,\alpha_{N})$ then we define
$\sin\bm{\alpha}=(\sin\alpha_{1},\sin\alpha_{2},\cdots,\sin\alpha_{N}).$

######  Definition 2.1.

For $1\leq P<\infty,$ the Lebesgue space $L^{P}(\mathbb{R}^{N})$ is a Banach
space and for every complex valued measurable function $f\in\mathbb{R}^{N}$
such that

$\displaystyle\int_{\mathbb{R}^{N}}|f(\bm{t})|^{P}d\bm{t}<\infty,$

where norm is given by

$\displaystyle\|f\|_{L^{P}(\mathbb{R}^{N})}=\left(\int_{\mathbb{R}^{N}}|f(\bm{t})|^{P}d\bm{t}\right)^{\frac{1}{P}}.$

In particular, $L^{2}(\mathbb{R}^{N})$ is a Hilbert space in which the inner
product is stated by

$\displaystyle\langle
f,g\rangle_{L^{2}(\mathbb{R}^{N})}=\int_{\mathbb{R}^{N}}f(\bm{t})\overline{g(\bm{t})}d\bm{t},$

here $\overline{g(\bm{t})}$ denotes the complex conjugate of $g(\bm{t}).$

######  Definition 2.2.

Assume $X$ is a complete inner product space of complex-valued functions
defined on $S,$ where $S$ is an arbitrary set with the inner product defined
by $\langle.,.\rangle_{X}$. Then a complex-valued function $\mathcal{K}$
defined on $S\times S$ is known as reproducing kernel of $X$ if it fulfill the
necessary conditions,

for a particular $p\in S$, we’ve $\mathcal{K}(\cdot,p)$also in $X$ and
$f(p)=\langle f(\cdot),\mathcal{K}(\cdot,p)\rangle_{X}$ for every $f\in X.$

We’ll go through the definition of the MFrFT ([12]) in the next section.

######  Definition 2.3.

The MFrFT of $f\in L^{2}(\mathbb{R}^{N}),$ of order
$\bm{\alpha}=(\alpha_{1},\alpha_{2},\cdots,\alpha_{N}),$
${\alpha_{i}}\in\left(-\pi,\pi\right)\backslash\\{0\\},$ for
$i=1,2,\cdots,{N}$ and $\lambda\in\mathbb{R}-\\{0\\},$ is given by

$\displaystyle\mathfrak{F}_{\bm{\alpha},\lambda}\left(f\right)\left(\bm{\xi}\right)=F_{\bm{\alpha},\lambda}\left(\bm{\xi}\right)=\int_{\mathbb{R}^{N}}f\left(\mathbf{x}\right)K_{\bm{\alpha},\lambda}\left(\mathbf{x},\bm{\xi}\right)\,d\mathbf{x},$
(1)

where
$K_{{\bm{\alpha}},\lambda}\left(\bm{x},\bm{\xi}\right)=\displaystyle\prod_{i=1}^{N}K_{\alpha_{k},\lambda}\left(x_{k},\xi_{k}\right)$
and
$K_{\alpha_{k},\lambda}\left(x_{k},\xi_{k}\right),~{}\mbox{for}~{}k=1,2,\cdots,{N},$
are defined by

$\displaystyle
K_{\alpha_{k},\lambda}\left(x_{k},\xi_{k}\right)=\begin{cases}\frac{c(\alpha_{k})}{(\sqrt{2\pi})}e^{i\lambda^{2}\\{a(\alpha_{k})[{x_{k}}^{2}+{\xi_{k}}^{2}-2b(\alpha_{k})x_{k}\xi_{k}]\\}},\alpha_{k}\notin\pi\mathbb{Z}\\\
\Delta(x_{k}-\xi_{k}),\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\alpha_{k}\in
2\pi\mathbb{Z}\\\
\Delta(x_{k}+\xi_{k}),\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\alpha_{k}\in
2\pi\mathbb{Z}+\pi.\end{cases}$

Here, $\mathbf{x}=(x_{1},x_{2},x_{3},...,x_{N}),\quad
a(\alpha_{k})=\frac{\cot(\alpha_{k})}{2},\quad
b(\alpha_{k})=\sec(\alpha_{k}),\quad c(\alpha_{k})=\sqrt{1-i\cot\alpha_{k}}.$

The kernel $K_{{\bm{\alpha}},\lambda}\left(\mathbf{x},\bm{\xi}\right)$ can be
re-written as

$\displaystyle
K_{\bm{\alpha},\lambda}\left(\mathbf{x},\bm{\eta}\right)=\frac{c(\bm{\alpha}_{\lambda})}{(\sqrt{2\pi})^{N}}e_{\bm{\alpha},\lambda^{2}}(\mathbf{x})e_{\bm{\alpha},\lambda^{2}}(\bm{\eta})e^{-i\lambda^{2}\sum_{k=1}^{N}x_{k}\eta_{k}\csc\alpha_{k}},$

where

$\displaystyle e_{\bm{\alpha},\lambda^{2}}(x)$ $\displaystyle=$ $\displaystyle
e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){x_{k}}^{2}},\quad
c(\bm{\alpha}_{\lambda})=c(\alpha_{1})c(\alpha_{2})\cdots c(\alpha_{N})$ (2)

and

$\bm{\alpha}_{\lambda}=\left(\cot^{-1}(\lambda^{2}\cot\alpha_{1}),\cot^{-1}(\lambda^{2}\cot\alpha_{2}),\cdots,\cot^{-1}(\lambda^{2}\cot\alpha_{N})\right).$

##  3 Multidimensional Fractional Wavelet Transform (MFrWT)

######  Definition 3.1.

A non-zero function $\psi\in L^{2}(\mathbb{R}^{N})$ is wavelet admissible if

$\displaystyle\displaystyle\int_{\mathbb{R}^{N}_{0}}|(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{u})|^{2}\frac{d\bm{u}}{|\bm{u}|_{m}}<\infty.$
(3)

A wavelet admissible function is also known as a fractional wavelet or simply
a wavelet.

Now, we define our new MFrWT.

Let $f\in L^{2}(\mathbb{R}^{N})$ and $\psi$ be an admissible wavelet, then
MFrWT is given by

$\displaystyle(W_{\psi}^{\bm{\alpha},\lambda}f)(\bm{a},\bm{b})=c(\bm{\alpha}_{\lambda})e_{\bm{\alpha},-\lambda^{2}}(\tilde{f}\ast\breve{g})(\bm{b}),\quad\bm{a}\in\mathbb{R}^{N}_{0},\quad\bm{b}\in\mathbb{R}^{N},$

where $\ast$ is the convolution given by

$(\mu\ast\nu)(\mathbf{x})=\int_{\mathbb{R}^{N}}\mu(\mathbf{y})\nu(\mathbf{x}-\mathbf{y})d\mathbf{y},$

$\tilde{\psi}=\psi{e}_{\bm{\alpha},\lambda^{2}},$
$\tilde{f}=f{e}_{\bm{\alpha},\lambda^{2}}$ and
${\breve{g}}(\bm{t})=\overline{\frac{1}{\sqrt{|\bm{a}|_{m}}}\tilde{\psi}\left(\frac{-\bm{t}}{\bm{a}}\right)}.$
Therefore,

$\displaystyle(W_{\psi}^{\bm{\alpha},\lambda}f)(\bm{a},\bm{b})$
$\displaystyle=$ $\displaystyle
c(\bm{\alpha}_{\lambda})e_{\bm{\alpha},-\lambda^{2}}(\bm{b})(\tilde{f}*{\breve{g}})(\bm{b})$
$\displaystyle=$ $\displaystyle
c(\bm{\alpha}_{\lambda})e_{\bm{\alpha},-\lambda^{2}}(\bm{b})\int_{\mathbb{R}^{N}}\tilde{f}(\bm{t})\overline{\frac{1}{\sqrt{|\bm{a}|_{m}}}\tilde{\psi}\left(\frac{\bm{t}-\bm{b}}{\bm{a}}\right)}d\bm{t}$
$\displaystyle=$ $\displaystyle
c(\bm{\alpha}_{\lambda})e_{\alpha,-\lambda^{2}}(\bm{b})\int_{\mathbb{R}^{N}}f(\bm{t})e_{\bm{\alpha},\lambda^{2}}(\bm{t})\overline{\frac{1}{\sqrt{|\bm{a}|_{m}}}\tilde{\psi}\left(\frac{\bm{t}-\bm{b}}{\bm{a}}\right)}d\bm{t}$
$\displaystyle=$ $\displaystyle\left\langle
f(\bm{t}),\overline{c(\bm{\alpha}_{\lambda})}e_{\bm{\alpha},\lambda^{2}}(\bm{b})e_{\bm{\alpha},-\lambda^{2}}(\bm{t}){\frac{1}{\sqrt{|\bm{a}|_{m}}}\tilde{\psi}}\left(\frac{\bm{t}-\bm{b}}{\bm{a}}\right)\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle
f(\bm{t}),\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})\right\rangle,$

where
${\mathbf{\Psi}}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})=\overline{c(\bm{\alpha}_{\lambda})}e_{\bm{\alpha},\lambda^{2}}(\bm{b})e_{\bm{\alpha},-\lambda^{2}}(\bm{t}){\frac{1}{\sqrt{|\bm{a}|_{m}}}\tilde{\psi}}\left(\frac{\bm{t}-\bm{b}}{\bm{a}}\right).$
We now prove some properties of the proposed MFrWT.

######  Theorem 3.1.

Let $\psi,\phi\in L^{2}(\mathbb{R}^{N})$ be two wavelets and for any two
functions $f,g\in L^{2}(\mathbb{R}^{N}).$ Also let $\sigma>0$ and
$r,s\in\mathbb{C},$ then

  1. (i)

Linearity:
$W^{\bm{\alpha},\lambda}_{\psi}(r{f}+s{g})=r(W_{\psi}^{\bm{\alpha},\lambda}{f})+s(W_{\psi}^{\bm{\alpha},\lambda}{g})$.
This property shows that the MFrWT upholds the superposition principle, which
is useful for multicomponent signal analysis.

  2. (ii)

Anti-linearity:
$W_{\left(r{\psi}+s{\phi}\right)}^{\bm{\alpha},\lambda}{f}=\overline{r}W_{{\psi}}{f}+\overline{s}W_{\phi}{f}.$

  3. (iii)

Dilation:
$(W_{\psi}^{\bm{\alpha},\lambda}D_{\sigma}f)(\bm{a},\bm{b})=\displaystyle\frac{C^{\prime}}{\sigma^{N}}W_{\psi}^{\bm{\alpha},\frac{\lambda}{\sigma}}f(\sigma\bm{a},\sigma\bm{b}),$
where $D_{\sigma}$ is a dilation operator, which is given by
$D_{\sigma}f(\bm{t})=f(\sigma\bm{t})$ and
$C^{\prime}=\sigma^{\frac{N}{2}}\overline{\left(\frac{c(\alpha_{\lambda})}{c\left(\alpha_{\frac{\lambda}{\sigma}}\right)}\right)},$

  4. (iv)

Conjugacy:
$(W_{\psi}^{\bm{\alpha},\lambda}\overline{f})(\bm{a},\bm{b})=\overline{W_{\overline{\psi}}^{-\bm{\alpha},\lambda}f(\bm{a},\bm{b})}.$

  5. (v)

Parity:
$(W_{\psi}^{\bm{\alpha},\lambda}Pf)(\bm{a},\bm{b})=W_{\psi}^{\bm{\alpha},\lambda}f(-\bm{a},-\bm{b}),$
Here, $P$ is the parity operator, which is defined by
$Pf(\mathbf{x})=f(-\mathbf{x}).$

  6. (vi)

Translation:
$(W_{\psi}^{\bm{\alpha},\lambda}\tau_{\mathbf{y}}f)(\bm{a},\bm{b})=e_{\bm{\alpha},-\lambda^{2}}(\bm{b})e_{\bm{\alpha},\lambda^{2}}(\bm{b}-\mathbf{y})e_{\bm{\alpha},\lambda^{2}}(\mathbf{y})(W_{\psi}^{\bm{\alpha},\lambda}\check{f})(\bm{a},\bm{b}-\mathbf{y}),$
where fractional translation operator $\tau_{\mathbf{y}}$ is given by
$\tau_{\mathbf{y}}f(\mathbf{x})=f(\mathbf{x}-\mathbf{y})$ and
$\displaystyle\check{f}(\bm{t})=f(\bm{t})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k})2{t_{k}}y_{k}}.$

###### Proof.

The proof of (i) and (ii) are straight forward and thus can be omitted.

(iii) Using the definition of $D_{\sigma}f,$ we have

$\displaystyle(W_{\psi}^{\bm{\alpha},\lambda}D_{\sigma}f)(\bm{a},\bm{b})=\int_{\mathbb{R}^{N}}(D_{\sigma}f)(\bm{t})\overline{{\mathbf{\Psi}}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})}d\bm{t}$
$\displaystyle=\displaystyle{\int_{\mathbb{R}^{N}}}f(\sigma\bm{t})\overline{{\mathbf{\Psi}}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})}d\bm{t},$
(4)

on putting $\sigma\bm{t}=\bm{u}$ and
$d\bm{t}=\displaystyle\frac{d\bm{u}}{\sigma}$ in equation (3), we get

$\displaystyle(W_{\psi}^{\bm{\alpha},\lambda}f_{\sigma})(\bm{a},\bm{b})=\frac{1}{\sigma^{N}}\displaystyle{\int_{\mathbb{R}^{N}}}f(\bm{u})\overline{{\mathbf{\Psi}}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}\left(\frac{\bm{u}}{\sigma}\right)}d\bm{u}.$
(5)

Now,

$\displaystyle{\mathbf{\Psi}}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})$
$\displaystyle=$
$\displaystyle\overline{c(\bm{\alpha}_{\lambda})}e_{\bm{\alpha},\lambda^{2}}(\bm{b})e_{\bm{\alpha},-\lambda^{2}}(\bm{t})\frac{1}{\sqrt{|\bm{a}|_{m}}}\tilde{\psi}\left(\frac{\bm{t}-\bm{b}}{\bm{a}}\right)$
$\displaystyle\Rightarrow{\mathbf{\Psi}}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}\left(\frac{\bm{u}}{\sigma}\right)$
$\displaystyle=$
$\displaystyle\overline{c(\bm{\alpha}_{\lambda})}e_{\bm{\alpha},\lambda^{2}}(\bm{b})e_{\bm{\alpha},-\lambda^{2}}\left(\frac{\bm{u}}{\sigma}\right)\frac{1}{\sqrt{|\bm{a}|_{m}}}\tilde{\psi}\left(\frac{(\frac{\bm{u}}{\sigma})-\bm{b}}{\bm{a}}\right)$
$\displaystyle=$
$\displaystyle\overline{c(\bm{\alpha}_{\lambda})}e_{\bm{\alpha},\frac{\lambda^{2}}{\sigma^{2}}}(\sigma\bm{b})e_{\bm{\alpha},\frac{-\lambda^{2}}{\sigma^{2}}}(\bm{u})\frac{1}{\sqrt{|\bm{a}|_{m}}}\tilde{\psi}\left(\frac{\bm{u}-\sigma\bm{b}}{\sigma{\bm{a}}}\right).$
$\displaystyle=$
$\displaystyle\sigma^{\frac{{N}}{2}}\overline{c(\bm{\alpha}_{\lambda})}e_{\bm{\alpha},\frac{\lambda^{2}}{\sigma^{2}}}(\sigma\bm{b})e_{\bm{\alpha},\frac{-\lambda^{2}}{\sigma^{2}}}(\bm{u})\frac{1}{\sqrt{|\sigma\bm{a}|_{m}}}\tilde{\psi}\left(\frac{\bm{u}-\sigma\bm{b}}{\sigma{\bm{a}}}\right).$

Therefore,

$\displaystyle\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}\left(\frac{\bm{u}}{\sigma}\right)=C^{\prime}{\mathbf{\Psi}}_{\bm{\alpha},\frac{\lambda}{\sigma},\sigma\bm{a},\sigma\bm{b}}(\bm{u}),$
(6)

where
$C^{\prime}=\sigma^{\frac{N}{2}}\overline{\left(\frac{c(\alpha_{\lambda})}{c\left(\alpha_{\frac{\lambda}{\sigma}}\right)}\right)}.$
We now get the required result by putting equation (6) into equation (5).

(iv)

$\displaystyle(W_{\psi}^{\bm{\alpha},\lambda}\overline{f})(\bm{a},\bm{b})$
$\displaystyle=$
$\displaystyle\left\langle\overline{f}(\bm{t}),\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})\right\rangle_{L^{2}(\mathbb{R}^{N})}$
$\displaystyle=$
$\displaystyle\left\langle\overline{f}(\bm{t}),{c((\bm{-}\alpha)_{\lambda})}e_{\bm{\alpha},\lambda^{2}}(\bm{b})e_{\bm{\alpha},-\lambda^{2}}(\bm{t}){\frac{1}{\sqrt{|\bm{a}|_{m}}}\tilde{\psi}}\left(\frac{\bm{t}-\bm{b}}{\bm{a}}\right)\right\rangle_{L^{2}(\mathbb{R}^{N})}$
$\displaystyle=$ $\displaystyle\overline{\left\langle
f(\bm{t}),\overline{c((\bm{-}\alpha)_{\lambda})}e_{\bm{-\alpha},\lambda^{2}}(\bm{b})e_{\bm{-\alpha},-\lambda^{2}}(\bm{t}){\frac{1}{\sqrt{|\bm{a}|_{m}}}\overline{\tilde{\psi}\left(\frac{\bm{t}-\bm{b}}{\bm{a}}\right)}}\right\rangle_{L^{2}(\mathbb{R}^{N})}}$
$\displaystyle=$ $\displaystyle\overline{\left\langle
f(\bm{t}),\overline{c((\bm{-}\alpha)_{\lambda})}e_{\bm{-\alpha},\lambda^{2}}(\bm{b})e_{\bm{-\alpha},-\lambda^{2}}(\bm{t})\frac{1}{\sqrt{|\bm{a}|_{m}}}\overline{({\psi
e_{\alpha,\lambda^{2}})}\left(\frac{\bm{t}-\bm{b}}{\bm{a}}\right)}\right\rangle_{L^{2}(\mathbb{R}^{N})}}$
$\displaystyle=$ $\displaystyle\overline{\left\langle
f(\bm{t}),\overline{c((\bm{-}\alpha)_{\lambda})}e_{\bm{-\alpha},\lambda^{2}}(\bm{b})e_{\bm{-\alpha},-\lambda^{2}}(\bm{t})\frac{1}{\sqrt{|\bm{a}|_{m}}}(\overline{\psi}e_{-\alpha,\lambda^{2}})\left(\frac{\bm{t}-\bm{b}}{\bm{a}}\right)\right\rangle_{L^{2}(\mathbb{R}^{N})}}$
$\displaystyle=$
$\displaystyle\overline{W_{\overline{\psi}}^{-\bm{\alpha},\lambda}f(\bm{a},\bm{b})}.$

(v) Using the definition of $Pf,$ we have

$\displaystyle(W_{\psi}^{\bm{\alpha},\lambda}Pf)(\bm{a},\bm{b})=\displaystyle\int_{\mathbb{R}^{N}}Pf(\bm{t})\overline{\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})}d\bm{t}$
$\displaystyle=\displaystyle{\int_{\mathbb{R}^{N}}}f(-\bm{t})\overline{{\mathbf{\Psi}}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})}d\bm{t}.$
(7)

Putting $-\bm{t}=\mathbf{x}$ in equation (3), we get

$\displaystyle(W_{\psi}^{\bm{\alpha},\lambda}Pf)(\bm{a},\bm{b})=\displaystyle{\int_{\mathbb{R}^{N}}}f(\mathbf{x})\overline{\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(-\mathbf{x}})d\mathbf{x}.$
(8)

Now,

$\displaystyle\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(-\bm{t})$
$\displaystyle=$
$\displaystyle\overline{c(\bm{\alpha}_{\lambda})}e_{\bm{\alpha},\lambda^{2}}(\bm{b})e_{\bm{\alpha},-\lambda^{2}}(-\bm{t})\frac{1}{\sqrt{|\bm{a}|_{m}}}\tilde{\psi}\left(\frac{-\bm{t}-\bm{b}}{\bm{a}}\right)$
(9) $\displaystyle=$
$\displaystyle\overline{c(\bm{\alpha}_{\lambda})}e_{\bm{\alpha},\lambda^{2}}(-\bm{b})e_{\bm{\alpha},-\lambda^{2}}(\bm{t})\frac{1}{\sqrt{|\bm{a}|_{m}}}\tilde{\psi}\left(\frac{\bm{t}-(-\bm{b})}{-\bm{a}}\right)$
$\displaystyle=$
$\displaystyle{\mathbf{\Psi}}_{\bm{\alpha},\lambda,-\bm{a},-\bm{b}}(\bm{t}).$

We get the required result by putting equation (9) into equation (8).

(vi) Using the definition of $\tau_{\mathbf{y}}f,$ we get

$\displaystyle(W_{\psi}^{\bm{\alpha},\lambda}\tau_{\mathbf{y}}f)(\bm{a},\bm{b})$
$\displaystyle=$
$\displaystyle\displaystyle\int_{\mathbb{R}^{N}}\tau_{\mathbf{y}}f(\mathbf{x})\overline{\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\mathbf{x})}d\mathbf{x}$
(10) $\displaystyle=$
$\displaystyle\displaystyle\int_{\mathbb{R}^{N}}f(\mathbf{x}-\mathbf{y})\overline{\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\mathbf{x})}d\mathbf{x},$

and by putting
$\mathbf{x}-\mathbf{y}=\bm{t},\mathbf{x}=\bm{t}+\mathbf{y},d\mathbf{x}=d\bm{t}$
in equation (10), we get

$\displaystyle(W_{\psi}^{\bm{\alpha},\lambda}\tau_{\mathbf{y}}f)(\bm{a},\bm{b})=\displaystyle\int_{\mathbb{R}^{N}}f(\bm{t})\overline{\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t}+\mathbf{y})}d\bm{t}.$
(11)

Now,

$\displaystyle\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t}+\mathbf{y})=$
$\displaystyle\overline{c(\bm{\alpha}_{\lambda})}e_{\bm{\alpha},\lambda^{2}}(\bm{b})e_{\bm{\alpha},-\lambda^{2}}(\bm{t}+\mathbf{y}){\frac{1}{\sqrt{|\bm{a}|_{m}}}\tilde{\psi}}\left(\frac{\bm{t}+\mathbf{y}-\bm{b}}{\bm{a}}\right)$
$\displaystyle=$
$\displaystyle\overline{c(\bm{\alpha}_{\lambda})}e_{\bm{\alpha},\lambda^{2}}(\bm{b})e_{\bm{\alpha},-\lambda^{2}}(\bm{t}+\mathbf{y}){\frac{1}{\sqrt{|\bm{a}|_{m}}}\tilde{\psi}}\left(\frac{\bm{t}-(\bm{b}-\mathbf{y})}{\bm{a}}\right)$
$\displaystyle=$ $\displaystyle
e_{\bm{\alpha},\lambda^{2}}(\bm{b})e_{\bm{\alpha},-\lambda^{2}}(\bm{b}-\mathbf{y})e_{\bm{\alpha},-\lambda^{2}}(\mathbf{y})e^{-i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k})2{t_{k}}y_{k}}\Big{[}\overline{c(\alpha_{\lambda})}e_{\bm{\alpha},\lambda^{2}}(\bm{b}-\mathbf{y})e_{\bm{\alpha},-\lambda^{2}}(\bm{t})$
$\displaystyle~{}~{}~{}{\frac{1}{\sqrt{|\bm{a}|_{m}}}}\tilde{\psi}\left(\frac{\bm{t}-(\bm{b}-\mathbf{y})}{\bm{a}}\right)\Big{]}$
$\displaystyle=$ $\displaystyle
e_{\bm{\alpha},\lambda^{2}}(\bm{b})e_{\bm{\alpha},-\lambda^{2}}(\bm{b}-\mathbf{y})e_{\bm{\alpha},-\lambda^{2}}(\mathbf{y})e^{-i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k})2{t_{k}}y_{k}}{\mathbf{\Psi}}_{\bm{\alpha},\lambda,\bm{a},\bm{b}-\mathbf{y}}(\bm{t}).$
(12)

We get the required result by putting equation(3) into equation (11). ∎

Now we will derive some lemmas that will help us to demonstrate the inner
product relation for MFrWT .

######  Lemma 3.1.

(FrFT of $\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})$) Suppose
a non-zero function $\psi\in L^{2}(\mathbb{R}^{N})$ is given, then

$\displaystyle(\mathfrak{F}_{\bm{\alpha},\lambda}{\mathbf{\Psi}}_{\bm{\alpha},\lambda,\bm{a},\bm{b}})(\bm{\xi})$
$\displaystyle=$
$\displaystyle\overline{c(\bm{\alpha}_{\lambda})}\sqrt{|\bm{a}|_{m}}e_{\bm{\alpha},\lambda^{2}}(\bm{b})e_{\alpha,\lambda^{2}}(\bm{\xi})e_{\bm{\alpha},-\lambda^{2}}(\bm{a}\bm{\xi})e^{-i\lambda^{2}\sum_{k=1}^{N}b_{k}\xi_{k}\csc\alpha_{k}}(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{a}\bm{\xi}).$
(13)

###### Proof.

For any $\bm{\xi}\in\mathbb{R}^{N},$

$\displaystyle(\mathfrak{F}_{\bm{\alpha},\lambda}{\mathbf{\Psi}}_{\bm{\alpha},\lambda,\bm{a},\bm{b}})(\bm{\xi})$
$\displaystyle=$
$\displaystyle\displaystyle\int_{\mathbb{R}^{N}}\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})K_{\bm{\alpha},\lambda}\left(\bm{t},\bm{\xi}\right)d\bm{t}$
(14) $\displaystyle=$
$\displaystyle\overline{c(\bm{\alpha}_{\lambda})}e_{\bm{\alpha},\lambda^{2}}(\bm{b})\displaystyle\int_{\mathbb{R}^{N}}e_{\bm{\alpha},-\lambda^{2}}(\bm{t})\frac{1}{\sqrt{|\bm{a}|_{m}}}\tilde{\psi}\left(\frac{\bm{t}-\bm{b}}{\bm{a}}\right)\frac{c(\bm{\alpha}_{\lambda})}{(\sqrt{2\pi})^{N}}e_{\bm{\alpha},\lambda^{2}}(\bm{t})e_{\bm{\alpha},\lambda^{2}}(\bm{\xi})e^{-i\lambda^{2}\sum_{k=1}^{N}t_{k}\xi_{k}\csc\alpha_{k}}d\bm{t}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{|\bm{a}|_{m}}}\frac{|c({\bm{\alpha}_{\lambda}})|^{2}}{(\sqrt{2\pi})^{N}}e_{\bm{\alpha},\lambda^{2}}(\bm{b})\displaystyle\int_{\mathbb{R}^{N}}\tilde{\psi}\left(\frac{\bm{t}-\bm{b}}{\bm{a}}\right)e_{\bm{\alpha},\lambda^{2}}(\bm{\xi})e^{-i\lambda^{2}\sum_{k=1}^{N}t_{k}\xi_{k}\csc\alpha_{k}}d\bm{t}.$

Putting $\bm{t}=\bm{a}\bm{u}+\bm{b}$ and $d\bm{t}=|\bm{a}|_{m}d\bm{u}$ in
equation (14), we’ve

$\displaystyle(\mathfrak{F}_{\bm{\alpha},\lambda}$
$\displaystyle{\mathbf{\Psi}}_{\bm{\alpha},\lambda,\bm{a},\bm{b}})(\bm{\xi})$
$\displaystyle=\frac{|c({\bm{\alpha}_{\lambda}})|^{2}}{(\sqrt{2\pi})^{N}}\sqrt{|\bm{a}|_{m}}e_{\bm{\alpha},\lambda^{2}}(\bm{b})e_{\bm{\alpha},\lambda^{2}}(\bm{\xi})e_{\bm{\alpha},-\lambda^{2}}(\bm{a}\bm{\xi})e^{-i\lambda^{2}\sum_{k=1}^{N}b_{k}\xi_{k}\csc\alpha_{k}}\int_{\mathbb{R}^{N}}\tilde{\psi}(\bm{u})e_{\bm{\alpha},\lambda^{2}}(\bm{a}\bm{\xi})e^{-i\lambda^{2}\sum_{k=1}^{N}a_{k}u_{k}\xi_{k}\csc\alpha_{k}}d\bm{u}$
$\displaystyle=\overline{c(\bm{\alpha}_{\lambda})}\sqrt{|\bm{a}|_{m}}e_{\bm{\alpha},\lambda^{2}}(\bm{b})e_{\bm{\alpha},\lambda^{2}}(\bm{\xi})e_{\bm{\alpha},-\lambda^{2}}(\bm{a}\bm{\xi})e^{-i\lambda^{2}\sum_{k=1}^{N}b_{k}\xi_{k}\csc\alpha_{k}}\int_{\mathbb{R}^{N}}\psi(\bm{u})K_{\bm{\alpha},\lambda}(\bm{u},\bm{a}\bm{\xi})d\bm{u}$
$\displaystyle=\overline{c(\bm{\alpha}_{\lambda})}\sqrt{|\bm{a}|_{m}}e_{\bm{\alpha},\lambda^{2}}(\bm{b})e_{\bm{\alpha},\lambda^{2}}(\bm{\xi})e_{\bm{\alpha},-\lambda^{2}}(\bm{a}\bm{\xi})e^{-i\lambda^{2}\sum_{k=1}^{N}b_{k}\xi_{k}\csc\alpha_{k}}(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{a}\bm{\xi}).$

This completes the proof. ∎

######  Lemma 3.2.

(FrFT of $W_{\psi}^{\bm{\alpha},\lambda}{f}(\bm{a},\bm{b})$) Suppose $\psi$ be
an admissible wavelet and for any arbitrary function $f\in
L^{2}(\mathbb{R}^{N})$, then

$\displaystyle\mathfrak{F}_{\bm{\alpha},\lambda}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})(\bm{\xi})=\sqrt{|\bm{a}|_{m}}(\sqrt{2\pi})^{N}\frac{c(\bm{\alpha}_{\lambda})}{\overline{c(\bm{\alpha}_{\lambda})}}F(\bm{\xi}),$

where
$F(\bm{\xi})=e_{\bm{\alpha},\lambda^{2}}(\bm{a}\bm{\xi})(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})\overline{(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{a}\bm{\xi})}.$

###### Proof.

Here, we know

$\displaystyle(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})$
$\displaystyle=\left\langle
f(\bm{t}),\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})\right\rangle$

Therefore,

$\displaystyle(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})$
$\displaystyle=\left\langle(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi}),(\mathfrak{F}_{\bm{\alpha},\lambda}{\mathbf{\Psi}}_{\bm{\alpha},\lambda,\bm{a},\bm{b}})(\bm{\xi}))\right\rangle$
$\displaystyle=\displaystyle\int_{\mathbb{R}^{N}}(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})\overline{(\mathfrak{F}_{\bm{\alpha},\lambda}\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}})(\bm{\xi})}d\bm{\xi}$
$\displaystyle=\displaystyle\int_{\mathbb{R}^{N}}(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi}){c(\bm{\alpha}_{\lambda})}\sqrt{|\bm{a}|_{m}}e_{\bm{\alpha},-\lambda^{2}}(\bm{b})e_{\bm{\alpha},-\lambda^{2}}(\bm{\xi})e_{\bm{\alpha},\lambda^{2}}(\bm{a}\bm{\xi})e^{i\lambda^{2}\sum_{k=1}^{N}b_{k}\xi_{k}\csc\alpha_{k}}\overline{(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{a}\bm{\xi})}d\bm{\xi}.$
(15)

From equation (3), we have

$(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})={c(\bm{\alpha}_{\lambda})}\sqrt{|\bm{a}|_{m}}\int_{\mathbb{R}^{N}}e_{\bm{\alpha},-\lambda^{2}}(\bm{b})e_{\bm{\alpha},-\lambda^{2}}(\bm{\xi})e^{i\lambda^{2}\sum_{k=1}^{N}b_{k}\xi_{k}\csc\alpha_{k}}F(\bm{\xi})d\bm{\xi},$

where

$\displaystyle
F(\bm{\xi})=e_{\bm{\alpha},\lambda^{2}}(\bm{a}\bm{\xi})(\mathfrak{F}_{\bm{\alpha},\lambda})(\bm{\xi})\overline{(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{a}\bm{\xi})}.$
(16)

Thus, we have

$\displaystyle(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})$
$\displaystyle=\sqrt{|\bm{a}|_{m}}(\sqrt{2\pi})^{N}\frac{c(\bm{\alpha}_{\lambda})}{\overline{c(\bm{\alpha}_{\lambda})}}\int_{\mathbb{R}^{N}}K_{-\bm{\alpha},\lambda}\left(\bm{b},\bm{\xi}\right)F(\bm{\xi})d\bm{\xi}$
$\displaystyle=\sqrt{|\bm{a}|_{m}}(\sqrt{2\pi})^{N}\frac{c(\bm{\alpha}_{\lambda})}{\overline{c(\bm{\alpha}_{\lambda})}}\mathfrak{F}_{-\bm{\alpha},\lambda}(F(\bm{\xi}))(\bm{b}),$

which implies that

$\displaystyle{}\mathfrak{F}_{\bm{\alpha},\lambda}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})(\bm{\xi})$
$\displaystyle=$
$\displaystyle\sqrt{|\bm{a}|_{m}}(\sqrt{2\pi})^{N}\frac{c(\bm{\alpha}_{\lambda})}{\overline{c(\bm{\alpha}_{\lambda})}}F(\bm{\xi}).$
(17)

We get the required result by putting equation (16) into equation (17). ∎

######  Theorem 3.2.

(Inner product relation for MFrWT) Let $W_{\psi}^{\bm{\alpha},\lambda}{f}$ and
$W_{\psi}^{\bm{\alpha},\lambda}{g}$ represent the MFrWT of functions $f$ and
$g$ with respect to admissible wavelet $\psi$ respectively and then

$\displaystyle\displaystyle\int_{\mathbb{R}^{N}_{0}}\displaystyle\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\overline{(W_{\psi}^{\bm{\alpha},\lambda}{g})(\bm{a},\bm{b})}\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}=C_{\bm{\alpha},\lambda}\langle
f,g\rangle_{L^{2}(\mathbb{R}^{N})},$ (18)

where

$\displaystyle
C_{\bm{\alpha},\lambda}={(2\pi)^{N}}\displaystyle\int_{\mathbb{R}^{N}_{0}}|(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{u})|^{2}\frac{d\bm{u}}{|\bm{u}|_{m}}.$
(19)

###### Proof.

Treating $(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})$,
$(W_{\psi}^{\bm{\alpha},\lambda}{g})(\bm{a},\bm{b})$ as
$L^{2}(\mathbb{R}^{N})$ are functions of some variable $\bm{b}$ and using the
Parseval’s theorem for the MFrFT ([12], Theorem 3.3) we have

$\displaystyle\displaystyle\int_{\mathbb{R}^{N}_{0}}\displaystyle\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\overline{(W_{\psi}^{\bm{\alpha},\lambda}{g})(\bm{a},\bm{b})}\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}=\displaystyle\int_{\mathbb{R}^{N}_{0}}\left(\displaystyle\int_{\mathbb{R}^{N}}\mathfrak{F}_{\bm{\alpha},\lambda}((W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b}))(\bm{\xi})\overline{\mathfrak{F}_{\bm{\alpha},\lambda}((W_{\psi}^{\bm{\alpha},\lambda}{g})(\bm{a},\bm{b}))(\bm{\xi})}d\bm{\xi}\right)\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}.$
(20)

Now, using equation (17) in equation (20), we get

$\displaystyle\displaystyle\int_{\mathbb{R}^{N}_{0}}\displaystyle\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\overline{(W_{\psi}^{\bm{\alpha},\lambda}{g})(\bm{a},\bm{b})}\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}$
$\displaystyle=\displaystyle\int_{\mathbb{R}^{N}_{0}}\left(\int_{\mathbb{R}^{N}}\sqrt{|\bm{a}|_{m}}{(\sqrt{2\pi})^{N}}e_{\bm{\alpha},\lambda^{2}}(\bm{a}\bm{\xi})\overline{(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{a}\bm{\xi})}(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})\overline{\sqrt{|\bm{a}}|_{m}(\sqrt{2\pi})^{N}e_{\bm{\alpha},\lambda^{2}}(\bm{a}\bm{\xi})\overline{(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{a}\bm{\xi})}(\mathfrak{F}_{\bm{\alpha},\lambda}g)(\bm{\xi})}d\bm{\xi}\right)\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}$
$\displaystyle=(2\pi)^{N}\displaystyle\int_{\mathbb{R}^{N}_{0}}\displaystyle\int_{\mathbb{R}^{N}}\overline{(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{a}\bm{\xi})}(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{a}\bm{\xi})(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})\overline{(\mathfrak{F}_{\bm{\alpha},\lambda}g)(\bm{\xi})}d\bm{\xi}\frac{d\bm{a}}{|\bm{a}|_{m}}$
$\displaystyle=\int_{\mathbb{R}^{N}}(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})\overline{(\mathfrak{F}_{\bm{\alpha},\lambda}g)(\bm{\xi})}\left(\displaystyle\int_{\mathbb{R}^{N}_{0}}(2\pi)^{N}(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{a}\bm{\xi})\overline{(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{a}\bm{\xi})}\frac{d\bm{a}}{|\bm{a}|_{m}}\right)d\bm{\xi}.$

Thus we have

$\displaystyle\displaystyle\int_{\mathbb{R}^{N}_{0}}\displaystyle\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\overline{(W_{\psi}^{\bm{\alpha},\lambda}{g})(\bm{a},\bm{b})}\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}$
$\displaystyle=\displaystyle\int_{\mathbb{R}^{N}}C_{\bm{\alpha},\lambda}(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})\overline{(\mathfrak{F}_{\bm{\alpha},\lambda}g)(\bm{\xi})}d\bm{\xi}$
$\displaystyle=C_{\bm{\alpha},\lambda}\left\langle(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi}),(\mathfrak{F}_{\bm{\alpha},\lambda}g)(\bm{\xi})\right\rangle_{L^{2}(\mathbb{R}^{N})}.$

Again, applying Parseval’s inequality ([12], Theorem 3.3), we get the required
result. Fubini’s theorem has been used to justify changing the order of
integration. ∎

######  Corollary 3.1.

Suppose $\psi$ be an admissible wavelet and for any arbitrary function $f\in
L^{2}(\mathbb{R}^{N})$. Then

$\displaystyle\int_{\mathbb{R}^{N}_{0}}\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\overline{(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})}\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}=C_{\bm{\alpha},\lambda}\|f\|^{2}_{L^{2}(\mathbb{R}^{N})}.$

######  Theorem 3.3.

(Reconstruction formula) Suppose there exist an arbitrary function $f\in
L^{2}(\mathbb{R}^{N})$ and an wavelet admissible function $\psi.$ Also let
$C_{\bm{\alpha},\lambda}\neq 0$ as defined in equation (19). The
reconstruction formula for $f$ is then as follows:

$\displaystyle
f(\bm{t})=\frac{1}{C_{\bm{\alpha},\lambda}}\int_{\mathbb{R}^{N}_{0}}\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}.$

###### Proof.

We have

$\displaystyle\left\langle
W_{\psi}^{\bm{\alpha},\lambda}{f}(\bm{a},\bm{b}),W_{\psi}^{\bm{\alpha},\lambda}{g}(\bm{a},\bm{b})\right\rangle_{L^{2}(\mathbb{R}^{N}_{0}\times\mathbb{R}^{N})}$
$\displaystyle=\int_{\mathbb{R}^{N}_{0}}\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\overline{(W_{\psi}^{\bm{\alpha},\lambda}{g}(\bm{a},\bm{b}))}\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}$
$\displaystyle=\int_{\mathbb{R}^{N}_{0}}\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\overline{\int_{\mathbb{R}^{N}}g(\bm{t})\overline{\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})}d\bm{t}}\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}$
$\displaystyle=\int_{\mathbb{R}^{N}_{0}}\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\int_{\mathbb{R}^{N}}\overline{g(\bm{t})}\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})d\bm{t}\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}$
$\displaystyle=\int_{\mathbb{R}^{N}}\overline{g(\bm{t})}\int_{\mathbb{R}^{N}_{0}}\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}d\bm{t}$
$\displaystyle=\left\langle\int_{\mathbb{R}^{N}_{0}}\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}},g(\bm{t})\right\rangle_{L^{2}(\mathbb{R}^{N})}.$

Using theorem (3.2)

$\displaystyle\langle f,g\rangle_{L^{2}(\mathbb{R}^{N})}$ $\displaystyle=$
$\displaystyle\frac{1}{C_{\bm{\alpha},\lambda}}\left\langle(W_{\psi}^{\bm{\alpha},\lambda}{f}),(W_{\psi}^{\bm{\alpha},\lambda}{g})\right\rangle_{L^{2}(\mathbb{R}^{N}_{0}\times\mathbb{R}^{N})}$
$\displaystyle=$
$\displaystyle\frac{1}{C_{\bm{\alpha},\lambda}}\left\langle\int_{\mathbb{R}^{N}_{0}}\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}},g(\bm{t})\right\rangle_{L^{2}(\mathbb{R}^{N})}$
$\displaystyle\Rightarrow\quad f(\bm{t})$ $\displaystyle=$
$\displaystyle\frac{1}{C_{\bm{\alpha},\lambda}}\int_{\mathbb{R}^{N}_{0}}\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}.$

Hence, the theorem follows. ∎

Now, we will derive a theorem which gives reproducing kernel for the range of
the MFrWT.

######  Theorem 3.4.

(Reproducing kernel for the range of MFrWT) Let $\psi$ be a wavelet. Also let
$C_{\alpha,\lambda}\neq 0$ as defined in equation (19) and
$(\bm{a}_{0},\bm{b}_{0})\in\mathbb{R}^{N}\times\mathbb{R}^{N}_{0}$. Then,
$F\in
L^{2}\left(\mathbb{R}^{N}\times\mathbb{R}^{N}_{0},\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}\right)$
be the MFrWT of some $f\in L^{2}(\mathbb{R}^{N})$ iff

$\displaystyle
F(\bm{a}_{0},\bm{b}_{0})=\int_{\mathbb{R}^{N}_{0}}\int_{\mathbb{R}^{N}}F(\bm{a},\bm{b})K_{\bm{\alpha},\lambda,\mathbf{\Psi}}(\bm{a}_{0},\bm{b}_{0};\bm{a},\bm{b})\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}},$

where
$K_{\bm{\alpha},\lambda,\mathbf{\Psi}}(\bm{a}_{0},\bm{b}_{0}:\bm{a},\bm{b})$
is the reproducing kernel which satisfies,

$\displaystyle
K_{\bm{\alpha},\lambda,\mathbf{\Psi}}(\bm{a}_{0},\bm{b}_{0};\bm{a},\bm{b})=\frac{1}{C_{\alpha,\lambda}}\int_{\mathbb{R}^{N}}\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})\overline{\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a}_{0},\bm{b}_{0}}(\bm{t})}d\bm{t}.$

###### Proof.

Suppose for an arbitrary function $f\in L^{2}(\mathbb{R}^{N})$ be such that
$W_{\psi}^{\bm{\alpha},\lambda}{f}=F$, then we can write,

$\displaystyle F(\bm{a}_{0},\bm{b}_{0})$ $\displaystyle=$
$\displaystyle(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a}_{0},\bm{b}_{0})$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}f(\bm{t})\overline{\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a}_{0},\bm{b}_{0}}(\bm{t})}d\bm{t}$
$\displaystyle=$
$\displaystyle\frac{1}{C_{\bm{\alpha},\lambda}}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}_{0}}\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}\overline{\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a}_{0},\bm{b}_{0}}(\bm{t})}d\bm{t}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}_{0}}\int_{\mathbb{R}^{N}}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\left(\frac{1}{C_{\bm{\alpha},\lambda}}\int_{\mathbb{R}^{N}}\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})\overline{\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a}_{0},\bm{b}_{0}}(\bm{t})}d\bm{t}\right)\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}.$

Hence,

$\displaystyle
F(\bm{a}_{0},\bm{b}_{0})=\int_{\mathbb{R}^{N}_{0}}\int_{\mathbb{R}^{N}}F(\bm{a},\bm{b})K_{\bm{\alpha},\lambda,\mathbf{\Psi}}(\bm{a}_{0},\bm{b}_{0};\bm{a},\bm{b})\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}},$

where $\displaystyle
K_{\bm{\alpha},\lambda,\mathbf{\Psi}}(\bm{a}_{0},\bm{b}_{0};\bm{a},\bm{b})=\frac{1}{C_{\bm{\alpha},\lambda}}\displaystyle\int_{\mathbb{R}^{N}}\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})\overline{\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a}_{0},\bm{b}_{0}}(\bm{t})}d\bm{t}$.
On the other way round, Let’s assume that the context holds only for the
provided $F\in
L^{2}\left(\mathbb{R}^{N}\times\mathbb{R}^{N}_{0},\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}\right)$
then required $f$ is determined by

$\displaystyle\displaystyle\frac{1}{C_{\bm{\alpha},\lambda}}\int_{\mathbb{R}^{N}_{0}}\int_{\mathbb{R}^{N}}F(\bm{a},\bm{b})\mathbf{\Psi}_{\bm{\alpha},\lambda,\bm{a},\bm{b}}(\bm{t})\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}.$

This completes the proof. ∎

##  4 Heisenberg’s Uncertainty Inequality for MFrWT

The uncertainty principle, first proposed by a German physicist named Werner
Heisenberg in 1927, states that it is impossible to measure position and
momentum at any arbitrarily well localised state at the same time. That is,
measuring position without disrupting momentum is impossible, and vice versa.
[5]. Similarly, in terms of time and frequency, it is impossible to measure
time and frequency at any state simultaneously. Here we will derive
Heisenberg’s uncertainty inequality for MFrFT.

###  4.1 Relation between Classical and Fractional Fourier Transform

Now we derive a relationship between classical FT and MFrFT which will further
help us to prove Heisenberg’s uncertainity inequality for MFrFT

$\displaystyle(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})=\int_{\mathbb{R}^{N}}f(\bm{t})K_{\bm{\alpha},\lambda}(\bm{t},\bm{\xi})d\bm{t},$

here $\bm{\alpha}=(\alpha_{1},\alpha_{2},\alpha_{3},\cdots,\alpha_{N})$ and
$\bm{\xi}=(\xi_{1},\xi_{2},\xi_{3},\cdots,\xi_{N}).$
We have

$\displaystyle(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}f(\bm{t})\frac{c(\bm{\alpha}_{\lambda})}{{(\sqrt{2\pi})^{N}}}e_{\bm{\alpha},\lambda^{2}}(\bm{\xi})e_{\bm{\alpha},\lambda^{2}}(\bm{t})e^{-i\lambda^{2}\sum_{k=1}^{N}t_{k}\xi_{k}\csc\alpha_{k}}d\bm{t}$
$\displaystyle=$
$\displaystyle\frac{c(\bm{\alpha}_{\lambda})}{(\sqrt{2\pi})^{N}}e_{\bm{\alpha},\lambda^{2}}(\bm{\xi})\int_{\mathbb{R}^{N}}f(\bm{t})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){t_{k}}^{2}}e^{-i\lambda^{2}\sum_{k=1}^{N}t_{k}\xi_{k}\csc\alpha_{k}}d\bm{t}.$

Therefore,

$\displaystyle(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})$
$\displaystyle=$
$\displaystyle\frac{c(\bm{\alpha}_{\lambda})}{(\sqrt{2\pi})^{N}}e_{\bm{\alpha},\lambda^{2}}(\bm{\xi})\int_{\mathbb{R}^{N}}f(\bm{t})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){t_{k}}^{2}}e^{-i\lambda^{2}\sum_{k=1}^{N}t_{k}\xi_{k}\csc\alpha_{k}}d\bm{t}.$
(21)

As we know traditional Fourier transform is stated as

$\displaystyle\mathfrak{F}f(\bm{\xi})$ $\displaystyle=$
$\displaystyle\frac{1}{(\sqrt{2\pi})^{N}}\int_{\mathbb{R}^{N}}f(\bm{t})e^{-i\sum_{k=1}^{N}t_{k}\xi_{k}}d\bm{t},\quad\bm{\xi}=(\xi_{1},\xi_{2},\cdots\xi_{N})\in\mathbb{R}^{N}.$

Replacing $f(\bm{t})$ by
$f(\bm{t})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){t_{k}}^{2}}$, and on
putting $\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}$ in place of $\bm{\xi}$,
we get

$\displaystyle\mathfrak{F}(f(\bm{t})e^{i\lambda^{2}\sum_{k=1}^{{N}}a(\alpha_{k}){t_{k}}^{2}})\left(\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}\right)$
$\displaystyle=\frac{1}{(\sqrt{2\pi})^{N}}\int_{\mathbb{R}^{N}}f(\bm{t})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){t_{k}}^{2}}e^{-i\lambda^{2}\sum_{k=1}^{N}t_{k}\xi_{k}\csc\alpha_{k}}d\bm{t}.$
(22)

From equation (21) and (22), we have

$\displaystyle(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})=c(\bm{\alpha}_{\lambda})e_{\bm{\alpha},\lambda^{2}}(\bm{\xi})\mathfrak{F}(f(\bm{t})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){t_{k}}^{2}})\left(\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}\right).$

This gives

$\displaystyle\mathfrak{F}(f(\bm{t})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){t_{k}}^{2}})(\bm{\eta})=\frac{1}{c(\bm{\alpha}_{\lambda})e_{\bm{\alpha},\lambda^{2}}\left(\frac{1}{\lambda^{2}}\bm{\eta}\sin\bm{\alpha}\right)}(\mathfrak{F}_{\bm{\alpha},\lambda}f)\left(\frac{1}{\lambda^{2}}\bm{\eta}\sin\bm{\alpha}\right).$
(23)

Equation (23) gives the relation between the classical and fractional FT. For
the MFrFT, we now derive Heisenberg’s uncertainty principle.
If $f\in L^{2}(\mathbb{R}^{N})$ be any arbitrary function, then Heisenberg’s
uncertainty inequality for dimension ${N}$ is described by the following [28].

$\displaystyle{}\left(\int_{\mathbb{R}^{N}}{\|\bm{t}\|}^{2}\left|{f(\bm{t})}\right|^{2}d\bm{t}\right)\left(\int_{\mathbb{R}^{N}}{\|\bm{\eta}\|}^{2}|\mathfrak{F}f(\bm{\eta})|^{2}d\bm{\eta}\right)\geq\frac{{N}^{2}}{4}\|f\|^{4}_{L^{2}(\mathbb{R}^{N})}.$
(24)

Replace $f(\bm{t})$ by
$f(\bm{t})e^{i\lambda^{2}{\sum}_{k=1}^{N}a(\alpha_{k}){t_{k}}^{2}}$ and noting
that
$\left|f(\bm{t})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){t_{k}}^{2}}\right|=|f(\bm{t})|,$
we’ve

$\displaystyle{}\left(\int_{\mathbb{R}^{N}}{\|\bm{t}\|}^{2}|{f(\bm{t})|}^{2}d\bm{t}\right)\left(\int_{\mathbb{R}^{N}}{\|\bm{\eta}\|}^{2}|\mathfrak{F}(f(\bm{t})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){t_{k}}^{2}})(\bm{\eta})|^{2}d\bm{\eta}\right)\geq\frac{{N}^{2}}{4}\|f\|^{4}_{L^{2}(\mathbb{R}^{N})}.$
(25)

On putting
$\displaystyle{\bm{\eta}=\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}}$ in
$\displaystyle\int_{\mathbb{R}^{N}}{\|\bm{\eta}\|}^{2}|\mathfrak{F}(f(\bm{t})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){t_{k}}^{2}})(\bm{\eta})|^{2}d\bm{\eta}$
and using equation (23), we get

$\displaystyle\int_{\mathbb{R}^{N}}{\|\bm{\eta}\|}^{2}|\mathfrak{F}(f(\bm{t})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){t_{k}}^{2}})(\bm{\eta})|^{2}d\bm{\eta}=\frac{\lambda^{2N}}{|\sin\bm{\alpha}|_{m}}\int_{\mathbb{R}^{N}}{\left\|\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}\right\|^{2}}\left|\frac{1}{c(\bm{\alpha}_{\lambda})e_{\alpha,\lambda^{2}}(\bm{\xi})}(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})\right|^{2}d\bm{\xi}.$

Now, consider
$\displaystyle{\left\|\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}\right\|}=\left\|\left(\frac{\lambda^{2}\xi_{1}}{\sin\alpha_{1}},\frac{\lambda^{2}\xi_{2}}{\sin\alpha_{2}},\cdots,\frac{\lambda^{2}\xi_{N}}{\sin\alpha_{N}}\right)\right\|$.
Let $\displaystyle\frac{1}{\sin\alpha_{i}}=m_{i}$ for $i=1,2,\cdots,{N},$ then

$\displaystyle{\left\|\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}\right\|}$
$\displaystyle=\lambda^{2}\|\left(m_{1}\xi_{1},m_{2}\xi_{2},\cdots,m_{N}\xi_{N}\right)\|$
$\displaystyle=\lambda^{2}\left({m_{1}}^{2}\xi_{1}^{2}+{m_{2}}^{2}\xi_{2}^{2}+\cdots+{m_{N}}^{2}\xi_{N}^{2}\right)^{\frac{1}{2}}.$

Therefore,

$\displaystyle{}\left\|\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}\right\|\leq\lambda^{2}M\|\bm{\xi}\|,$
(27)

where $M^{2}=\max\\{m_{i}^{2}:i=1,2\cdots,{N}\\}.$
Therefore,

$\displaystyle{}\int_{\mathbb{R}^{N}}{\|\bm{\eta}\|}^{2}|\mathfrak{F}(f(\bm{t})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){t_{k}}^{2}})(\bm{\eta})|^{2}d\bm{\eta}\leq\frac{M^{2}}{|c(\bm{\alpha}_{\lambda})|^{2}}\frac{\lambda^{2N+4}}{\left|\sin\bm{\alpha}\right|_{m}}\int_{\mathbb{R}^{N}}{\|\bm{\xi}\|}^{2}|\mathfrak{F}_{\bm{\alpha},\lambda}f(\bm{\xi})|^{2}d\bm{\xi}.$
(28)

Using inequality (28) in (25), we get

$\displaystyle\left(\int_{\mathbb{R}^{N}}{\|\bm{t}\|}^{2}|{f(\bm{t})|}^{2}d\bm{t}\right)\left(\int_{\mathbb{R}^{N}}{\|\bm{\xi}\|}^{2}|\mathfrak{F}_{\bm{\alpha},\lambda}f(\bm{\xi})|^{2}d\bm{\xi}\right)\geq\frac{|c(\bm{\alpha}_{\lambda})|^{2}|\sin\bm{\alpha}|_{m}}{M^{2}(\lambda^{2})^{{N}+2}}\frac{{N}^{2}}{4}\|f\|^{4}_{L^{2}(\mathbb{R}^{N})}.$

Which implies

$\displaystyle{}\left(\int_{\mathbb{R}^{N}}{\|\bm{t}\|}^{2}|{f(\bm{t})|}^{2}d\bm{t}\right)\left(\int_{\mathbb{R}^{N}}{\|\bm{\xi}\|}^{2}|\mathfrak{F}_{\bm{\alpha},\lambda}f(\bm{\xi})|^{2}d\bm{\xi}\right)\geq
P_{\bm{\alpha},\lambda}\frac{{N}^{2}}{4}\|f\|^{4}_{L^{2}(\mathbb{R}^{N})},$
(29)

where

$\displaystyle
P_{\bm{\alpha},\lambda}=\frac{|c(\bm{\alpha}_{\lambda})|^{2}|\sin\bm{\alpha}|_{m}}{M^{2}(\lambda^{2})^{{N}+2}}.$
(30)

######  Lemma 4.1.

Let $\psi$ be an admissible wavelet and for any function $f\in
L^{2}(\mathbb{R}^{N}).$ Then,

$\displaystyle\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}\|\bm{\xi}\|^{2}|\mathfrak{F}_{\bm{\alpha},\lambda}((W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\cdot))(\bm{\xi})|^{2}d\bm{\xi}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}$
$\displaystyle=$ $\displaystyle
C_{\bm{\alpha},\lambda}\int_{\mathbb{R}^{N}}\|\bm{\xi}\|^{2}|(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})|^{2}d\bm{\xi},$

where $C_{\bm{\alpha},\lambda}$ is as defined in equation (19).

###### Proof.

Using lemma (3.2) and theorem (3.2), we get

$\displaystyle\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}\|\bm{\xi}\|^{2}$
$\displaystyle|\mathfrak{F}_{\bm{\alpha},\lambda}((W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b}))(\bm{\xi})|^{2}d\bm{\xi}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}$
$\displaystyle=\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}\|\bm{\xi}\|^{2}{|\bm{a}|_{m}({2\pi})^{N}}(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{a}\bm{\xi})(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})\overline{(\mathfrak{F}_{\bm{\alpha},\lambda}\psi)(\bm{a}\bm{\xi})}\overline{\mathfrak{F}_{\bm{\alpha},\lambda}f(\bm{\xi})}d\bm{\xi}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}$
$\displaystyle=C_{\bm{\alpha},\lambda}\int_{\mathbb{R}^{N}}\|\bm{\xi}\|^{2}|(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})|^{2}d\bm{\xi}.$

This concludes the proof. ∎

Equation (29) is the Heisenberg’s uncertainty inequality for the MFrFT. Now we
prove the Heisenberg’s uncertainty principle for the MFrWT following the idea
of [24],[29].

######  Theorem 4.1.

(Heisenberg’s inequality of uncertainty for MFrWT) Let $\psi$ be an admissible
wavelet and for any function $f\in L^{2}(\mathbb{R}^{N}),$ the MFrWT of $f$
with respect to the variable $\bm{\alpha}$ is specified by
$(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})$ . Then

$\displaystyle\left(\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}\|\bm{b}\|^{2}|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}d\bm{b}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}\right)\left(\int_{\mathbb{R}^{N}}\|\bm{\xi}\|^{2}|(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})|^{2}d\bm{\xi}\right)\geq
P_{\bm{\alpha},\lambda}C_{\bm{\alpha},\lambda}\frac{{N}^{2}}{4}\|f\|^{2}_{L^{2}(\mathbb{R}^{N})},$

where $C_{\bm{\alpha},\lambda}$ and $P_{\bm{\alpha},\lambda}$ are defined in
equation (19) and equation (30) respectively.

###### Proof.

By substituting $(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\cdot)$ for
$f(\cdot)$ in equation (29), we obtain

$\displaystyle\left(\int_{\mathbb{R}^{N}}\|\bm{b}\|^{2}|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}d\bm{b}\right)\left(\int_{\mathbb{R}^{N}}\|\bm{\xi}\|^{2}|\mathfrak{F}_{\bm{\alpha},\lambda}((W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},.))(\bm{\xi})|^{2}d\bm{\xi}\right)\geq
P_{\bm{\alpha},\lambda}\frac{{N}^{2}}{4}\|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},.)\|^{4}_{L^{2}(\mathbb{R}^{N})}.$

Taking the square root of the above equation on both sides yields,

$\displaystyle\left(\int_{\mathbb{R}^{N}}\|\bm{b}\|^{2}|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}d\bm{b}\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{N}}\|\bm{\xi}\|^{2}|\mathfrak{F}_{\bm{\alpha},\lambda}((W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b}))(\bm{\xi})|^{2}d\bm{\xi}\right)^{\frac{1}{2}}\geq{P^{\frac{1}{2}}_{\bm{\alpha},\lambda}}\frac{{N}}{2}\int_{\mathbb{R}^{N}}|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}d\bm{b}.$

Now, we Integrate above equation with respect to the measure
$\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}$, we’ve

$\displaystyle\int_{\mathbb{R}_{0}^{N}}\left(\int_{\mathbb{R}^{N}}\|\bm{b}\|^{2}|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}d\bm{b}\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{N}}\|\bm{\xi}\|^{2}|\mathfrak{F}_{\bm{\alpha},\lambda}(W_{\psi}^{\bm{\alpha},\lambda}{f}(\bm{a},\bm{b}))(\bm{\xi})|^{2}d\bm{\xi}\right)^{\frac{1}{2}}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}$
$\displaystyle\geq{P^{\frac{1}{2}}_{\bm{\alpha},\lambda}}\frac{{N}}{2}\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}\frac{d\bm{a}d\bm{b}}{|\bm{a}|_{m}^{2}}.$

Using Holder’s inequality and corollary (3.1), we have

$\displaystyle\left(\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}\|\bm{b}\|^{2}|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}d\bm{b}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}\right)^{\frac{1}{2}}\left(\displaystyle\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}\|\bm{\xi}\|^{2}|\mathfrak{F}_{\bm{\alpha},\lambda}((W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b}))(\bm{\xi})|^{2}d\bm{\xi}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}\right)^{\frac{1}{2}}$
$\displaystyle\geq{P^{\frac{1}{2}}_{\bm{\alpha},\lambda}}\frac{{N}}{2}C_{\bm{\alpha},\lambda}\|f\|^{2}_{L^{2}(\mathbb{R}^{N})}.$

Squaring both sides and applying lemma (4.1), we obtain

$\displaystyle\left(\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}\|\bm{b}\|^{2}|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}d\bm{b}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}\right)\left(C_{\bm{\alpha},\lambda}\int_{\mathbb{R}^{N}}\|\bm{\xi}\|^{2}|(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})|^{2}d\bm{\xi}\right)\geq
P_{\bm{\alpha},\lambda}\frac{{N}^{2}}{4}C^{2}_{\bm{\alpha},\lambda}\|f\|^{4}_{L^{2}(\mathbb{R}^{N})},$

i.e.,

$\displaystyle\left(\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}\|\bm{b}\|^{2}|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}d\bm{b}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}\right)\left(\int_{\mathbb{R}^{N}}\|\bm{\xi}\|^{2}|(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})|^{2}d\bm{\xi}\right)\geq
P_{\bm{\alpha},\lambda}C_{\bm{\alpha},\lambda}\frac{{N}^{2}}{4}\|f\|^{4}_{L^{2}(\mathbb{R}^{N})}.$

This concludes the proof for this section. ∎

###  4.2 Logarithmic Uncertainty Principle for MFrWT

Basic idea about logarithmic uncertainty principle for fractional wavelet
transform is given by Mawardi Bahri in [2]. In this context, we formally
derive a logarithmic uncertainty principle for the MFrFT using MFrFT
properties and the logarithmic uncertainty principle for the fractional
Fourier transform.
In [3], the logarithmic uncertainty inequality for classical Fourier transform
is given by

$\displaystyle{}\int_{\mathbb{R}^{N}}\ln\|\mathbf{x}\||f(\mathbf{x})|^{2}d\mathbf{x}+\int_{\mathbb{R}^{N}}\ln\|\bm{\eta}\||\mathfrak{F}f(\bm{\eta})|^{2}d\bm{\eta}\geq
D\int_{\mathbb{R}^{N}}|f(\mathbf{x})|^{2}d\mathbf{x},$ (31)

where $D=\psi\left(\frac{{N}}{4}\right)-\ln 2,$
$\psi(t)=\frac{d}{dt}[\ln\Gamma(t)]$.

Replacing $f(\mathbf{x})$ with
$f(\mathbf{x})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){x_{k}}^{2}}$ in
inequality (31) and observing that
$\left|f(\mathbf{x})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){x_{k}}^{2}}\right|=\left|f(\mathbf{x})\right|,$
we get

$\displaystyle\int_{\mathbb{R}^{N}}\ln\|\mathbf{x}\|\left|f(\mathbf{x})\right|^{2}d\mathbf{x}+\int_{\mathbb{R}^{N}}\ln\|\bm{\eta}\||\mathfrak{F}(f(\mathbf{x})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){x_{k}}^{2}})(\bm{\eta})|^{2}d\bm{\eta}\geq
D\int_{\mathbb{R}^{N}}|f(\mathbf{x})|^{2}d\mathbf{x}.$ (32)

Put $\bm{\eta}=\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}$ in
$\displaystyle{\int_{\mathbb{R}^{N}}\ln\|\bm{\eta}\||\mathfrak{F}f(\mathbf{x})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){x_{k}}^{2}}(\bm{\eta})|^{2}d\bm{\eta}}$
and using equation (23), we get

$\displaystyle\int_{\mathbb{R}^{N}}\ln\|\bm{\eta}\||\mathfrak{F}(f(\mathbf{x})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){x_{k}}^{2}})(\bm{\eta})|^{2}d\bm{\eta}$
$\displaystyle=\frac{\lambda^{2N}}{\left|\sin\bm{\alpha}\right|_{m}}\int_{\mathbb{R}^{N}}\ln\left\|\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}\right\|\left|\frac{1}{c(\bm{\alpha}_{\lambda})e_{\bm{\alpha},\lambda^{2}}(\bm{\xi})}\mathfrak{F}_{\bm{\alpha},\lambda}f(\bm{t})(\bm{\xi})\right|^{2}d\bm{\xi}$
$\displaystyle=\frac{1}{|c(\bm{\alpha}_{\lambda})|^{2}}\frac{\lambda^{2N}}{\left|\sin\bm{\alpha}\right|_{m}}\int_{\mathbb{R}^{N}}\ln\left\|\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}\right\|\left|(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})\right|^{2}d\bm{\xi}.$

From equation (32), we get

$\displaystyle\int_{\mathbb{R}^{N}}\ln\|\mathbf{x}\||f(\mathbf{x})|^{2}d\mathbf{x}+\frac{1}{|c(\bm{\alpha}_{\lambda})|^{2}}\frac{\lambda^{2N}}{\left|\sin\bm{\alpha}\right|_{m}}\int_{\mathbb{R}^{N}}\ln\left\|\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}\right\||(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})|^{2}d\bm{\xi}\geq
D\int_{\mathbb{R}^{N}}|f(\mathbf{x})|^{2}d\mathbf{x}.$

Then we have

$\displaystyle\int_{\mathbb{R}^{N}}\ln\|\mathbf{x}\||f(\mathbf{x})|^{2}d\mathbf{x}+P^{\prime}_{\bm{\alpha},\lambda}\int_{\mathbb{R}^{N}}\ln\left\|\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}\right\||(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})|^{2}d\bm{\xi}\geq
D\int_{\mathbb{R}^{N}}|f(\mathbf{x})|^{2}d\mathbf{x},$ (33)

where

$P^{\prime}_{\bm{\alpha},\lambda}=\frac{\lambda^{2N}}{{|c(\bm{\alpha}_{\lambda})|^{2}}|\sin\bm{\alpha}|_{m}}.$
(34)

From equation (27), we know

$\displaystyle\left\|\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}\right\|\leq\lambda^{2}M\|\bm{\xi}\|.$

Taking $\ln$ on both sides of above equation, we get

$\displaystyle\ln\left\|\frac{\lambda^{2}\bm{\xi}}{\sin\bm{\alpha}}\right\|\leq\ln(\lambda^{2}M)+\ln\|\bm{\xi}\|.$

Therefore, equation(33) can be re-written as,

$\displaystyle\int_{\mathbb{R}^{N}}\ln\|\mathbf{x}\||f(\mathbf{x})|^{2}d\mathbf{x}+P^{\prime}_{\bm{\alpha},\lambda}\int_{\mathbb{R}^{N}}[\ln(\lambda^{2}M)+\ln\|\bm{\xi}\|]|(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})|^{2}d\bm{\xi}\geq
D\int_{\mathbb{R}^{N}}|f(\mathbf{x})|^{2}d\mathbf{x}$
$\displaystyle\int_{\mathbb{R}^{N}}\ln\|\mathbf{x}\||f(\mathbf{x})|^{2}d\mathbf{x}+P^{\prime}_{\bm{\alpha},\lambda}\ln(\lambda^{2}M)\int_{\mathbb{R}^{N}}|(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})|^{2}d\bm{\xi}+P^{\prime}_{\bm{\alpha},\lambda}\int_{\mathbb{R}^{N}}\ln\|\bm{\xi}\||(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})|^{2}d\bm{\xi}\geq
D\int_{\mathbb{R}^{N}}|f(\mathbf{x})|^{2}d\mathbf{x}.$

Using Parseval’s relation, we get

$\displaystyle\int_{\mathbb{R}^{N}}\ln\|\mathbf{x}\||f(\mathbf{x})|^{2}d\mathbf{x}+P^{\prime}_{\bm{\alpha},\lambda}\int_{\mathbb{R}^{N}}\ln\|\bm{\xi}\||(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})|^{2}d\bm{\xi}\geq(D-P^{\prime}_{\alpha,\lambda}\ln(\lambda^{2}M))\int_{\mathbb{R}^{N}}|f(\mathbf{x})|^{2}d\mathbf{x}.$
(35)

With the help of logarithmic uncertainty inequality for MFrFT (35), we will
prove the following for the MFrWT.

######  Theorem 4.2.

(Logarithmic uncertainty inequality for MFrWT) Let $\psi$ be an admissible
wavelet and for any $f\in L^{2}(\mathbb{R}^{N})$ and for some $M>0,$ Let MFrWT
of $f$ with parameter $\bm{\alpha}$ is given by
$(W_{\psi}^{\bm{\alpha},\lambda}f)(\bm{a},\bm{b}).$ Then

$\displaystyle\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}\ln\|\bm{b}\||(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}d\bm{b}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}+C_{\bm{\alpha},\lambda}P^{\prime}_{\bm{\alpha},\lambda}\int_{\mathbb{R}^{N}}\ln\|\bm{\xi}\||$
$\displaystyle\mathfrak{F}_{\bm{\alpha},\lambda}f)|^{2}(\bm{\xi})d\bm{\xi}$
$\displaystyle\geq\left(D-P^{\prime}_{\bm{\alpha,\lambda}}\ln(\lambda^{2}M)\right)C_{\alpha,\lambda}\|f\|_{L^{2}(\mathbb{R}^{N})}^{2},$

where $P^{\prime}_{\bm{\alpha},\lambda}$ is given by equation (34).

###### Proof.

Replace $f(.)$ with $(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},.)$ on either
side of equation (35), we have

$\displaystyle\int_{\mathbb{R}^{N}}\ln\|\bm{b}\||(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}d\bm{b}+P^{\prime}_{\bm{\alpha},\lambda}\int_{\mathbb{R}^{N}}\ln\left\|\bm{\xi}\right\||(\mathfrak{F}_{\bm{\alpha},\lambda}(W_{\psi}^{\bm{\alpha},\lambda}{f})$
$\displaystyle(\bm{a},\bm{b}))(\bm{\xi})|^{2}d\bm{\xi}$
$\displaystyle\geq\left(D-P^{\prime}_{\bm{\alpha,\lambda}}\ln(\lambda^{2}M)\right)\int_{\mathbb{R}^{N}}|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}d\bm{b}.$

Now, we integrate above equation with respect to measure
$\frac{d\bm{a}}{|\bm{a}|_{m}^{2}},$ we get

$\displaystyle\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}\ln\|\bm{b}\||(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}d\bm{b}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}+P^{\prime}_{\bm{\alpha},\lambda}\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}$
$\displaystyle\ln\left\|\bm{\xi}\right\||(\mathfrak{F}_{\bm{\alpha},\lambda}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b}))(\bm{\xi})|^{2}d\bm{\xi}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}$
$\displaystyle\geq\left(D-P^{\prime}_{\bm{\alpha,\lambda}}\ln(\lambda^{2}M)\right)\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}d\bm{b}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}.$
(36)

Hence, using corollary (3.1) and lemma (3.2) in equation (4.2), we have

$\displaystyle\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}\ln\|\bm{b}\||(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})|^{2}d\bm{b}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}+P^{\prime}_{\bm{\alpha},\lambda}C_{\bm{\alpha},\lambda}\int_{\mathbb{R}^{N}}\ln\|\bm{\xi}\|$
$\displaystyle|(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\xi})|^{2}d\bm{\xi}$
$\displaystyle\geq\left(D-P^{\prime}_{\bm{\alpha,\lambda}}\ln(\lambda^{2}M)\right)C_{\alpha,\lambda}\int_{\mathbb{R}^{N}}|f(\mathbf{x})|^{2}d\mathbf{x}.$

where
$M^{2}=\displaystyle\max\left\\{\frac{1}{\sin^{2}\alpha_{i}}:i=1,2,\cdots,{N}\right\\}.$
This concludes the proof for this section. ∎

###  4.3 Local Uncertainty Inequality for MFrWT

In harmonic analysis, Heisenberg’s uncertainty principle states that if any
arbitrary function $f$ is specifically restricted, the corresponding Fourier
transform of $f$ cannot be reduced to the neighborhood of a point, But this
never prevents the Fourier transform of $f$ from being focused approximately
at two or more distinct points. In fact, it cannot occur either, and this is
the purpose of local uncertainty inequalities to make this specific. The local
uncertainty principle for the classical Fourier transform can be stated using
the definition of the traditional Fourier transform for
$\alpha=\frac{\pi}{2},$ as follows [30] :
(i)There exists a invariable $A_{\theta}$ for $0<\theta<\frac{{N}}{2}$ such
that for every measurable subsets $K$ of $\mathbb{R}^{N}$ and for all $f\in
L^{2}(\mathbb{R}^{N})$

$\displaystyle{}\int_{K}|(\mathfrak{F}f)(\bm{\xi})|^{2}d\bm{\xi}\leq
A_{\theta}(\lambda(K))^{\frac{2\theta}{N}}\|\|\mathbf{x}\|^{\theta}f\|^{2}_{L^{2}(\mathbb{R}^{N})}.$
(37)

(ii)There exists a invariable $A_{\theta}$ for $\theta>\frac{N}{2}$ such that
for every measurable subsets $K$ of $\mathbb{R}^{N}$ and for all $f\in
L^{2}(\mathbb{R}^{N})$

$\displaystyle{}\int_{K}|(\mathfrak{F}f)(\bm{\xi})|^{2}d\bm{\xi}\leq
A_{\theta}\lambda(K)\|f\|^{2-\frac{N}{\theta}}_{L^{2}(\mathbb{R}^{N})}\|\|\mathbf{x}\|^{\theta}f\|^{\frac{N}{\theta}}_{L^{2}(\mathbb{R}^{N})}.$
(38)

Replacing $f(\mathbf{x})$ by
$f(\mathbf{x})e^{i\lambda^{2}\sum_{k=1}^{N}a(\alpha_{k}){x_{k}}^{2}}$ in
equation (37) and using equation (23) we have

$\displaystyle\int_{K}\left|(\mathfrak{F}_{\bm{\alpha},\lambda}f)\left(\frac{1}{\lambda^{2}}\bm{\xi}\sin\bm{\alpha}\right)\right|^{2}d\bm{\xi}\leq|c(\bm{\alpha}_{\lambda})|^{2}A_{\theta}(\lambda(K))^{\frac{2\theta}{N}}\|\|\mathbf{x}\|^{\theta}f\|^{2}_{L^{2}(\mathbb{R}^{N})}.$
(39)

Put $\frac{1}{\lambda^{2}}\bm{\xi}\sin\bm{\alpha}=\bm{\eta}$ in equation (39),
we have

$\displaystyle\frac{(\lambda^{2})^{N}}{|\sin\bm{\alpha}|_{m}}\int_{K}|(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\eta})|^{2}d\bm{\eta}\leq|c(\bm{\alpha}_{\lambda})|^{2}A_{\theta}(\lambda(K))^{\frac{2\theta}{N}}\|\|\mathbf{x}\|^{\theta}f\|^{2}_{L^{2}(\mathbb{R}^{N})}.$

That is,

$\displaystyle\int_{K}|(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\eta})|^{2}d\bm{\eta}\leq|c(\bm{\alpha}_{\lambda})|^{2}A_{\theta}\frac{|\sin\bm{\alpha}|_{m}}{(\lambda^{2})^{N}}(\lambda(K))^{\frac{2\theta}{N}}\|\|\mathbf{x}\|^{\theta}f\|^{2}_{L^{2}(\mathbb{R}^{N})}.$
(40)

Similarly, from equation (38), we get

$\displaystyle\int_{K}|(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\eta})|^{2}d\bm{\eta}\leq|c(\bm{\alpha}_{\lambda})|^{2}A_{\theta}\frac{|\sin\bm{\alpha}|_{m}}{(\lambda^{2})^{N}}\lambda(K)\|f\|^{2-\frac{N}{\theta}}_{L^{2}(\mathbb{R}^{N})}\|\|\mathbf{x}\|^{\theta}f\|^{\frac{N}{\theta}}_{L^{2}(\mathbb{R}^{N})}.$
(41)

Now, we will illustrate the major outcome of this section.

######  Theorem 4.3.

(Local uncertainty inequality for MFrWT) If $\psi$ is an admissible wavelet
and $\theta$ be such that $0<\theta<\frac{N}{2}.$ Then, for every measurable
subsets $K$ of $\mathbb{R}^{N},$ there exist constant $A_{\theta}$ and for all
$f\in L^{2}(\mathbb{R}^{N})$ such that

$\displaystyle\int_{K}|(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\eta})|^{2}d\bm{\eta}\leq\frac{|c(\bm{\alpha}_{\lambda})|^{2}}{C_{\bm{\alpha},\lambda}}A_{\theta}\frac{|\sin\bm{\alpha}|_{m}}{(\lambda^{2})^{N}}(\lambda(K))^{\frac{2\theta}{N}}\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}\|\bm{b}\|^{2\theta}\left|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\right|^{2}d\bm{b}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}.$

###### Proof.

By substituting $W_{\psi}^{\bm{\alpha},\lambda}{f}(\bm{a},\cdot)$ for
$f(\cdot)$ in equation (40), we obtain

$\displaystyle\int_{K}|\mathfrak{F}_{\bm{\alpha},\lambda}((W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b}))(\bm{\eta})|^{2}d\bm{\eta}\leq|c(\bm{\alpha}_{\lambda})|^{2}A_{\theta}\frac{|\sin\bm{\alpha}|_{m}}{(\lambda^{2})^{N}}(\lambda(K))^{\frac{2\theta}{N}}\|\|\bm{b}\|^{\theta}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\|^{2}_{L^{2}(\mathbb{R}^{N})}.$

Now we integrate either side of above equation with respect to
$\frac{d\bm{a}}{|\bm{a}|_{m}^{2}},$ we get

$\displaystyle\int_{\mathbb{R}_{0}^{N}}\int_{K}|\mathfrak{F}_{\bm{\alpha},\lambda}((W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b}))(\bm{\eta})|^{2}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}d\bm{\eta}\leq|c(\bm{\alpha}_{\lambda})|^{2}A_{\theta}\frac{|\sin\bm{\alpha}|_{m}}{(\lambda^{2})^{N}}(\lambda(K))^{\frac{2\theta}{N}}\int_{\mathbb{R}_{0}^{N}}\|\|\bm{b}\|^{\theta}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\|^{2}_{L^{2}(\mathbb{R}^{N})}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}},$

and using theorem (3.2),

$\displaystyle{C_{\bm{\alpha},\lambda}}\int_{K}|(\mathfrak{F}_{\bm{\alpha},\lambda}f)|^{2}d\bm{\eta}\leq|c(\bm{\alpha}_{\lambda})|^{2}A_{\theta}\frac{|\sin\bm{\alpha}|_{m}}{(\lambda^{2})^{N}}(\lambda(K))^{\frac{2\theta}{N}}\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}\left|\|\bm{b}\|^{\theta}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\right|^{2}d\bm{b}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}.$

This gives

$\displaystyle\int_{K}|(\mathfrak{F}_{\bm{\alpha},\lambda}f)|^{2}d\bm{\eta}\leq\frac{|c(\bm{\alpha}_{\lambda})|^{2}}{C_{\bm{\alpha},\lambda}}A_{\theta}\frac{|\sin\bm{\alpha}|_{m}}{(\lambda^{2})^{N}}(\lambda(K))^{\frac{2\theta}{N}}\int_{\mathbb{R}_{0}^{N}}\int_{\mathbb{R}^{N}}\|\bm{b}\|^{2\theta}\left|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\bm{b})\right|^{2}d\bm{b}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}.$

This concludes the proof. ∎

The following theorem expresses the local uncertainty inequality for MFrWT
when $\theta\textgreater\frac{N}{2}.$

######  Theorem 4.4.

(Local uncertainty inequality for MFrWT ) Assume $\psi$ is an admissible
wavelet and $\theta$ is such that $\theta\textgreater\frac{N}{2}$. Then, for
every measurable subset $K$ of $\mathbb{R}^{N}$, there exist a constant
$A_{\theta}$ and for all $f\in L^{2}(\mathbb{R}^{N})$ such that

$\displaystyle\int_{K}|(\mathfrak{F}_{\bm{\alpha},\lambda}f)(\bm{\eta})|^{2}d\bm{\eta}\leq\frac{|c(\bm{\alpha}_{\lambda})|^{2}}{C_{\bm{\alpha},\lambda}}A_{\theta}\frac{|\sin\bm{\alpha}|_{m}}{(\lambda^{2})^{N}}\lambda(K)\int_{\mathbb{R}_{0}^{N}}\|(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\cdot)\|^{2-\frac{N}{\theta}}_{L^{2}(\mathbb{R}^{N})}\|\|\cdot\|^{\theta}(W_{\psi}^{\bm{\alpha},\lambda}{f})(\bm{a},\cdot)\|_{L^{2}(\mathbb{R}^{N})}^{\frac{N}{\theta}}\frac{d\bm{a}}{|\bm{a}|_{m}^{2}}.$

###### Proof.

Using equation (41), the proof follows similar to that of theorem 4.3. ∎

##  5 Conclusions

We offer a novel definition of the wavelet transformation of a function
specified in $\mathbb{R}^{N}$, i.e. MFrWT, and investigate some of its
fundamental aspects, such as linearity, anti-linearity, parity, conjugation,
and so on, in this study. For the image space of the suggested transformation,
we established the inner product relation and inversion formula for MFrWT with
reproducible kernel function. Finally, Heisenberg’s uncertainty inequality and
the logarithmic uncertainty principle are obtained from the relationship
between classical FT and MFrFT. The commonalities between different transform
domains can be better understood using logarithmic, Heisenberg, and local
uncertainty principles. The three uncertainty principles can be useful in the
future because they are features of signal processing.

##  6 Acknowledgement

The work is partly supported by Council of Scientific and Industrial Research
(CSIR), New Delhi, India (File No. 09/1023(0035)/2020-EMR-I) and UGC File No.
16-9 (June 2017)/2018(NET/CSIR), New Delhi, India.

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