Novel Uncertainty Principles Related to Quaternion Linear Canonical S-Transform

Abstract

In this work, we introduce the quaternion linear canonical S-transform, which is a generalization of the linear canonical S-transform using quaternion. We investigate its properties and seek the different types of uncertainty principles related to this transformation. The obtained results can be looked as an extension of the uncertainty principles for the quaternion linear canonical transform and the quaternion windowed linear canonical transform.

1. Introduction

Over the last few years, the study of the quaternion Fourier transform and many other generalized transforms has become a new research topic. There are several published papers that try to study various transformations using quaternion algebra. For example, the authors of [1,2,3,4] have studied the quaternion windowed transform and corresponding uncertainty principle. In [5,6], the authors have developed the shearlet transform and the linear canonical using wavelet transform over a quaternion field, which are the generalizations of shearlet transform and the linear canonical wavelet transform. The authors of [7,8,9] have studied the quaternion Wigner–Ville distribution and quaternion ambiguity function, respectively. Further, in [10,11,12], several researchers have studied the Wigner–Ville distribution associated with the linear canonical transform and its application to noisy linear frequency-modulated signals. In [13,14], the authors have proposed the linear canonical S-transform (LCST), which is a non-trivial generalization of the classical S-transform in the context of the linear canonical transform (LCT). It is well-established that several fundamental properties of the classical S-transform, such as linearity, modulation, shifting, and uncertainty principles, can be applied to the LCST (see, e.g., [14,15]). Recently, in [16], the authors made significant strides in studying the quaternion linear canonical S-transform (QLCST), which can be looked as the extension of the linear canonical S-transform. Additionally, they also have derived several uncertainty principles, such as the Heisenberg-type and logarithmic uncertainty principles. However, many other uncertainty principles related to the new extended transformation like Pitt’s inequality and the sharp Hausdorff–Young inequality remain unexplored.

Therefore, in the present work, we extend the investigation initiated in [16] to gain a deeper understanding of the quaternion linear canonical S-transform (QLCST). We thoroughly demonstrate several fundamental properties related to this proposed transformation. It is known that the uncertainty principles are important properties and their specifics form each transformation, including the QLCST. Based on the properties, we build different types of the uncertainty principles associated with the QLCST, such as Pitt’s inequality and the Hausdorff–Young inequality. The important relations can be regarded as a non-trivial generalization of the uncertainty principles concerning the quaternion linear canonical transform (QLCT) and the quaternion windowed linear canonical transform (QWLCT). We emphasize that the considered uncertainty principles associated with the QLCST are a continuation the ones proposed in [16].

The main body of this paper is organized as follows: Section 2 compiles essential facts about quaternion algebra and its properties. This section also defines the quaternion Fourier transform (QFT) and outlines its useful properties. Section 3 provides a brief overview of the quaternion linear canonical transform (QLCT) and its connection to the QFT. Section 4 introduces the quaternion linear canonical S-transform (QLCST) and derives several fundamental properties and uncertainty principles related to this transformation. Lastly, Section 5 concludes this work.

2. Preliminaries

In this section, we discuss quaternion algebra and the quaternion Fourier transform, which serve as a backbone throughout this paper.

2.1. Quaternion Algebra

Let $\mathbb{H}$ be the associative algebra of real quaternions. The quaternion $q\in \mathbb{H}$ can be written in the form [17]

$$\mathbb{H}=\{q=q_0+\mathbf{i}{q}_1+\mathbf{j}{q}_2+\mathbf{k}{q}_3\mid q_0,q_1,q_2,q_3\in \mathbb{R}\},$$

(1)

where we use $\mathbb{R}$ to denote the set of real numbers. Here, the three different imaginary quaternion units $\mathbf{i},\mathbf{j}$ and $\mathbf{k}$ satisfy the multiplication rules:

$$\mathbf{ij}=-\mathbf{ji}=\mathbf{k},\mathbf{jk}=-\mathbf{kj}=\mathbf{i},\mathbf{ki}=-\mathbf{ik}=\mathbf{j},{\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=\mathbf{ijk}=-1.$$

(2)

Equation (2) tells us that the multiplication in $\mathbb{H}$ is not commutative. For any $q\in \mathbb{H}$, it takes the form

$$q=q_0+\mathit{q}=Sc(q)+V(q),$$

(3)

where $Sc(q)=q_0$ and $V(q)=\mathit{q}=\mathbf{i}{q}_1+\mathbf{j}{q}_2+\mathbf{k}{q}_3$ denote the scalar part and vector part or pure quaternion, respectively.

The quaternion conjugate $\overline{q}$ is described through

$$\overline{q}=q_0-\mathbf{i}{q}_1-\mathbf{j}{q}_2-\mathbf{k}{q}_3.$$

(4)

It satisfies

$$\overline{qp}=\overline{p}\overline{q}.$$

(5)

For every $q\in \mathbb{H}$, we may write the scalar and vector parts in the form

$$Sc(q)=\frac{1}{2}(q+\overline{q})\mathrm{and}\mathrm{V}(q)=\frac{1}{2}(q-\overline{q}).$$

(6)

The module (norm) of quaternion q is defined by the formula

$$|q|=\sqrt{q\overline{q}}=\sqrt{q_0^2+q_1^2+q_2^2+q_3^2}.$$

(7)

It is straightforward to verify that for every $q,t,p\in \mathbb{H}$, the following holds:

$$Sc(q)\le |q|,|\mathit{q}|=|\mathrm{V}(q)|\le |q|,\mathrm{and}Sc(qpt)=Sc(ptq)=Sc(qpt).$$

(8)

Now, define the inner product of two quaternion functions $f,g:{\mathbb{R}}^2⟶\mathbb{H}$ as

$$(f,g)={\int }_{{\mathbb{R}}^2}f(\mathit{x})\overline{g(\mathit{x})}d\mathit{x},d\mathit{x}=d{x}_{1}d{x}_2.$$

(9)

with the symmetric scalar product

$$\begin{array}{cc}\hfill ⟨f,g⟩& =Sc(f,g)\hfill \\ \hfill & =\frac{1}{2}\left[(f,g)+(g,f)\right].\hfill \end{array}$$

(10)

Based on Equation (10), we have the property:

$$Sc(f,g,h)=Sc(g,f,h)=Sc(h,g,f),\forall f,g,h\in \mathbb{H}.$$

(11)

For $f=g$ in (9), we get $L^2({\mathbb{R}}^2;\mathbb{H})$ norm:

$${\parallel f\parallel }_{L^2({\mathbb{R}}^2;\mathbb{H})}={\left({\int }_{{\mathbb{R}}^2}{|f(\mathit{x})|}^{2}d\mathit{x}\right)}^{1/2}.$$

(12)

2.2. Two-Sided Quaternion Fourier Transform

In this subsection, we start by defining the two-sided quaternion Fourier transform (QFT). We summarize several properties, which will be needed later. A complete account of the QFT and its properties can be found in [18,19,20,21,22].

Definition 1

([8,19]). Given g in $L^2({\mathbb{R}}^2;\mathbb{H})$, the definition of the two-sided quaternion Fourier transform of g is described through

$${\mathcal{F}}_Q\left\{g\right\}(\mathit{\omega })={\int }_{{\mathbb{R}}^2}{e}^{-\mathbf{i}2\pi {\omega }_1{x}_1}g(\mathit{x})e^{-\mathbf{j}2\pi {\omega }_2{x}_2}d\mathit{x}.$$

(13)

The following definition presents the pointwise inversion of the QFT, which states that the original quaternion function can be generated using its QFT, as shown below.

Definition 2

([8,19]). Let $g\in L^1({\mathbb{R}}^2;\mathbb{H})$ and ${\mathcal{F}}_Q\left\{g\right\}\in L^1({\mathbb{R}}^2;\mathbb{H})$. The inverse quaternion Fourier transform for g is described by

$${\mathcal{F}}_Q^{-1}\left[{\mathcal{F}}_Q\left\{g\right\}\right](\mathit{x})=g(\mathit{x})={\int }_{{\mathbb{R}}^2}{e}^{\mathbf{i}2\pi {\omega }_1{x}_1}{\mathcal{F}}_Q\left\{g\right\}(\mathit{\omega })e^{\mathbf{j}2\pi {\omega }_2{x}_2}d\mathit{\omega }.$$

(14)

From relation (14), for every $g\in L^2({\mathbb{R}}^2;\mathbb{H})$ one gets

$$\begin{array}{c}\hfill {\mathcal{F}}_Q\left\{{\mathcal{F}}_Q\left\{g\right\}\right\}(\mathit{x})=g(-\mathit{\omega }).\end{array}$$

(15)

3. Quaternion Linear Canonical Transform (QLCT)

In this section, we recall the definition of the two-sided quaternion linear canonical transform (shortly QLCT) and then present its relation to the quaternion Fourier transform (QFT). We develop this relation in order to obtain the main result in the next section. For a detailed discussion of this transform, we refer the readers to [23,24,25,26].

Definition 3

([25,26]). Given $A_1=(a_1,b_1,c_1,d_1)=\left[\begin{array}{cc}{a}_1& b_1\\ c_1& d_1\end{array}\right]$ and $A_2=(a_2,b_2,c_2,d_2)=\left[\begin{array}{cc}{a}_2& b_2\\ c_2& d_2\end{array}\right]$ in $SL(2,\mathbb{R})$. The QLCT of $f\in L({\mathbb{R}}^2;\mathbb{H})\cap L^2({\mathbb{R}}^2;\mathbb{H})$ is defined through

$$\begin{array}{c}\hfill {\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega })=\left\{\begin{array}{cc}{\int }_{{\mathbb{R}}^2}{K}_{A_1}(x_1,{\omega }_1)f(\mathit{x})K_{A_2}(x_2,{\omega }_2)d\mathit{x},\hfill & \mathrm{for}{b}_1{b}_2\ne 0\hfill \\ \sqrt{d_1}{e}^{\mathbf{i}\left(\frac{c_1{d}_1}{2}\right){\omega }_1^2}f(d_1{\omega }_1,d_2{\omega }_2)\sqrt{d_2}{e}^{\mathbf{j}\left(\frac{c_2{d}_2}{2}\right){\omega }_2^2},\hfill & \mathrm{for}{b}_1{b}_2=0,\hfill \end{array}\right.\end{array}$$

(16)

where the kernel functions of the QLCT above are described by

$$\begin{array}{c}\hfill K_{A_1}(x_1,{\omega }_1)=\frac{1}{\sqrt{2\pi b_1}}{e}^{\frac{\mathbf{i}}{2}\left(\frac{a_1}{b_1}{x}_1^2-\frac{2}{b_1}{x}_1{\omega }_1+\frac{d_1}{b_1}{\omega }_1^2-\frac{\pi }{2}\right)},\end{array}$$

(17)

and

$$\begin{array}{c}\hfill K_{A_2}(x_2,{\omega }_2)=\frac{1}{\sqrt{2\pi b_2}}{e}^{\frac{\mathbf{j}}{2}\left(\frac{a_2}{b_2}{x}_2^2-\frac{2}{b_2}{x}_2{\omega }_2+\frac{d_2}{b_2}{\omega }_2^2-\frac{\pi }{2}\right)}.\end{array}$$

(18)

In this work, we consider the QLCT in the case of $b_1{b}_2\ne 0$. For specific parameter matrices $A_1=A_2=(a_i,b_i,c_i,d_i)=(0,1,-1,0)$ with $i=1,2$, the two-sided QLCT Definition (16) boils down to the two-sided QFT definition:

$${\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega })={\int }_{{\mathbb{R}}^2}\frac{e^{-\mathbf{i}\frac{\pi }{4}}}{\sqrt{2\pi }}{e}^{-\mathbf{i}{\omega }_1{x}_1}f(\mathit{x})e^{-\mathbf{j}{\omega }_2{x}_2}\frac{e^{-\mathbf{j}\frac{\pi }{4}}}{\sqrt{2\pi }}d\mathit{x}=\frac{e^{-\mathbf{i}\frac{\pi }{4}}}{\sqrt{2\pi }}{\mathcal{F}}_Q\left\{f\right\}\left(\frac{\mathit{\omega }}{2\pi }\right)\frac{e^{-\mathbf{j}\frac{\pi }{4}}}{\sqrt{2\pi }}.$$

(19)

where ${\mathcal{F}}_Q\left\{f\right\}$ is defined by (13).

Definition 4

([25,26]). For any $f\in L^1({\mathbb{R}}^2;\mathbb{H})$ with ${\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}\in L({\mathbb{R}}^2;\mathbb{H})$, the inversion theorem of the QLCT is described by

$$\begin{array}{cc}\hfill f(\mathit{x})& ={\left({\mathcal{L}}_{A_1,A_2}^Q\right)}^{-1}\left[{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}\right](\mathit{x})\hfill \\ \hfill & ={\int }_{\mathbb{R}}\frac{1}{\sqrt{2\pi b_1}}{e}^{-\frac{\mathbf{i}}{2}\left(\frac{a_1}{b_1}{x}_1^2-\frac{2}{b_1}{x}_1{\omega }_1+\frac{d_1}{b_1}{\omega }_1^2-\frac{\pi }{2}\right)}{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega })\hfill \\ \hfill & \times \frac{1}{\sqrt{2\pi b_2}}{e}^{-\frac{\mathbf{j}}{2}\left(\frac{a_2}{b_2}{x}_2^2-\frac{2}{b_2}{x}_2{\omega }_2+\frac{d_2}{b_2}{\omega }_2^2-\frac{\pi }{2}\right)}d\mathit{\omega }.\hfill \end{array}$$

(20)

From the definition of the QLCT (16), we see that

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega })\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_1}}{e}^{\frac{\mathbf{i}}{2}\left(\frac{a_1}{b_1}{x}_1^2-\frac{2}{b_1}{x}_1{\omega }_1+\frac{d_1}{b_1}{\omega }_1^2-\frac{\pi }{2}\right)}f(\mathit{x})\frac{1}{\sqrt{2\pi b_2}}{e}^{\frac{\mathbf{j}}{2}\left(\frac{a_2}{b_2}{x}_2^2-\frac{2}{b_2}{x}_2{\omega }_2+\frac{d_2}{b_2}{\omega }_2^2-\frac{\pi }{2}\right)}d\mathit{x}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}\frac{e^{-\mathbf{i}\frac{\pi }{4}}}{\sqrt{2\pi b_1}}{e}^{\mathbf{i}\frac{d_1}{2b_1}{\omega }_1^2}{e}^{-\mathbf{i}\frac{x_1{\omega }_1}{b_1}}{e}^{\mathbf{i}\frac{a_1}{2b_1}{x}_1^2}f(\mathit{x})\frac{e^{-\mathbf{j}\frac{\pi }{4}}}{\sqrt{2\pi b_2}}{e}^{\mathbf{j}\frac{d_2}{2b_2}{\omega }_2^2}{e}^{-\mathbf{j}\frac{x_2{\omega }_2}{b_2}}{e}^{\mathbf{j}\frac{a_2}{2b_2}{x}_2^2}d\mathit{x}.\hfill \end{array}$$

(21)

Hence,

$$\begin{array}{cc}\hfill \sqrt{2\pi b_1}{e}^{\mathbf{i}\frac{\pi }{4}}{e}^{-\mathbf{i}\frac{d_1}{2b_1}{\omega }_1^2}{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega })& e^{-\mathbf{j}\frac{d_2}{2b_2}{\omega }_2^2}\sqrt{2\pi b_2}{e}^{\mathbf{j}\frac{\pi }{4}}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{e}^{-\mathbf{i}\frac{x_1{\omega }_1}{b_1}}{e}^{\mathbf{i}\frac{a_1}{2b_1}{x}_1^2}f(\mathit{x})e^{\mathbf{j}\frac{a_2}{2b_2}{x}_2^2}{e}^{-\mathbf{j}\frac{x_2{\omega }_2}{b_2}}d\mathit{x}\hfill \\ \hfill & ={\mathcal{F}}_Q\left\{h\right\}\left(\frac{\mathit{\omega }}{2\pi \mathit{b}}\right),\hfill \end{array}$$

(22)

where

$$\begin{array}{c}\hfill h(\mathit{x})=e^{\mathbf{i}\frac{a_1}{2b_1}{x}_1^2}f(\mathit{x})e^{\mathbf{j}\frac{a_2}{2b_2}{x}_2^2}\end{array}$$

(23)

and

$$\begin{array}{c}\hfill \left(\frac{\mathit{\omega }}{2\pi \mathit{b}}\right)=\left(\frac{{\omega }_1}{2\pi b_1},\frac{{\omega }_2}{2\pi b_2}\right).\end{array}$$

For every $f\in L^2({\mathbb{R}}^2;\mathbb{H})$, there is

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega }){|}^{2}d\mathit{\omega }={\int }_{{\mathbb{R}}^2}{|f(\mathit{x})|}^{2}d\mathit{x},\end{array}$$

(24)

which is known as Parseval’s theorem for the two-sided QLCT.

4. Quaternion Linear Canonical S-Transform and Uncertainty Principles

In this section, we introduce the quaternion linear canonical S-transform (QLCST). We are going to investigate its main properties, which will be useful to derive the main result in this work.

Definition 5

(QLCST definition). Let $\varphi \in L^2({\mathbb{R}}^2;\mathbb{H})$ be a non-zero quaternion window function. The quaternion linear canonical S-transform ${\mathcal{S}}_{\varphi }^Q$, is defined for every function $h\in L^2({\mathbb{R}}^2;\mathbb{H})$ with respect to ϕ, by

$$\begin{array}{cc}\hfill {\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })& ={\int }_{{\mathbb{R}}^2}{K}_{A_1}(x_1,{\omega }_1)h(\mathit{x})\overline{\varphi (\mathit{u}-\mathit{x},\mathit{\omega })}{K}_{A_2}(x_2,{\omega }_2)d\mathit{x}.\hfill \end{array}$$

(25)

We collect the following facts related to the above definition.

• It is readily checked that

  $${\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })={\mathcal{L}}_{A_1,A_2}^Q\left\{h(\mathit{x})\overline{\varphi (\mathit{u}-\mathit{x},\mathit{\omega })}\right\}.$$

  (26)
• Using the inverse QLCT (20) to (26) results in

  $$\begin{array}{c}\hfill h(\mathit{x})\overline{\varphi (\mathit{u}-\mathit{x},\mathit{\omega })}={\left({\mathcal{L}}_{A_1,A_2}^Q\right)}^{-1}\left[{\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })\right].\end{array}$$

  (27)

Corollary 1.

Let $\varphi \in L^1({\mathbb{R}}^2;\mathbb{H})\cap L^2({\mathbb{R}}^2;\mathbb{H})$ satisfying

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}\varphi (\mathit{u}-\mathit{x},\mathit{\omega })d\mathit{u}=1,\end{array}$$

(28)

then for any $h\in L^2({\mathbb{R}}^2;\mathbb{H})$

$${\int }_{{\mathbb{R}}^2}{\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })d\mathit{u}={\mathcal{L}}_{A_1,A_2}^Q\left\{h\right\}(\mathit{\omega }).$$

(29)

If $\varphi (\mathit{u}-\mathit{x},\mathit{\omega })=m(\mathit{\omega })$, then

$${\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })=\overline{m(\mathit{\omega })}{\mathcal{L}}_{A_1,A_2}^Q\left\{h\right\}(\mathit{\omega }).$$

(30)

If we assume that $\varphi (\mathit{u}-\mathit{x},\mathit{\omega })=\varphi (\mathit{x}-\mathit{u})$, then

$${\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })={\mathcal{G}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega }),$$

(31)

where ${\mathcal{G}}_{\varphi }^Q$ is the quaternion windowed linear canonical transform (QWLCT) defined by

$${\mathcal{G}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })={\int }_{{\mathbb{R}}^2}{K}_{A_1}(x_1,{\omega }_1)h(\mathit{x})\overline{\varphi (\mathit{x}-\mathit{u})}{K}_{A_2}(x_2,{\omega }_2)d\mathit{x}.$$

(32)

Proof. 

We just derive Equation (29), and the others are similar. In fact, we have

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^2}{\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })d\mathit{u}& ={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{K}_{A_1}(x_1,{\omega }_1)h(\mathit{x})\overline{\varphi (\mathit{u}-\mathit{x},\mathit{\omega })}{K}_{A_2}(x_2,{\omega }_2)d\mathit{u}d\mathit{x}\hfill \\ & ={\int }_{{\mathbb{R}}^2}{K}_{A_1}(x_1,{\omega }_1)h(\mathit{x})K_{A_2}(x_2,{\omega }_2){\int }_{{\mathbb{R}}^2}\overline{\varphi (\mathit{u}-\mathit{x},\mathit{\omega })}d\mathit{u}d\mathit{x}\hfill \\ & ={\mathcal{L}}_{A_1,A_2}^Q\left\{h\right\}(\mathit{\omega }),\hfill \end{array}$$

and the proof is complete. □

Theorem 1.

Let ϕ be a quaternion window function. If $h,g\in L^2({\mathbb{R}}^2;\mathbb{H})$ are two quaternion functions, then the following orthogonality relation holds:

$${\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}Sc\left({\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })\overline{{\mathcal{S}}_{\varphi }^{Q}g(\mathit{u},\mathit{\omega })}\right)d\mathit{\omega }d\mathit{u}={⟨h{\varphi }_{\mathit{u},\mathit{\omega }},g⟩}_{L^2({\mathbb{R}}^2;\mathbb{H})},$$

(33)

where

$${\varphi }_{\mathit{u},\mathit{\omega }}={\int }_{{\mathbb{R}}^2}|\varphi (\mathit{u},\mathit{\omega }){|}^{2}d\mathit{u}.$$

(34)

In particular, for $h=g$, we obtain:

$${\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|{\mathcal{S}}_{\varphi }^Q{h(\mathit{u},\mathit{\omega })|}^{2}d\mathit{\omega }d\mathit{u}={\varphi }_{\mathit{u},\mathit{\omega }}{\parallel h\parallel }_{L^2({\mathbb{R}}^2;\mathbb{H})}^2.$$

(35)

Proof. 

By the QLCST Definition (32) and Equation (5), we have

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}Sc\left({\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })\overline{{\mathcal{S}}_{\varphi }^{Q}g(\mathit{u},\mathit{\omega })}\right)d\mathit{u}d\mathit{\omega }\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}Sc(\frac{1}{\sqrt{2\pi b_1}}{e}^{-\frac{\mathbf{i}}{2}\left(\frac{a_1}{b_1}{x}_1^2-\frac{2}{b_1}{x}_1{\omega }_1+\frac{d_1}{b_1}{\omega }_1^2-\frac{\pi }{2}\right)}h(\mathit{x})\overline{\varphi (\mathit{u}-\mathit{x},\mathit{\omega })}\hfill \\ \hfill & \hspace{1em}\times \frac{1}{\sqrt{2\pi b_2}}{e}^{-\frac{\mathbf{j}}{2}\left(\frac{a_1}{b_2}{x}_2^2-\frac{2}{b_2}{x}_2{\omega }_2+\frac{d_2}{b_2}{\omega }_2^2-\frac{\pi }{2}\right)}\times {\int }_{{\mathbb{R}}^2}\frac{1}{\sqrt{2\pi b_2}}{e}^{\frac{\mathbf{j}}{2}\left(\frac{a_2}{b_2}{t}_2^2-\frac{2}{b_2}{t}_2{\omega }_2+\frac{d_2}{b_2}{\omega }_2^2-\frac{\pi }{2}\right)}\hfill \\ \hfill & \hspace{1em}\times \varphi (\mathit{u}-\mathit{t},\mathit{\omega })\overline{g(\mathit{x})}\frac{1}{\sqrt{2\pi b_1}}{e}^{\frac{\mathbf{i}}{2}(\frac{a_1}{b_1}{t}_1^2-\frac{2}{b_1}{t}_1{\omega }_1+\frac{d_1}{b_1}{\omega }_1^2-\frac{\pi }{2})})d\mathit{t}d\mathit{x}d\mathit{u}d\mathit{\omega }.\hfill \end{array}$$

Using Equation (11), we obtain

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}Sc\left({\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })\overline{{\mathcal{S}}_{\varphi }^{Q}g(\mathit{u},\mathit{\omega })}\right)d\mathit{u}d\mathit{\omega }\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}Sc(\frac{1}{\sqrt{2\pi b_1}}{e}^{-\frac{\mathbf{i}{a}_1}{2b_1}(x_1^2-t_1^2)}\frac{1}{\sqrt{2\pi b_1}}{e}^{\frac{\mathbf{i}}{b_1}(t_1-x_1){\omega }_1}h(\mathit{x})\overline{\varphi (\mathit{u}-\mathit{x},\mathit{\omega })}\hfill \\ \hfill & \hspace{1em}\times \frac{1}{\sqrt{2\pi b_2}}{e}^{-\frac{\mathbf{j}{a}_2}{2b_2}(x_2^2-t_2^2)}\frac{1}{\sqrt{2\pi b_2}}{e}^{\frac{\mathbf{j}}{b_2}(t_2-x_2){\omega }_2}{\int }_{{\mathbb{R}}^2}\varphi (\mathit{u}-\mathit{t},\mathit{\omega })\overline{g(\mathit{x})})d\mathit{t}d\mathit{x}d\mathit{u}d\mathit{\omega }\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}Sc(e^{-\frac{\mathbf{i}{a}_1}{2b_1}(x_1^2-t_1^2)}h(\mathit{x})\overline{\varphi (\mathit{u}-\mathit{x},\mathit{\omega })}\hfill \\ \hfill & \hspace{1em}\times e^{-\frac{\mathbf{j}{a}_2}{2b_2}(x_2^2-t_2^2)}\overline{g(\mathit{x})}\varphi (\mathit{u}-\mathit{t},\mathit{\omega })\delta (\mathit{t}-\mathit{x}))d\mathit{t}d\mathit{x}d\mathit{u}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}Sc\left(h(\mathit{t})\overline{\varphi (\mathit{u}-\mathit{t},\mathit{\omega })}\varphi (\mathit{u}-\mathit{t},\mathit{\omega })\overline{g(\mathit{t})}\right)d\mathit{u}d\mathit{t}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}Sc\left(h(\mathit{t})\overline{g(\mathit{t})}{\int }_{{\mathbb{R}}^2}|\varphi (\mathit{u},\mathit{\omega }){|}^{2}d\mathit{u}\right)d\mathit{t}\hfill \\ \hfill & ={⟨h{\varphi }_{\mathit{u},\mathit{\omega }},g⟩}_{L^2({\mathbb{R}}^2;\mathbb{H})}.\hfill \end{array}$$

The proof is complete. □

Remark 1.

Based on (31) we obtain

$${\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}Sc\left({\mathcal{G}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })\overline{{\mathcal{G}}_{\varphi }^{Q}g(\mathit{u},\mathit{\omega })}\right)d\mathit{\omega }d\mathit{u}={\parallel \varphi \parallel }_{L^2({\mathbb{R}}^2;\mathbb{H})}{⟨h,g⟩}_{L^2({\mathbb{R}}^2;\mathbb{H})},$$

(36)

and

$${\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|{\mathcal{G}}_{\varphi }^Q{h(\mathit{u},\mathit{\omega })|}^{2}d\mathit{\omega }d\mathit{u}={\parallel \varphi \parallel }_{L^2({\mathbb{R}}^2;\mathbb{H})}^2{\parallel h\parallel }_{L^2({\mathbb{R}}^2;\mathbb{H})}^2.$$

(37)

In the next theorem, we will derive a reconstruction formula associated with the linear canonical S-transform.

Theorem 2.

Let $\varphi \in L^2({\mathbb{R}}^2;\mathbb{H})$ be a quaternion window functions that ϕ satisfies the condition:

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}{|\varphi (\mathit{u},\mathit{\omega })|}^{2}d\mathit{u}={\varphi }_{\mathit{u},\mathit{\omega }},0<{\varphi }_{\mathit{u},\mathit{\omega }}<\infty .\end{array}$$

(38)

Then, every quaternion signal $h\in L^2({\mathbb{R}}^2;\mathbb{H})$ can be generated using inversion formula for the QLCST:

$$h(\mathit{x})=\frac{1}{{\varphi }_{\mathit{u},\mathit{\omega }}}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })\varphi (\mathit{u}-\mathit{x},\mathit{\omega })K_{A_1}(x_1,{\omega }_1)K_{A_2}(x_2,{\omega }_2)d\mathit{u}d\mathit{\omega }.$$

(39)

Proof. 

An application of (33) and (8) yields

$$\begin{array}{cc}\hfill & {⟨h{\int }_{{\mathbb{R}}^2}|\varphi (\mathit{u},\mathit{\omega }){|}^{2}d\mathit{u},g⟩}_{L^2({\mathbb{R}}^2;\mathbb{H})}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}Sc\left({\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })\overline{{\mathcal{S}}_{\varphi }^{Q}g(\mathit{u},\mathit{\omega })}\right)d\mathit{u}d\mathit{\omega }\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}Sc\left({\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega }){\int }_{{\mathbb{R}}^2}{K}_{A_1}(x_1,{\omega }_1)\varphi (\mathit{u}-\mathit{x},\mathit{\omega })\overline{g(\mathit{x})}{K}_{A_2}(x_2,{\omega }_2)\right)d\mathit{x}d\mathit{u}d\mathit{\omega }\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}Sc\left({\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })\varphi (\mathit{u}-\mathit{x},\mathit{\omega })K_{A_1}(x_1,{\omega }_1)K_{A_2}(x_2,{\omega }_2)\overline{g(\mathit{x})}\right)d\mathit{u}d\mathit{\omega }d\mathit{x}\hfill \\ \hfill & =⟨{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })\varphi (\mathit{u}-\mathit{x},\mathit{\omega })K_{A_1}(x_1,{\omega }_1)K_{A_2}(x_2,{\omega }_2)d\mathit{u}d\mathit{\omega },g⟩.\hfill \end{array}$$

(40)

Since relation (40) holds for any $g\in L^2({\mathbb{R}}^2;\mathbb{H})$, we have

$$\begin{array}{c}\hfill h(\mathit{x}){\int }_{{\mathbb{R}}^2}|\varphi (\mathit{u},\mathit{\omega }){|}^{2}d\mathit{u}={\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })\varphi (\mathit{u}-\mathit{x},\mathit{\omega })K_{A_1}(x_1,{\omega }_1)K_{A_2}(x_2,{\omega }_2)d\mathit{u}d\mathit{\omega }.\end{array}$$

(41)

This means that

$$\begin{array}{c}\hfill h(\mathit{x})=\frac{1}{{\varphi }_{\mathit{u},\mathit{\omega }}}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{\mathcal{S}}_{\varphi }^{Q}h(\mathit{u},\mathit{\omega })\varphi (\mathit{u}-\mathit{x},\mathit{\omega })K_{A_1}(x_1,{\omega }_1)K_{A_2}(x_2,{\omega }_2)d\mathit{u}d\mathit{\omega }.\end{array}$$

(42)

Therefore, the proof is complete. □

4.1. Heisenberg-Type Uncertainty Principle for QLCST

The following result presents the Heisenberg-type uncertainty principle for the QLCT, which is a non-trivial generalization of the QFT inequality studied by the authors in [25].

Theorem 3.

For any $f\in L^2({\mathbb{R}}^2;\mathbb{H})$ and $\alpha ,\gamma \ge 1$, we have

$$\begin{array}{c}\hfill {\left({\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }|f(\mathit{x}){|}^{2}d\mathit{x}\right)}^{\frac{\alpha }{\alpha +\gamma }}{\left({\int }_{{\mathbb{R}}^2}{\left|\mathit{\omega }\right|}^{2\alpha }|{\mathcal{F}}_Q\left\{f\right\}(\mathit{\omega }){|}^{2}d\xi \right)}^{\frac{\gamma }{\alpha +\gamma }}\ge {\int }_{{\mathbb{R}}^2}|f(\mathit{x}){|}^{2}d\mathit{x}.\end{array}$$

(43)

Its generalization in framework of the QLCT will lead to the following important result.

Theorem 4.

Under the same assumption as in Theorem 3, we have, for all ${\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}\in L^2({\mathbb{R}}^2;\mathbb{H})$,

$$\begin{array}{cc}\hfill ({\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }{|f(\mathit{x})|}^{2}d\mathit{x}& {)}^{\frac{\alpha }{\alpha +\gamma }}{\left({\int }_{{\mathbb{R}}^2}|\mathit{\omega }{|}^{2\alpha }|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega }){|}^{2}d\mathit{\omega }\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ & \ge {|2\pi \mathit{b}|}^{\frac{2\alpha \gamma }{\alpha +\gamma }}{\int }_{{\mathbb{R}}^2}{|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega })|}^{2}d\mathit{\omega }.\hfill \end{array}$$

(44)

Proof. 

Replacing $f(\mathit{x})$ with $h(\mathit{x})$ defined by (23) in (43) we immediately get

$$\begin{array}{cc}\hfill ({\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }|e^{\mathbf{i}\frac{a_1}{2b_1}{x}_1^2}f(\mathit{x})e^{\mathbf{j}\frac{a_2}{2b_2}{x}_2^2}{|}^{2}d\mathit{x}& {)}^{\frac{\alpha }{\alpha +\gamma }}{\left({\int }_{{\mathbb{R}}^2}{|\mathit{\omega }|}^{2\alpha }|{\mathcal{F}}_Q\left\{h\right\}(\mathit{\omega }){|}^{2}d\mathit{\omega }\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ & \ge {\int }_{{\mathbb{R}}^2}|e^{\mathbf{i}\frac{a_1}{2b_1}{x}_1^2}f(\mathit{x})e^{\mathbf{j}\frac{a_2}{2b_2}{x}_2^2}{|}^{2}d\mathit{x}.\hfill \end{array}$$

(45)

Simplifying it results in

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }{|f(\mathit{x})|}^{2}d\mathit{x}{)}^{\frac{\alpha }{\alpha +\gamma }}{\left({\int }_{{\mathbb{R}}^2}|\mathit{\omega }{|}^{2\alpha }|\frac{1}{2\pi \mathit{b}}{|}^{2\alpha +1}|{\mathcal{F}}_Q\left\{h\right\}\left(\frac{\mathit{\omega }}{2\pi \mathit{b}}\right){|}^{2}d\mathit{\omega }\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge {\int }_{{\mathbb{R}}^2}{|f(\mathit{x})|}^{2}d\mathit{x}.\hfill \end{array}$$

(46)

By virtue of (22) we get

$$\begin{array}{c}{\left({\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }{|f(\mathit{x})|}^{2}d\mathit{x}\right)}^{\frac{\alpha }{\alpha +\gamma }}\hfill \\ {\left({\int }_{{\mathbb{R}}^2}|\mathit{\omega }{|}^{2\alpha }|\frac{1}{2\pi \mathit{b}}{|}^{2\alpha }\frac{1}{4\pi |b_1||b_2|}|\sqrt{2\pi b_1}{e}^{\mathbf{i}\frac{\pi }{4}}{e}^{-\mathbf{i}\frac{d_1}{2b_1}{\omega }_1^2}{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega })e^{-\mathbf{j}\frac{d_2}{2b_2}{\omega }_2^2}\sqrt{2\pi b_2}{e}^{\mathbf{j}\frac{\pi }{4}}{|}^{2}d\mathit{\omega }\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ \ge {\int }_{{\mathbb{R}}^2}{|f(\mathit{x})|}^{2}d\mathit{x}.\hfill \end{array}$$

Hence,

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }{|f(\mathit{x})|}^{2}d\mathit{x}{)}^{\frac{\alpha }{\alpha +\gamma }}{\left({\int }_{{\mathbb{R}}^2}|\mathit{\omega }{|}^{2\alpha }|\frac{1}{2\pi \mathit{b}}{|}^{2\alpha }|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega }){|}^{2}d\mathit{\omega }\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge {\int }_{{\mathbb{R}}^2}{|f(\mathit{x})|}^{2}d\mathit{x}.\hfill \end{array}$$

(47)

Using Parseval’s theorem for the QLCT (24) in the right-hand side of the above identity yields

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }{|f(\mathit{x})|}^{2}d\mathit{x}{)}^{\frac{\alpha }{\alpha +\gamma }}{\left({\int }_{{\mathbb{R}}^2}|2\pi \mathit{b}{|}^{-2\alpha }|\mathit{\omega }{|}^{2\alpha }|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega }){|}^{2}d\mathit{\omega }\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge {\int }_{{\mathbb{R}}^2}{|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega })|}^{2}d\mathit{\omega },\hfill \end{array}$$

which gives the required result. □

Remark 2.

For $\alpha =\gamma =1$, one gets

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^2|f(\mathit{x}){|}^{2}d\mathit{x}{\int }_{{\mathbb{R}}^2}{|\mathit{\omega }|}^2|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}{|}^{2}d\mathit{\omega }\ge 4{\pi }^2{|\mathit{b}|}^2{\left({\int }_{{\mathbb{R}}^2}|f(\mathit{x}){|}^{2}d\mathit{x}\right)}^2.\end{array}$$

(48)

The generalization of Theorem 3 in the framework of the QLCST is given by the following result.

Theorem 5.

For any $f\in L^2({\mathbb{R}}^2;\mathbb{H})$ and $\alpha ,\gamma \ge 1$, we have

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }|f(\mathit{x}){|}^{2}d\mathit{x}{)}^{\frac{\alpha }{\alpha +\gamma }}{\left({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|\mathit{\omega }{|}^{2\alpha }|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{2}d\mathit{\omega }d\mathit{u}\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge {|2\pi \mathit{b}|}^{\frac{2\alpha \gamma }{\alpha +\gamma }}{\varphi }_{\mathit{u},\mathit{\omega }}^{\frac{\gamma }{\alpha +\gamma }}{\parallel f\parallel }_{L^2({\mathbb{R}}^2;\mathbb{H})}^2.\hfill \end{array}$$

(49)

Proof. 

From Equation (47), it follows that

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }{|f(\mathit{x})|}^{2}d\mathit{x}{)}^{\frac{\alpha }{\alpha +\gamma }}{\left({\int }_{{\mathbb{R}}^2}|\mathit{\omega }{|}^{2\alpha }|2\pi \mathit{b}{|}^{-2\alpha }|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega }){|}^{2}d\mathit{\omega }\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge {\int }_{{\mathbb{R}}^2}{|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega })|}^{2}d\mathit{\omega }.\hfill \end{array}$$

(50)

Including (20) into the first term of the above identity results in

$$\begin{array}{cc}\hfill ({\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }|{\left({\mathcal{L}}_{A_1,A_2}^Q\right)}^{-1}& \left[{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}\right](\mathit{x}){|}^{2}d\mathit{x}{)}^{\frac{\alpha }{\alpha +\gamma }}\hfill \\ & \times {\left({\int }_{{\mathbb{R}}^2}|\mathit{\omega }{|}^{2\alpha }|2\pi \mathit{b}{|}^{-2\alpha }|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega }){|}^{2}d\mathit{\omega }\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ & \ge {\int }_{{\mathbb{R}}^2}{|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega })|}^{2}d\mathit{\omega }.\hfill \end{array}$$

(51)

Integrating the above relation with respect to $d\mathit{u}$ will lead to

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^2}\left[\right({\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }& |{\left({\mathcal{L}}_{A_1,A_2}^Q\right)}^{-1}\left[{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}\right](\mathit{x}){|}^{2}d\mathit{x}{)}^{\frac{\alpha }{\alpha +\gamma }}\hfill \\ & \times {\left({\int }_{{\mathbb{R}}^2}|\mathit{\omega }{|}^{2\alpha }|2\pi \mathit{b}{|}^{-2\alpha }|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega }){|}^{2}d\mathit{\omega }\right)}^{\frac{\gamma }{\alpha +\gamma }}]d\mathit{u}\hfill \\ & \ge {\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega })|}^{2}d\mathit{\omega }d\mathit{u}.\hfill \end{array}$$

(52)

Therefore,

$$\begin{array}{cc}\hfill ({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }& |{\left({\mathcal{L}}_{A_1,A_2}^Q\right)}^{-1}\left[{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}\right](\mathit{x}){|}^{2}d\mathit{x}d\mathit{u}{)}^{\frac{\alpha }{\alpha +\gamma }}\hfill \\ & \times {\left({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|\mathit{\omega }{|}^{2\alpha }|2\pi \mathit{b}{|}^{-2\alpha }|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega }){|}^{2}d\mathit{\omega }d\mathit{u}\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ & \ge {\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega })|}^{2}d\mathit{\omega }d\mathit{u}.\hfill \end{array}$$

(53)

We replace ${\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}$ by ${\mathcal{S}}_{\varphi }^Q\left\{f\right\}$ and get

$$\begin{array}{cc}\hfill ({\int }_{{\mathbb{R}}^2}& {\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }|{\left({\mathcal{L}}_{A_1,A_2}^Q\right)}^{-1}\left[{\mathcal{S}}_{\varphi }^Q\left\{f\right\}\right](\mathit{x}){|}^{2}d\mathit{x}d\mathit{u}{)}^{\frac{\alpha }{\alpha +\gamma }}\hfill \\ & \times {\left({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|\mathit{\omega }{|}^{2\alpha }|2\pi \mathit{b}{|}^{-2\alpha }|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{2}d\mathit{\omega }d\mathit{u}\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ & \hspace{1em}\ge {\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega })|}^{2}d\mathit{\omega }d\mathit{u}.\hfill \end{array}$$

(54)

Applying relations (27) and (35), we further obtain

$$\begin{array}{cc}\hfill ({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}& {|\mathit{x}|}^{2\gamma }|f(\mathit{x})\overline{\varphi (\mathit{u}-\mathit{x},\mathit{\omega })}{|}^{2}d\mathit{x}d\mathit{u}{)}^{\frac{\alpha }{\alpha +\gamma }}\hfill \\ & \times {\left({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|\mathit{\omega }{|}^{2\alpha }|2\pi \mathit{b}{|}^{-2\alpha }|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{2}d\mathit{\omega }d\mathit{u}\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ & \ge {\parallel f\parallel }_{L^2({\mathbb{R}}^2;\mathbb{H})}^2{\int }_{{\mathbb{R}}^2}|\varphi (\mathit{u},\mathit{\omega }){|}^{2}d\mathit{u}.\hfill \end{array}$$

(55)

This is equivalent to

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }|f(\mathit{x}){|}^2{\varphi }_{\mathit{u},\mathit{\omega }}d\mathit{x}{)}^{\frac{\alpha }{\alpha +\gamma }}{\left({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|\mathit{\omega }{|}^{2\alpha }|2\pi \mathit{b}{|}^{-2\alpha }|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{2}d\mathit{\omega }d\mathit{u}\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge {\varphi }_{\mathit{u},\mathit{\omega }}{\parallel f\parallel }_{L^2({\mathbb{R}}^2;\mathbb{H})}^2.\hfill \end{array}$$

(56)

Simplifying the above relation yields

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }|f(\mathit{x}){|}^{2}d\mathit{x}{)}^{\frac{\alpha }{\alpha +\gamma }}{\left({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|\mathit{\omega }{|}^{2\alpha }|2\pi \mathit{b}{|}^{-2\alpha }|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{2}d\mathit{\omega }d\mathit{u}\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge {\varphi }_{\mathit{u},\mathit{\omega }}^{\frac{\gamma }{\alpha +\gamma }}{\parallel f\parallel }_{L^2({\mathbb{R}}^2;\mathbb{H})}^2.\hfill \end{array}$$

(57)

The proof is complete. □

Remark 3.

Based on (31) and (37), we obtain

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{2\gamma }|f(\mathit{x}){|}^{2}d\mathit{x}{)}^{\frac{\alpha }{\alpha +\gamma }}{\left({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|\mathit{\omega }{|}^{2\alpha }|{\mathcal{G}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{2}d\mathit{\omega }d\mathit{u}\right)}^{\frac{\gamma }{\alpha +\gamma }}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge {|2\pi \mathit{b}|}^{\frac{2\alpha \gamma }{\alpha +\gamma }}{\parallel \varphi \parallel }_{L^2({\mathbb{R}}^2;\mathbb{H})}^{\frac{2\gamma }{\alpha +\gamma }}{\parallel f\parallel }_{L^2({\mathbb{R}}^2;\mathbb{H})}^2.\hfill \end{array}$$

(58)

which is known as the Heisenberg-type uncertainty principle for the QWLCT [3]. If the Relation (28) is satisfied, Theorem 5 boils down to Theorem 4.

4.2. Pitt’S Inequality for QLCST

In this subsection, we establish Pitt’s inequality from the QLCT domains to the QLCST domains. For this purpose, we first introduce Pitt’s inequality related to the QLCT.

Theorem 6

(QLCT Pitt’s inequality [23]). Let $f\in \mathcal{S}({\mathbb{R}}^2;\mathbb{H})$ and $0\le \alpha <2$. Then, we have

$${\int }_{{\mathbb{R}}^2}{|\mathit{\omega }|}^{-\alpha }|{\mathcal{L}}_{A_1,A_2}^Q{\left\{f\right\}(\mathit{\omega })|}^{2}d\mathit{\omega }\le \frac{M_{\alpha }}{|b_1{b}_2{|}^{\alpha }}{\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{\alpha }{|f(\mathit{x})|}^{2}d\mathit{x},$$

(59)

where

$$M_{\alpha }={\pi }^{\alpha }{\left[\frac{\Gamma \left(\frac{2-\alpha }{4}\right)}{\Gamma \left(\frac{2+\alpha }{4}\right)}\right]}^2.$$

Here $\mathcal{S}({\mathbb{R}}^2;\mathbb{H})$ is quaternion Schwartz space, and $\Gamma (t)$ is the gamma function.

The following theorem explores an extension of Pitt’s inequality within the context of the QLCST.

Theorem 7

(QLCST Pitt’s Inequality). Let $\psi \in \mathcal{S}({\mathbb{R}}^2;\mathbb{H})$ be a non-zero quaternion window function. Then, for any $f\in \mathcal{S}({\mathbb{R}}^2;\mathbb{H})$ such that ${\mathcal{S}}_{\varphi }^Q\left\{f\right\}\in \mathcal{S}({\mathbb{R}}^2;\mathbb{H})$, one gets

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{|\mathit{\omega }|}^{-\alpha }|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{2}d\mathit{\omega }d\mathit{u}\le \frac{{\varphi }_{\mathit{u},\mathit{\omega }}{M}_{\alpha }}{|b_1{b}_2{|}^{\alpha }}{\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{\alpha }|f(\mathit{x}){|}^{2}d\mathit{x}.\end{array}$$

(60)

Proof. 

It follows with (59) that

$${\int }_{{\mathbb{R}}^2}{|\mathit{\omega }|}^{-\alpha }|{\mathcal{L}}_{A_1,A_2}^Q{\left\{f\right\}(\mathit{\omega })|}^{2}d\mathit{\omega }\le \frac{M_{\alpha }}{|b_1{b}_2{|}^{\alpha }}{\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{\alpha }{|f(\mathit{x})|}^{2}d\mathit{x}.$$

(61)

Applying (20) into the right hand-side of the identity (61) results in

$${\int }_{{\mathbb{R}}^2}{|\mathit{\omega }|}^{-\alpha }|{\mathcal{L}}_{A_1,A_2}^Q{\left\{f\right\}(\mathit{\omega })|}^{2}d\mathit{\omega }\le \frac{M_{\alpha }}{|b_1{b}_2{|}^{\alpha }}{\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{\alpha }|{({\mathcal{L}}_{A_1,A_2}^Q)}^{-1}\left[{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}\right](\mathit{x}){|}^{2}d\mathit{x},$$

(62)

Integrating the above relation with respect to $d\mathit{u}$ yields

$${\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{|\mathit{\omega }|}^{-\alpha }|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{2}d\mathit{\omega }d\mathit{u}\le \frac{M_{\alpha }}{|b_1{b}_2{|}^{\alpha }}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{\alpha }|{({\mathcal{L}}_{A_1,A_2}^Q)}^{-1}\left[{\mathcal{S}}_{\varphi }^Q\left\{f\right\}\right](\mathit{x}){|}^{2}d\mathit{x}d\mathit{u}.$$

(63)

With the help of Relation (27), we obtain

$${\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{|\mathit{\omega }|}^{-\alpha }|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{2}d\mathit{\omega }d\mathit{u}\le \frac{M_{\alpha }}{|b_1{b}_2{|}^{\alpha }}{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{\alpha }|f(\mathit{x})\overline{\varphi (\mathit{u}-\mathit{x},\mathit{\omega })}{|}^{2}d\mathit{x}d\mathit{u}.$$

(64)

Hence,

$${\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{|\mathit{\omega }|}^{-\alpha }|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{2}d\mathit{\omega }d\mathit{u}\le \frac{M_{\alpha }}{|b_1{b}_2{|}^{\alpha }}{\int }_{{\mathbb{R}}^2}|\varphi (\mathit{u},\mathit{\omega }){|}^{2}d\mathit{u}{\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{\alpha }|f(\mathit{x}){|}^{2}d\mathit{x},$$

(65)

which is the desired result. □

As an easy consequence of Pitt’s inequality related to the QLCST above, we deduce the following remarks:

Remark 4.

It should be noticed that by setting $\alpha =0$ in Theorem 7, we obtain the following inequality

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{2}d\mathit{\omega }d\mathit{u}\le {\varphi }_{\mathit{u},\mathit{\omega }}{\int }_{{\mathbb{R}}^2}|f(\mathit{x}){|}^{2}d\mathit{x}.\end{array}$$

(66)

Remark 5.

Due to Relations (31) and (37), Theorem 7 above will be reduced to

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}{|\mathit{\omega }|}^{-\alpha }|{\mathcal{G}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{2}d\mathit{\omega }d\mathit{u}\le \frac{M_{\alpha }}{|b_1{b}_2{|}^{\alpha }}{\parallel \varphi \parallel }_{L^2({\mathbb{R}}^2;\mathbb{H})}^2{\int }_{{\mathbb{R}}^2}{|\mathit{x}|}^{\alpha }|f(\mathit{x}){|}^{2}d\mathit{x},\end{array}$$

(67)

which is the Pitt’s uncertainty principle for the quaternion windowed linear canonical transform (compare to [3]).

Theorem 8

(Hausdorff–Young inequality for QLCST). Let $r\in [1,2]$ and s such that $\frac{1}{r}+\frac{1}{s}=1$. Then, for all $f\in L^r({\mathbb{R}}^2;\mathbb{H})$, we have

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{s}d\mathit{\omega }d\mathit{u}{)}^{1/s}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le C_r^2{(2\pi )}^{2/s-1}(|b_1||b_2{|)}^{1/s-1/2}{\left({\int }_{{\mathbb{R}}^2}|\varphi (\mathit{u},\mathit{\omega }){|}^{r}d\mathit{u}\right)}^{1/r}{\parallel f\parallel }_{L^r({\mathbb{R}}^2;\mathbb{H})},\hfill \end{array}$$

(68)

where

$$\begin{array}{c}\hfill C_r={\left(r^{1/r}{s}^{-1/s}\right)}^{1/2}.\end{array}$$

Proof. 

It follows from the Hausdorff–Young inequality for the QLCT that

$${\left({\int }_{{\mathbb{R}}^2}|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega }){|}^{s}d\mathit{\omega }\right)}^{1/s}\le C_r^2{(2\pi )}^{2/s-1}(|b_1||b_2{|)}^{1/s-1/2}{\left({\int }_{{\mathbb{R}}^2}{|f(\mathit{x})|}^{r}d\mathit{x}\right)}^{1/r}.$$

(69)

An application of Relation (20) into the right-hand side of Identity (69) above, we see that

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega }){|}^{s}d\mathit{\omega }{)}^{1/s}\hfill \\ \hspace{1em}\hspace{1em}\le C_r^2{(2\pi )}^{2/s-1}(|b_1||b_2{|)}^{1/s-1/2}{\left({\int }_{{\mathbb{R}}^2}|{({\mathcal{L}}_{A_1,A_2}^Q)}^{-1}\left[{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}\right](\mathit{x}){|}^{r}d\mathit{x}\right)}^{1/r}.\hfill \end{array}$$

(70)

This implies that

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega }){|}^{s}d\mathit{\omega }{)}^{1/s}\hfill \\ \hspace{1em}\hspace{1em}\le C_r^2{(2\pi )}^{2/s-1}(|b_1||b_2{|)}^{1/s-1/2}{\left({\int }_{{\mathbb{R}}^2}|{({\mathcal{L}}_{A_1,A_2}^Q)}^{-1}\left[{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}\right](\mathit{x}){|}^{r}d\mathit{x}\right)}^{1/r}.\hfill \end{array}$$

(71)

Therefore,

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}(\mathit{\omega }){|}^{s}d\mathit{\omega }d\mathit{u}{)}^{1/s}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le C_r^2{(2\pi )}^{2/s-1}(|b_1||b_2{|)}^{1/s-1/2}{\left({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|{({\mathcal{L}}_{A_1,A_2}^Q)}^{-1}\left[{\mathcal{L}}_{A_1,A_2}^Q\left\{f\right\}\right](\mathit{x}){|}^{r}d\mathit{x}d\mathit{u}\right)}^{1/r}.\hfill \end{array}$$

(72)

Consequently,

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{s}d\mathit{\omega }d\mathit{u}{)}^{1/s}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le C_r^2{(2\pi )}^{2/s-1}(|b_1||b_2{|)}^{1/s-1/2}{\left({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|{({\mathcal{L}}_{A_1,A_2}^Q)}^{-1}\left[{\mathcal{S}}_{\varphi }^Q\left\{f\right\}\right](\mathit{x}){|}^{r}d\mathit{x}d\mathit{u}\right)}^{1/r}.\hfill \end{array}$$

(73)

This will lead to

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{s}d\mathit{\omega }d\mathit{u}{)}^{1/s}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le C_r^2{(2\pi )}^{2/s-1}(|b_1||b_2{|)}^{1/s-1/2}{\left({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|f(\mathit{x})\overline{\varphi (\mathit{u}-\mathit{x},\mathit{\omega })}{|}^{r}d\mathit{x}d\mathit{u}\right)}^{1/r}.\hfill \end{array}$$

(74)

This equation is equivalent to

$$\begin{array}{c}({\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}|{\mathcal{S}}_{\varphi }^Q\left\{f\right\}(\mathit{u},\mathit{\omega }){|}^{s}d\mathit{\omega }d\mathit{u}{)}^{1/s}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le C_r^2{(2\pi )}^{2/s-1}(|b_1||b_2{|)}^{1/s-1/2}{\left({\int }_{{\mathbb{R}}^2}|\varphi (\mathit{u},\mathit{\omega }){|}^{r}d\mathit{u}\right)}^{1/r}{\left({\int }_{{\mathbb{R}}^2}|f(\mathit{x}){|}^{r}d\mathit{x}\right)}^{1/r},\hfill \end{array}$$

which is the required result. □

5. Conclusions

This study introduced the quaternion linear canonical S-transform as a natural extension of the linear canonical S-transform within the framework of quaternion algebra. Detailed exploration of its key properties was conducted alongside an investigation into various uncertainty principles associated with this generalized transformation.

Future work would be related to further investigations of several uncertainty principles associated with the quaternion linear canonical S-transform, such as Nazarov’s uncertainty principle, Hardy’s-type uncertainty principle, Beurling’s uncertainty principle, Lieb’s inequality and Matolcsi-Szücs uncertainty principles.