Novel Uncertainty Principles Concerning Linear Canonical Wavelet Transform

Abstract

The linear canonical wavelet transform is a nontrivial generalization of the classical wavelet transform in the context of the linear canonical transform. In this article, we first present a direct interaction between the linear canonical transform and Fourier transform to obtain the generalization of the uncertainty principles related to the linear canonical transform. We develop these principles for constructing some uncertainty principles concerning the linear canonical wavelet transform.

1. Introduction

It is well known that the uncertainty principles play prominent roles in quantum physics and Fourier analysis. In quantum physics they assert that there is a lower bound of the product of position and momentum for a system. In harmonic analysis they inform that a nonzero function and its Fourier transform cannot both be simultaneously sharply localized at the same time. The latter has inspired a number of researchers in the investigation of the generalizations of the uncertainty principles in various transformations. For instance, in the papers [1,2], the authors have reported some uncertainty principles related to the shearlet transform. The uncertainty principles for the wavelet transform have been carried out in papers [3,4,5,6].

Recently, some researchers have come up with the linear canonical transform (LCT), which is a natural extension of the Fourier transform (FT). Based on the LCT, they then have generalized the classical wavelet transform (WT) to the LCT, which is often called the linear canonical wavelet transform (LCWT). Compared to the WT, the LCWT is more flexible due to its three independent parameters, and can be considered as an effective tool for expanding applications of the WT in image processing [7,8,9,10,11,12]. Further, several of its general properties like the orthogonality relation, inversion formula, and reproducing kernel, were studied in some detail. However, to the best our knowledge, the uncertainty principle as the main result for the linear canonical wavelet transform has not yet been published. In the present paper, we first present an interesting relationship between the Fourier transform and the linear canonical transform. The relation allows us to derive a new generalization of uncertainty principle for the linear canonical transform. We also study a natural link between the linear canonical wavelet transform and classical wavelet transform, which permits us to provide a simple proof of the orthogonality relation of the linear canonical wavelet transform. Subsequently, based on the principles and relevant properties of the linear canonical wavelet transform we investigate in detail several versions of uncertainty principle associated with the linear canonical wavelet transform.

We display here the plan of this paper. In Section 2, we shortly review the definition of the linear canonical transform and basic notations that will be useful later. This section also describes a natural link between the Fourier transform and the linear canonical transform. Section 3 provides the definition of the linear canonical wavelet transform and collects its essential properties. In Section 4, we focus our attention on the derivation of an uncertainty principle for the linear canonical transform and extend to the investigation of several uncertainty principles involving the linear canonical wavelet transform. In Section 5, we conclude the article.

2. Definition of LCT

Let us first recall the linear canonical transform (LCT) and its basic properties, as this will be useful later. The definition of the LCT was initiated by Moshinsky and Collins [13,14] with their endeavor to generalize the Fourier transform in higher dimensions. For more results on the properties of the LCT and applications, see, e.g., [11,15,16,17,18].

Definition 1.

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\in {\mathbb{R}}^{2\times 2}$ be a matrix parameter satisfying $det\left(A\right)=ad-bc=1$. The LCT of a function $f\in L^1\left(\mathbb{R}\right)$ is defined by

(1) $\begin{array}{c}\hfill L_A\left\{f\right\}\left(\xi \right)=\left\{\begin{array}{c}{\int }_{\mathbb{R}}f\left(v\right)K_A(\xi ,v)dv,b\ne 0\hfill \\ \sqrt{d}{e}^{i\frac{cd}{2}{v}^2}f\left(dv\right),b=0,\hfill \end{array}\right.\end{array}$

where $K_A(\xi ,v)$ is so-called kernel of the LCT given by

$K_A(\xi ,v)=\frac{1}{\sqrt{2\pi b}}{e}^{\frac{i}{2}(\frac{a}{b}{v}^2-\frac{2}{b}\xi v+\frac{d}{b}{\xi }^2-\frac{\pi }{2})}.$

It is obvious that the LCT kernel fulfills the following property:

$-K_{A^{-1}}(v,\xi )=\overline{K_A(\xi ,v)}=\frac{1}{\sqrt{2\pi b}}{e}^{-\frac{i}{2}(\frac{a}{b}{v}^2-\frac{2}{b}\xi v+\frac{d}{b}{\xi }^2-\frac{\pi }{2})}.$

In this respect we only consider the case of $b\ne 0$, because the LCT of a signal is essentially a chirp multiplication for $b=0$. Without loss of generality throughout this article, we always assume that $b>0$. As a result of the LCT definition (1), for $A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$, we get the definition of the Fourier transform. The LCT inversion formula is described by

(2)$\begin{array}{cc}\hfill f\left(v\right)& ={\int }_{\mathbb{R}}{L}_A\left\{f\right\}\left(\xi \right)\overline{K_A(\xi ,v)}d\xi \hfill \\ \hfill & ={\int }_{\mathbb{R}}{L}_A\left\{f\right\}\left(\xi \right)\frac{1}{\sqrt{2\pi b}}{e}^{-\frac{i}{2}(\frac{a}{b}{v}^2-\frac{2}{b}\xi v+\frac{d}{b}{\xi }^2-\frac{\pi }{2})}d\xi .\hfill \end{array}$

The basic interaction between the LCT and the Fourier transform (FT) is described by

(3)$\begin{array}{c}\hfill L_A\left\{f\right\}\left(\xi \right)=\frac{e^{-i\frac{\pi }{4}}}{\sqrt{b}}{e}^{\frac{id}{2b}{\xi }^2}\mathcal{F}\left\{e^{\frac{ia}{2b}{v}^2}f\left(v\right)\right\}\left(\frac{\xi }{b}\right),\end{array}$

where $\mathcal{F}\left\{f\right\}\left(\xi \right)=\widehat{f}\left(\xi \right)$ is the Fourier transform of a function (or a signal) $f\in L^2\left(\mathbb{R}\right)$ defined by (see [19,20])

(4)$\mathcal{F}\left\{f\right\}\left(\xi \right)=\frac{1}{\sqrt{2\pi }}{\int }_{\mathbb{R}}f\left(v\right)e^{-iv\xi }dv.$

Now by letting

(5)$h\left(v\right)=e^{-i\frac{\pi }{4}}{e}^{\frac{ia}{2b}{v}^2}f\left(v\right),$

one can write (3) in the form

(6)$\begin{array}{c}\hfill e^{-\frac{id}{2b}{\xi }^2}{L}_A\left\{f\right\}\left(\xi \right)=\frac{1}{\sqrt{b}}\mathcal{F}\left\{h\right\}\left(\frac{\xi }{b}\right).\end{array}$

Due to (3), we also have

(7)$\begin{array}{c}\hfill L_A\left\{e^{-\frac{ia}{2b}{v}^2}f\right\}\left(\gamma \xi \right)=\frac{e^{-i\frac{\pi }{4}}}{\sqrt{b}}{e}^{\frac{id}{2b}{\left(\gamma \xi \right)}^2}\mathcal{F}\left\{f\left(v\right)\right\}\left(\frac{\gamma \xi }{b}\right).\end{array}$

Now, let $L^q\left(\mathbb{R}\right)$ denote the Banach space of integrable functions defined in $\mathbb{R}$ with the norm

(8)$\begin{array}{c}\hfill {\parallel f\parallel }_{L^q\left(\mathbb{R}\right)}={\left({\int }_{\mathbb{R}}{\left|f\left(v\right)\right|}^{q}dv\right)}^{1/q}<\infty ,\mathrm{for}q\in [1,\infty ),\end{array}$

and

(9)$\begin{array}{c}\hfill {\parallel f\parallel }_{L^{\infty }\left(\mathbb{R}\right)}={\mathrm{ess}sup}_{v\in \mathbb{R}}\left|f\left(v\right)\right|<\infty ,\mathrm{for}q=\infty .\end{array}$

As is known, Parseval’s formula for the LCT is given by

(10)$\begin{array}{c}\hfill {\int }_{\mathbb{R}}f\left(v\right)\overline{g\left(v\right)}dv=(f,g)=(L_A\left\{f\right\},L_A\left\{g\right\})={\int }_{\mathbb{R}}{L}_A\left\{f\right\}\left(\xi \right)\overline{L_A\left\{g\right\}\left(\xi \right)}d\xi .\end{array}$

When $f=g$ in (10), we obtain

(11)${\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2={\parallel L_A\left\{f\right\}\parallel }_{L^2\left(\mathbb{R}\right)}^2.$

3. Linear Canonical Wavelet Transform (LCWT)

Below, we discuss famous properties of the linear canonical wavelet transform. We also give a new proof of its orthogonality relation.

3.1. Definition

In this subsection, we introduce the linear canonical wavelet transform and its relationship to the classical wavelet transform. We also summarize its general properties, which are the generalization of the classical wavelet transform properties.

The family of wavelets ${\phi }_{\gamma ,\mu }$ constructed from translation and dilation of $\phi \in L^2\left(\mathbb{R}\right)$ is given by

(12)${\phi }_{\gamma ,\mu }\left(v\right)=\frac{1}{\sqrt{\gamma }}\phi \left(\frac{v-\mu }{\gamma }\right),\mu \in \mathbb{R},\gamma \in {\mathbb{R}}_{+}.$

where $\phi$ is often called the basic (mother) wavelets and $\mu$ is a parameter of dilation and $\gamma$ is the parameter of translation. The Fourier transform of (12) is given by

(13)${\widehat{\phi }}_{\gamma ,\mu }\left(\xi \right)=\sqrt{\gamma }{e}^{-i\mu \xi }\mathcal{F}\left\{\phi \right\}\left(\gamma \xi \right).$

The wavelet transform of function $f\in L^2\left(\mathbb{R}\right)$ with respect to the basic wavelet $\phi$ is defined by

(14)$\begin{array}{cc}\hfill T_{\phi }f(\gamma ,\mu )& ={\int }_{\mathbb{R}}f\left(v\right)\overline{{\phi }_{\gamma ,\mu }\left(v\right)}dv\hfill \\ \hfill & =\frac{1}{\sqrt{\gamma }}{\int }_{\mathbb{R}}f\left(v\right)\overline{\phi \left(\frac{v-\mu }{\gamma }\right)}dv.\hfill \end{array}$

From Equation (12), we further obtain the definition of the family of linear canonical basic wavelets as

(15)${\phi }_{\gamma ,\mu ,A}\left(v\right)=\frac{1}{\sqrt{\gamma }}\phi \left(\frac{v-\mu }{\gamma }\right)e^{-i\frac{a}{2b}(v^2-{\mu }^2)}.$

An essential property of the linear canonical basic wavelets (15) is the following condition given by:

Definition 2.

A basic wavelet $\phi \in L^2\left(\mathbb{R}\right)$ related to the LCT is known as the admissibility condition if it satisfies

(16) $0<C_{\phi ,A}={\int }_{{\mathbb{R}}^{+}}|L_A\left\{e^{-i\frac{a}{2b}{(·)}^2}\phi \right\}\left(\gamma \xi \right){|}^2\frac{d\gamma }{\gamma }<\infty .$

According to (3), for $A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$, the admissibility condition defined by (16) above will be reduced to

(17)$0<C_{\phi }={\int }_{{\mathbb{R}}^{+}}|\mathcal{F}\left\{\phi \right\}\left(\gamma \xi \right){|}^2\frac{d\gamma }{\gamma }<\infty .$

It is clear that (17) is the the admissibility condition for the classical wavelet transform (see, e.g., [21,22]).

Lemma 1.

The relationship between (16) and (17) is given by

(18) $\begin{array}{c}\hfill C_{\phi }=b{C}_{\phi ,A}.\end{array}$

Proof. 

By (7) we have

(19)$\begin{array}{cc}\hfill C_{\phi }& =\left({\int }_{{\mathbb{R}}^{+}}|\mathcal{F}\left\{\phi \right\}\left(\gamma \xi \right){|}^2\frac{d\gamma }{\gamma }\right)\hfill \\ \hfill & =\left({\int }_{{\mathbb{R}}^{+}}|\mathcal{F}\left\{\phi \right\}\left(\frac{\gamma \xi }{b}\right){|}^2\frac{d\gamma }{\gamma }\right)\hfill \\ \hfill & =\left({\int }_{{\mathbb{R}}^{+}}|e^{i\frac{\pi }{4}}\sqrt{b}{e}^{-\frac{id}{2b}{\left(\gamma \xi \right)}^2}{L}_A\left\{e^{-\frac{ia}{2b}{v}^2}\phi \right\}\left(\gamma \xi \right){|}^2\frac{d\gamma }{\gamma }\right)\hfill \\ \hfill & =b\left({\int }_{{\mathbb{R}}^{+}}|L_A\left\{e^{-\frac{ia}{2b}{v}^2}\phi \right\}\left(\gamma \xi \right){|}^2\frac{d\gamma }{\gamma }\right)\hfill \\ \hfill & =b{C}_{\phi ,A},\hfill \end{array}$

and the proof is complete. □

Lemma 2.

[23] Let φ be a basic wavelet. The family of the linear canonical basic wavelets (15) can be rewritten in terms of the LCT as

(20) $\begin{array}{cc}\hfill L_A\left\{{\phi }_{\gamma ,\mu ,A}\right\}\left(\xi \right)& =\sqrt{\gamma }{e}^{i\frac{d}{2b}{\xi }^2}{e}^{i\frac{a}{2b}{\mu }^2}{e}^{-\frac{i}{b}\mu \xi }{e}^{-i\frac{d}{2b}{\left(\gamma \xi \right)}^2}{L}_A\left\{e^{-i\frac{a}{2b}{(·)}^2}\phi \right\}\left(\gamma \xi \right).\hfill \end{array}$

3.2. Properties

Following [23], we introduce the definition of the linear canonical wavelet transform (LCWT) below.

Definition 3.

Let φ be basic wavelet in $L^2\left(\mathbb{R}\right)$. The LCWT of function $f\in L^2\left(\mathbb{R}\right)$ is given by

(21) $\begin{array}{cc}\hfill T_{\phi }^{A}f(\gamma ,\mu )& =(f,{\phi }_{\gamma ,\mu ,A})\hfill \\ \hfill & ={\int }_{\mathbb{R}}f\left(v\right)\overline{{\phi }_{\gamma ,\mu ,A}\left(v\right)}dv\hfill \\ \hfill & =\frac{1}{\sqrt{\gamma }}{\int }_{\mathbb{R}}f\left(v\right)\overline{\phi \left(\frac{v-\mu }{\gamma }\right)}{e}^{i\frac{a}{2b}(v^2-{\mu }^2)}dv.\hfill \end{array}$

We collect a few general properties of the LCWT (21). The first result is the following lemma. For the sake of the readers’convenience, we state it with a proof.

Lemma 3.

Suppose that $f,g,\varphi ,\phi \in L^2\left(\mathbb{R}\right)$. One has

(22) $\begin{array}{c}\hfill \parallel T_{\phi }^{A}f{T}_{\varphi }^A{g\parallel }_{L^{\infty }\left(\mathbb{R}\right)}\le {\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}{\parallel \phi \parallel }_{L^2\left(\mathbb{R}\right)}{\parallel g\parallel }_{L^2\left(\mathbb{R}\right)}{\parallel \varphi \parallel }_{L^2\left(\mathbb{R}\right)}.\end{array}$

Especially,

(23) $\begin{array}{c}\hfill \parallel T_{\phi }^A{f\parallel }_{L^{\infty }\left(\mathbb{R}\right)}^2\le {\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2{\parallel \phi \parallel }_{L^2\left(\mathbb{R}\right)}^2.\end{array}$

Proof. 

We have from the Cauchy–Schwartz inequality

(24)$\begin{array}{cc}\hfill |T_{\phi }^{A}f(\gamma ,\mu )|& =\left|\frac{1}{\sqrt{\gamma }}{\int }_{\mathbb{R}}f\left(v\right)\overline{\phi \left(\frac{v-\mu }{\gamma }\right)}{e}^{i\frac{a}{2b}(v^2-{\mu }^2)}dv\right|\hfill \\ \hfill & \le \frac{1}{\sqrt{\gamma }}{\int }_{\mathbb{R}}\left|f\left(v\right)\overline{\phi \left(\frac{v-\mu }{\gamma }\right)}{e}^{i\frac{a}{2b}(v^2-{\mu }^2)}\right|dv\hfill \\ \hfill & =\frac{1}{\sqrt{\gamma }}{\int }_{\mathbb{R}}\left|f\left(v\right)\overline{\phi \left(\frac{v-\mu }{\gamma }\right)}\right|dv\hfill \\ \hfill & \le \frac{1}{\sqrt{\gamma }}{\left({\int }_{\mathbb{R}}{\left|f\left(v\right)\right|}^{2}dv\right)}^{\frac{1}{2}}{\left({\int }_{\mathbb{R}}|\overline{\phi \left(\frac{v-\mu }{\gamma }\right)}{|}^{2}dv\right)}^{\frac{1}{2}}.\hfill \end{array}$

Let $r=\frac{v-\mu }{\gamma }$, then the above identity turns into

(25)$\begin{array}{cc}\hfill |T_{\phi }^{A}f(\gamma ,\mu )|& \le {\left({\int }_{\mathbb{R}}{\left|f\left(v\right)\right|}^{2}dv\right)}^{\frac{1}{2}}{\left({\int }_{\mathbb{R}}{\left|\phi \left(r\right)\right|}^{2}dr\right)}^{\frac{1}{2}}\hfill \\ \hfill & ={\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}{\parallel \phi \parallel }_{L^2\left(\mathbb{R}\right)}.\hfill \end{array}$

Then, we obtain (22). □

Lemma 4.

[24] Let Ψ, $f\in L^2\left(\mathbb{R}\right)$. Then, the LCWT of a function of $f\in L^2\left(\mathbb{R}\right)$ with matrix parameter $A=(a,b,c,d)$ can be changed to the the classical wavelet transform, that is

(26) $\begin{array}{c}\hfill T_{\Psi }^{A}f(\gamma ,\mu )=e^{-i\frac{a}{2b}{\mu }^2}{T}_{\Psi }\check{f}(\gamma ,\mu ),\end{array}$

where

(27) $\begin{array}{c}\hfill \check{f}\left(v\right)=f\left(v\right)e^{i\frac{a}{2b}{v}^2}.\end{array}$

Lemma 5.

[23] Let $f,\phi \in L^2\left(\mathbb{R}\right)$. Then, the LCWT (21) has a linear canonical Fourier representation form

(28) $\begin{array}{cc}\hfill T_{\phi }^{A}f(\gamma ,\mu )& =\sqrt{\gamma }{\int }_{\mathbb{R}}{e}^{-i\frac{d}{2b}{\xi }^2}{e}^{-i\frac{a}{2b}{\mu }^2}{e}^{\frac{i}{b}\mu \xi }{e}^{i\frac{d}{2b}{\left(\gamma \xi \right)}^2}\overline{L_A\left\{e^{-i\frac{a}{2b}{(·)}^2}\phi \right\}\left(\gamma \xi \right)}{L}_A\left\{f\right\}\left(\xi \right)d\xi .\hfill \end{array}$

Lemma 6.

[23] Suppose that $f,\phi \in L^2\left(\mathbb{R}\right)$. The LCT of the LCWT is given by

(29) $\begin{array}{cc}\hfill L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}& =\sqrt{\gamma }{e}^{i\frac{\pi }{4}}\sqrt{2\pi b}{e}^{i\frac{d}{2b}{\left(\gamma \xi \right)}^2}\overline{L_A\left\{e^{-i\frac{a}{2b}{(·)}^2}\phi \right\}\left(\gamma \xi \right)}{L}_A\left\{f\right\}\left(\xi \right),\hfill \end{array}$

and

(30) $\begin{array}{c}\hfill |L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}|=\sqrt{2\pi b\gamma }|L_A\left\{e^{-i\frac{a}{2b}{(·)}^2}\phi \right\}\left(\gamma \xi \right)L_A\left\{f\right\}\left(\xi \right)|.\end{array}$

Now we provide a new proof of the orthogonality relation for the LCWT using the relationship between the linear canonical transform and the classical wavelet transform.

Theorem 1

(LCWT orthogonality relation). Assume that the basic wavelet $\phi \in L^2\left(\mathbb{R}\right)$ related to the LCT satisfies the admissibility condition (16). For f, $g\in L^2\left(\mathbb{R}\right)\cap L^1\left(\mathbb{R}\right)$ we have

(31)${\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{T}_{\phi }^{A}f(\gamma ,\mu )\overline{T_{\phi }^{A}g(\gamma ,\mu )}d\mu \frac{d\gamma }{{\gamma }^2}=2\pi b{C}_{\phi ,A}{(f,g)}_{L^2\left(\mathbb{R}\right)},$

and

(32)${\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}={\int }_{{\mathbb{R}}^{+}}\parallel T_{\phi }^A{f(\gamma ,\mu )\parallel }_{L^2\left(\mathbb{R}\right)}^2\frac{d\gamma }{{\gamma }^2}=2\pi b{C}_{\phi ,A}{\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2.$

Proof. 

Because $\check{f}\left(v\right)$ and $\check{g}\left(v\right)$ defined by (27) are in $L^2\left(\mathbb{R}\right)$, then the orthogonality relation for the classical wavelet transform results in

(33)$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^{+}}\left({\int }_{\mathbb{R}}{T}_{\phi }\check{f}(\gamma ,\mu )\overline{T_{\phi }\check{g}(\gamma ,\mu )}d\mu \right)\frac{d\gamma }{{\gamma }^2}=2\pi C_{\phi }{(\check{f},\check{g})}_{L^2\left(\mathbb{R}\right)}.\end{array}$

where $C_{\Psi }$ is given by (17). With the aid of Equations (18) and (26), both sides of (33) may be expressed in the form

(34)$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^{+}}\left({\int }_{\mathbb{R}}{e}^{-i\frac{a}{2b}{\mu }^2}{e}^{i\frac{a}{2b}{\mu }^2}{T}_{\phi }\check{f}(\gamma ,\mu )\overline{T_{\phi }\check{g}(\gamma ,\mu )}d\mu \right)\frac{d\gamma }{{\gamma }^2}=2\pi C_{\phi }{(\check{f},\check{g})}_{L^2\left(\mathbb{R}\right)}\\ \hfill {\int }_{{\mathbb{R}}^{+}}\left({\int }_{\mathbb{R}}{e}^{-i\frac{a}{2b}{\mu }^2}{T}_{\phi }\check{f}(\gamma ,\mu )\overline{e^{-i\frac{a}{2b}{\mu }^2}{T}_{\phi }\check{g}(\gamma ,\mu )}d\mu \right)\frac{d\gamma }{{\gamma }^2}=2\pi C_{\phi }{\int }_{\mathbb{R}}f\left(v\right)e^{i\frac{a}{2b}{v}^2}\overline{g\left(v\right)e^{i\frac{a}{2b}{v}^2}}dv\\ \hfill {\int }_{{\mathbb{R}}^{+}}\left({\int }_{\mathbb{R}}{T}_{\phi }^{A}f(\gamma ,\mu )\overline{T_{\phi }^{A}g(\gamma ,\mu )}d\mu \right)\frac{d\gamma }{{\gamma }^2}=2\pi b{C}_{\phi ,A}{\int }_{\mathbb{R}}f\left(v\right)\overline{g\left(v\right)}dv,\end{array}$

and the proof is complete. □

Remark 1.

It is easily seen that the proof of Theorem 1 is simpler than the proof of ones given in the references [23,24].

4. Uncertainty Principles for the LCT and LCWT

The aim of this section is to establish several uncertainty principles related to the LCT and LCWT. Thus far, many variations and generalizations of the uncertainty principles were proposed by many researchers in the literature (see, e.g., [25,26,27,28,29,30]). The starting point for our result is a generalization of the uncertainty principle for the FT in the framework of the LCT as shown in Theorem 2 below. We apply this result to investigate several versions of uncertainty principles related to the LCWT.

4.1. Uncertainty Principles for the LCT

In this part, we study uncertainty principles for the LCT, which will be necessary to prove the uncertainty principles related to the LCWT.

Theorem 2

(LCT uncertainty principle). Let $f\in L^2\left(\mathbb{R}\right)$ be a complex function. If $L_A\left\{f\right\}\in L^2\left(\mathbb{R}\right)$, then the following inequality holds

(35)${\left({\int }_{\mathbb{R}}{v}^2{\left|f\left(v\right)\right|}^{2}dv\right)}^{1/2}{\left({\int }_{\mathbb{R}}{\xi }^2{\left|L_A\left\{f\right\}\left(\xi \right)\right|}^{2}d\xi \right)}^{1/2}\ge \frac{b}{2}{\int }_{\mathbb{R}}{\left|f\left(v\right)\right|}^{2}dv.$

Proof. 

The proof is similar to that of Theorem 3 below. □

The new generalization of the above uncertainty principle is described by the following result.

Theorem 3.

Under the same situation as in Theorem 2, we have

(36) $\begin{array}{cc}\hfill {\left({\int }_{\mathbb{R}}{v}^p|f\left(v\right){|}^{p}dv\right)}^{1/p}{\left({\int }_{\mathbb{R}}{\xi }^p|L_A\left\{f\right\}\left(\xi \right){|}^{p}d\xi \right)}^{1/p}& \ge \frac{b^{\frac{1}{2}+\frac{1}{p}}}{2}{\int }_{\mathbb{R}}|f\left(v\right){|}^{2}dv,\hfill \end{array}$

for $1\le p\le 2$.

Proof. 

It directly follows from the generalization of the Heisenberg’s inequality for the FT (see [31]) that

$\begin{array}{c}\hfill {\int }_{\mathbb{R}}{v}^p{\left|f\left(v\right)\right|}^{p}dv{\int }_{\mathbb{R}}{\xi }^p{\left|\mathcal{F}\left\{f\right\}\left(\xi \right)\right|}^{p}d\xi \ge {\left(\frac{1}{2}\right)}^p{\left({\int }_{\mathbb{R}}{\left|f\left(v\right)\right|}^{2}dv\right)}^p.\end{array}$

Since h defined by (5) belongs to $L^2\left(\mathbb{R}\right)$, then by replacing f by h into the above identity, we obtain

$\begin{array}{c}\hfill {\int }_{\mathbb{R}}{v}^p{\left|h\left(v\right)\right|}^{p}dv{\int }_{\mathbb{R}}{\xi }^p{\left|\mathcal{F}\left\{h\right\}\left(\xi \right)\right|}^{p}d\xi \ge {\left(\frac{1}{2}\right)}^p{\left({\int }_{\mathbb{R}}{\left|h\left(v\right)\right|}^{2}dv\right)}^p.\end{array}$

Substituting $\xi$ with $\frac{\xi }{b}$, we see from relations (5) and (6) that

$\begin{array}{cc}\hfill {\int }_{\mathbb{R}}{v}^p|e^{-i\frac{\pi }{4}}{e}^{\frac{ia}{2b}{v}^2}f\left(v\right){|}^{p}dv{\int }_{\mathbb{R}}\frac{{\xi }^p}{b^p}|\mathcal{F}\left\{h\right\}\left(\frac{\xi }{b}\right){|}^{p}d\frac{\xi }{b}& \ge {\left(\frac{1}{2}\right)}^p{\left({\int }_{\mathbb{R}}|e^{-i\frac{\pi }{4}}{e}^{\frac{ia}{2b}{v}^2}f\left(v\right){|}^{2}dv\right)}^p\hfill \\ \hfill {\int }_{\mathbb{R}}\frac{v^p}{b^{p+1}}|f\left(v\right){|}^{p}dv{\int }_{\mathbb{R}}{\xi }^p|\mathcal{F}\left\{h\right\}\left(\frac{\xi }{b}\right){|}^{p}d\xi & \ge \frac{1}{2^p}{\left({\int }_{\mathbb{R}}|f\left(v\right){|}^{2}dv\right)}^p\hfill \\ \hfill {\int }_{\mathbb{R}}{v}^p|f\left(v\right){|}^{p}dv{\int }_{\mathbb{R}}{\xi }^p|\mathcal{F}\left\{h\right\}\left(\frac{\xi }{b}\right){|}^{p}d\xi & \ge \frac{b^{p+1}}{2^p}{\left({\int }_{\mathbb{R}}|f\left(v\right){|}^{2}dv\right)}^p\hfill \\ \hfill {\left({\int }_{\mathbb{R}}{v}^p|f\left(v\right){|}^{p}dv\right)}^{1/p}{\left({\int }_{\mathbb{R}}{\xi }^p|\mathcal{F}\left\{h\right\}\left(\frac{\xi }{b}\right){|}^{p}d\xi \right)}^{1/p}& \ge \frac{b^{1+\frac{1}{p}}}{2}{\int }_{\mathbb{R}}|f\left(v\right){|}^{2}dv\hfill \\ \hfill {\left({\int }_{\mathbb{R}}{v}^p|f\left(v\right){|}^{p}dv\right)}^{1/p}{\left({\int }_{\mathbb{R}}{\xi }^p|\sqrt{b}{e}^{-\frac{id}{2b}{\xi }^p}{L}_A\left\{f\right\}\left(\xi \right){|}^{p}d\xi \right)}^{1/p}& \ge \frac{b^{1+\frac{1}{p}}}{2}{\int }_{\mathbb{R}}|f\left(v\right){|}^{2}dv\hfill \\ \hfill {\left({\int }_{\mathbb{R}}{v}^p|f\left(v\right){|}^{p}dv\right)}^{1/p}{\left({\int }_{\mathbb{R}}{\xi }^p|L_A\left\{f\right\}\left(\xi \right){|}^{p}d\xi \right)}^{1/p}& \ge \frac{b^{\frac{1}{2}+\frac{1}{p}}}{2}{\int }_{\mathbb{R}}|f\left(v\right){|}^{2}dv,\hfill \end{array}$

and the proof is complete. □

It can be easily seen that when $p=2$, Theorem 3 becomes Theorem 2. This shows that Theorem 3 is a general form of Theorem 2.

4.2. Uncertainty Principles for the LCWT

Let us first build an uncertainty principle for the LCWT, which explains how the LCWT interacts to the LCT of a function. In order to prove the theorem below, we need the help of the following lemma.

Lemma 7.

One has

(37) ${\int }_{{\mathbb{R}}^{+}}\parallel {\xi }^2{L}_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}{\parallel }_{L^2\left(\mathbb{R}\right)}^2\frac{d\gamma }{{\gamma }^2}=2\pi b{C}_{\phi ,A}{\parallel \xi L_A\left\{f\right\}\parallel }_{L^2\left(\mathbb{R}\right)}^2.$

Proof. 

According to (16), (30) and the Fubini’s theorem, we get

$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^{+}}& \parallel {\xi }^2{L}_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}{\parallel }_{L^2\left(\mathbb{R}\right)}^2\frac{d\gamma }{{\gamma }^2}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\xi }^2|L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}{|}^{2}d\xi \frac{d\gamma }{{\gamma }^2}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}2\pi b|L_A\left\{e^{-i\frac{a}{2b}{(·)}^2}\phi \right\}\left(\gamma \xi \right){|}^2|\xi L_A\left\{f\right\}\left(\xi \right){|}^{2}d\xi \frac{d\gamma }{\gamma }\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^{+}}2\pi b|L_A\left\{e^{-i\frac{a}{2b}{(·)}^2}\phi \right\}\left(\gamma \xi \right){|}^2\frac{d\gamma }{\gamma }{\int }_{\mathbb{R}}|\xi L_A\left\{f\right\}\left(\xi \right){|}^{2}d\xi \hfill \\ \hfill & =2\pi b{C}_{\phi ,A}{\int }_{\mathbb{R}}|\xi L_A\left\{f\right\}\left(\xi \right){|}^{2}d\xi ,\hfill \end{array}$

which completes the proof. □

Theorem 4.

Let $\phi \in L^2\left(\mathbb{R}\right)$ be an admissible wavelet that satisfies the admissibility condition (16); then, for every $f\in L^2\left(\mathbb{R}\right)$ we have the inequality

(38) $\begin{array}{c}\hfill {\left({\int }_{{\mathbb{R}}^{+}}{\parallel \mu T_{\phi }^{A}f(\gamma ,·)\parallel }_{L^2\left(\mathbb{R}\right)}^2\frac{d\gamma }{{\gamma }^2}\right)}^{1/2}{\left(\parallel \xi L_A{\left\{f\right\}\parallel }_{L^2\left(\mathbb{R}\right)}^2\right)}^{1/2}\ge b^{3/2}\sqrt{\frac{\pi C_{\phi ,A}}{2}}{\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2.\end{array}$

Proof. 

With the help of the uncertainty principle for the LCT we obtain

(39)$\begin{array}{c}\hfill {\left[\parallel \mu T_{\phi }^A{f(\gamma ,·)\parallel }_{L^2\left(\mathbb{R}\right)}^2\right]}^{1/2}{\left[\parallel \xi L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}{\parallel }_{L^2\left(\mathbb{R}\right)}^2\right]}^{1/2}\ge \frac{b}{2}{\parallel T_{\phi }^{A}f(\gamma ,·)\parallel }_{L^2\left(\mathbb{R}\right)}^2.\end{array}$

Now integrating both sides of (39) with respect to the Haar measure $\frac{d\gamma }{{\gamma }^2}$, we obtain

(40)$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^{+}}\left({\left[\parallel \mu T_{\phi }^A{f(\gamma ,·)\parallel }_{L^2\left(\mathbb{R}\right)}^2\right]}^{1/2}{\left[\parallel \xi L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}{\parallel }_{L^2\left(\mathbb{R}\right)}^2\right]}^{1/2}\right)\frac{d\gamma }{{\gamma }^2}\\ \hfill \ge \frac{b}{2}{\int }_{{\mathbb{R}}^{+}}{\parallel T_{\phi }^{A}f(\gamma ,·)\parallel }_{L^2\left(\mathbb{R}\right)}^2\frac{d\gamma }{{\gamma }^2}.\end{array}$

By applying the Cauchy–Schwartz inequality on the left-hand side of (40), we see that

(41)$\begin{array}{c}\hfill {\left({\int }_{{\mathbb{R}}^{+}}{\parallel \mu T_{\phi }^{A}f(\gamma ,·)\parallel }_{L^2\left(\mathbb{R}\right)}^2\frac{d\gamma }{{\gamma }^2}\right)}^{1/2}{\left({\int }_{{\mathbb{R}}^{+}}{\parallel \xi L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}\parallel }_{L^2\left(\mathbb{R}\right)}^2\frac{d\gamma }{{\gamma }^2}\right)}^{1/2}\\ \hfill \ge \frac{b}{2}{\int }_{{\mathbb{R}}^{+}}{\parallel T_{\phi }^{A}f(\gamma ,·)\parallel }_{L^2\left(\mathbb{R}\right)}^2\frac{d\gamma }{{\gamma }^2}.\end{array}$

Then, replacing (37) into the second term of (41), we easily obtain

(42)$\begin{array}{cc}\hfill {\left({\int }_{{\mathbb{R}}^{+}}{\parallel \mu T_{\phi }^{A}f(\gamma ,·)\parallel }_{L^2\left(\mathbb{R}\right)}^2\frac{d\gamma }{{\gamma }^2}\right)}^{1/2}& {\left(2\pi b{C}_{\phi ,A}{\parallel \xi L_A\left\{f\right\}\parallel }_{L^2\left(\mathbb{R}\right)}^2\right)}^{1/2}\hfill \\ \hfill & \ge \frac{b}{2}{\int }_{{\mathbb{R}}^{+}}{\parallel T_{\phi }^{A}f(\gamma ,·)\parallel }_{L^2\left(\mathbb{R}\right)}^2\frac{d\gamma }{{\gamma }^2}.\hfill \end{array}$

Placing (32) into the right-hand side of (42) and simplifying it, we finally find

(43)$\begin{array}{c}\hfill {\left({\int }_{{\mathbb{R}}^{+}}{\parallel \mu T_{\phi }^{A}f(\gamma ,·)\parallel }_{L^2\left(\mathbb{R}\right)}^2\frac{d\gamma }{{\gamma }^2}\right)}^{1/2}{\left(\parallel \xi L_A{\left\{f\right\}\parallel }_{L^2\left(\mathbb{R}\right)}^2\right)}^{1/2}\ge b^{3/2}\sqrt{\frac{\pi C_{\phi ,A}}{2}}{\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2,\end{array}$

which gives the desired result. □

Remark 2.

It can be observed that for $A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$, we obtain

(44) $\begin{array}{c}\hfill {\left({\int }_{{\mathbb{R}}^{+}}{\parallel \mu T_{\phi }f(\gamma ,·)\parallel }_{L^2\left(\mathbb{R}\right)}^2\frac{d\gamma }{{\gamma }^2}\right)}^{1/2}{\left({\parallel \xi \mathcal{F}\left\{f\right\}\parallel }_{L^2\left(\mathbb{R}\right)}^2\right)}^{1/2}\ge \sqrt{\frac{\pi C_{\phi }}{2}}{\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2,\end{array}$

which is the same as the uncertainty principle for the classical wavelet transform [6].

In the following, we explicitly derive a logarithmic uncertainty principle for the LCWT. In order to prove the result, we use the following lemma:

Lemma 8.

We have

(45) ${\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}ln\left|\xi \right||L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}{\left(\xi \right)|}^{2}d\xi \frac{d\gamma }{{\gamma }^2}=2\pi b{C}_{\phi ,A}{\int }_{\mathbb{R}}ln\left|\xi \right||L_A{\left\{f\right\}\left(\xi \right)|}^{2}d\xi .$

Proof. 

The proof of Lemma 8 is essentially the same as that of Lemma 7, so we omit it. □

Let us now state a fundamental result of this subsection.

Theorem 5.

Let $\phi \in L^2\left(\mathbb{R}\right)$ be an admissible wavelet. Then for every $f\in L^2\left(\mathbb{R}\right)$ such that $T_{\phi }^{A}f(\gamma ,·)\in L^2\left(\mathbb{R}\right)$ we have

(46) $\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}ln|\mu & \parallel T_{\phi }^A{f(\gamma ,·)|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}+2\pi b{C}_{\phi ,A}{\int }_{\mathbb{R}}ln\left|\xi \right||L_A{\left\{f\right\}\left(\xi \right)|}^{2}d\xi \hfill \\ \hfill & \ge (D+ln|b\left|\right)2\pi b{C}_{\phi ,A}{\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$

where $D=\psi \left(\frac{1}{2}\right)-ln\pi ,\psi \left(x\right)=\frac{d}{dx}ln[\Gamma \left(x\right)]$ and $\Gamma (·)$ is the gamma function.

Proof. 

According to the logarithmic uncertainty relation associated with the LCT [25], we have

(47)$\begin{array}{c}\hfill {\int }_{\mathbb{R}}{ln\left|v\right|\left|f\left(v\right)\right|}^{2}dv+{\int }_{\mathbb{R}}ln\left|\xi \right||L_A{\left\{f\right\}\left(\xi \right)|}^{2}d\xi \ge (D+ln|b\left|\right){\int }_{\mathbb{R}}{\left|f\left(v\right)\right|}^{2}dv.\end{array}$

Replacing f with $T_{\phi }^{A}f(\gamma ,\mu )$ on both sides of (47) results in

(48)$\begin{array}{c}\hfill {\int }_{\mathbb{R}}ln\left|\mu \right||T_{\phi }^A{f(\gamma ,·)|}^{2}d\mu +{\int }_{\mathbb{R}}ln|\xi \parallel L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}{\left(\xi \right)|}^{2}d\xi \ge (D+ln|b\left|\right){\int }_{\mathbb{R}}{\left|T_{\phi }^{A}f(\gamma ,·)\right|}^{2}d\mu .\end{array}$

Now integrating both sides of the above equation with respect to the Haar measure $\frac{d\gamma }{{\gamma }^2}$, we obtain

(49)$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}ln\left|\mu \right|& |T_{\phi }^A{f(\gamma ,·)|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}+{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}ln\left|\xi \right||L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}{\left(\xi \right)|}^{2}d\xi \frac{d\gamma }{{\gamma }^2}\hfill \\ \hfill & \ge (D+ln|b\left|\right){\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\left|T_{\phi }^{A}f(\gamma ,·)\right|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}.\hfill \end{array}$

Then, putting (45) into the left-hand side of (49), we obtain

(50)$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}ln\left|\mu \right|& |T_{\phi }^A{f(\gamma ,·)|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}+2\pi b{C}_{\phi ,A}{\int }_{\mathbb{R}}ln\left|\xi \right||L_A{\left\{f\right\}\left(\xi \right)|}^{2}d\xi \hfill \\ \hfill & \ge (D+ln|b\left|\right){\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\left|T_{\phi }^{A}f(\gamma ,·)\right|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}.\hfill \end{array}$

Substituting (32) into the right-hand side of (50), we finally obtain

(51)$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}ln\left|\mu \right|& |T_{\phi }^A{f(\gamma ,·)|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}+2\pi b{C}_{\phi ,A}{\int }_{\mathbb{R}}ln\left|\xi \right||L_A{\left\{f\right\}\left(\xi \right)|}^{2}d\xi \hfill \\ \hfill & \ge (D+ln|b\left|\right)2\pi b{C}_{\phi ,A}{\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2,\hfill \end{array}$

which completes the proof. □

We require the following lemma to derive Theorem 6 below.

Lemma 9.

Suppose that f and ψ both belong to $L^2\left(\mathbb{R}\right)$. Then

(52) $\frac{C_{\phi ,A}^{\frac{p}{2}}}{{\gamma }^{2-\frac{p}{2}}}{\int }_{\mathbb{R}}{\left(2\pi b\gamma \right)}^{\frac{p}{2}}{\xi }^p|L_A\left\{f\right\}\left(\xi \right){|}^{p}d\xi \ge {\int }_{\mathbb{R}}{\int }_{{\mathbb{R}}^{+}}{\xi }^p|L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}\left(\xi \right){|}^p\frac{d\gamma }{{\gamma }^2}d\xi .$

Proof. 

An application of equations (16) and (30) will lead to

$\begin{array}{cc}\hfill C_{\phi ,A}^{\frac{p}{2}}& {\int }_{\mathbb{R}}{\left(2\pi b\gamma \right)}^{\frac{p}{2}}{\xi }^p|L_A\left\{f\right\}\left(\xi \right){|}^{p}d\xi \hfill \\ \hfill & ={\left({\int }_{{\mathbb{R}}^{+}}|L_A\left\{e^{-i\frac{a}{2b}{(·)}^2}\phi \right\}\left(\gamma \xi \right){|}^2\frac{d\gamma }{\gamma }\right)}^{\frac{p}{2}}{\int }_{\mathbb{R}}{\left(2\pi b\gamma \right)}^{\frac{p}{2}}{\xi }^p|L_A\left\{f\right\}\left(\xi \right){|}^{p}d\xi \hfill \\ \hfill & \ge {\int }_{\mathbb{R}}{\int }_{{\mathbb{R}}^{+}}{\xi }^p{\left(2\pi b\gamma \right)}^{\frac{p}{2}}|L_A\left\{f\right\}\left(\xi \right){|}^p|L_A\left\{e^{-i\frac{a}{2b}{(·)}^2}\phi \right\}\left(\gamma \xi \right){|}^p\frac{d\gamma }{{\gamma }^{\frac{p}{2}}}d\xi \hfill \\ \hfill & ={\int }_{\mathbb{R}}{\int }_{{\mathbb{R}}^{+}}{\xi }^p{\left(2\pi b\gamma \right)}^{\frac{p}{2}}|L_A\left\{f\right\}\left(\xi \right){|}^p|L_A\left\{e^{-i\frac{a}{2b}{(·)}^2}\phi \right\}\left(\gamma \xi \right){|}^p\frac{d\gamma }{{\gamma }^{\frac{p}{2}}}d\xi \hfill \\ \hfill & ={\int }_{\mathbb{R}}{\int }_{{\mathbb{R}}^{+}}{\xi }^p|L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}\left(\xi \right){|}^p\frac{d\gamma }{{\gamma }^{\frac{p}{2}}}d\xi .\hfill \end{array}$

Consequently,

$\begin{array}{c}\hfill \frac{C_{\phi ,A}^{\frac{p}{2}}}{{\gamma }^{2-\frac{p}{2}}}{\int }_{\mathbb{R}}{\left(2\pi b\gamma \right)}^{\frac{p}{2}}{\xi }^p|L_A\left\{f\right\}\left(\xi \right){|}^{p}d\xi \ge {\int }_{\mathbb{R}}{\int }_{{\mathbb{R}}^{+}}{\xi }^p|L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}\left(\xi \right){|}^p\frac{d\gamma }{{\gamma }^2}d\xi .\end{array}$

The proof is complete. □

Theorem 6.

Let $\phi \in L^2\left(\mathbb{R}\right)$ be a linear canonical admissible wavelet. For every $\phi \in L^2\left(\mathbb{R}\right)$ with $1\le p\le 2$, we have

(53) $\begin{array}{cc}\hfill {\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\mu }^p|T_{\phi }^{A}f(\gamma ,·){|}^{p}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1/p}& {\left[{\int }_{\mathbb{R}}{\xi }^p|L_A\left\{f\right\}\left(\xi \right){|}^{p}d\xi \right]}^{1/p}\hfill \\ \hfill & \ge b^{1+\frac{1}{p}}{\gamma }^{\frac{2}{p}-1}\sqrt{\frac{\pi C_{\phi ,A}}{2}}{\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$

Proof. 

Using the generalization of the uncertainty principle associated with the LCT described by (36) we have

(54)$\begin{array}{c}\hfill {\left[{\int }_{\mathbb{R}}{\mu }^p|T_{\phi }^{A}f(\gamma ,·){|}^{p}d\mu \right]}^{1/p}{\left[{\int }_{\mathbb{R}}{\xi }^p|L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}{|}^{p}d\xi \right]}^{1/p}\ge \frac{b^{\frac{1}{2}+\frac{1}{p}}}{2}{\int }_{\mathbb{R}}|T_{\phi }^{A}f(\gamma ,·){|}^{2}d\mu .\end{array}$

Now integrating both sides of (54) with respect to the Haar measure $\frac{d\gamma }{{\gamma }^2}$, we see that

(55)$\begin{array}{cc}\hfill {\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\mu }^p|T_{\phi }^{A}f(\gamma ,·){|}^{p}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1/p}& {\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\xi }^p|L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}|d\xi \frac{d\gamma }{{\gamma }^2}\right]}^{1/p}\hfill \\ \hfill & \ge \frac{b^{\frac{1}{2}+\frac{1}{p}}}{2}{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}|T_{\phi }^{A}f(\gamma ,·){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}.\hfill \end{array}$

Then, including Lemma 9 into the second term of (55), we easily obtain

(56)$\begin{array}{cc}\hfill {\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\mu }^p|T_{\phi }^{A}f(\gamma ,·){|}^{p}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1/p}& {\left[\frac{C_{\phi ,A}^{\frac{p}{2}}}{{\gamma }^{2-\frac{p}{2}}}{\int }_{\mathbb{R}}{\left(2\pi b\gamma \right)}^{\frac{p}{2}}{\xi }^p|L_A\left\{f\right\}\left(\xi \right){|}^{p}d\xi \right]}^{1/p}\hfill \\ \hfill & \ge \frac{b^{\frac{1}{2}+\frac{1}{p}}}{2}{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}|T_{\phi }^{A}f(\gamma ,·){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}.\hfill \end{array}$

Inserting (32) into the right-hand side of (56), we arrive at

(57)$\begin{array}{cc}\hfill {\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\mu }^p|T_{\phi }^{A}f(\gamma ,·){|}^{p}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1/p}& {\left[\frac{C_{\phi ,A}^{\frac{p}{2}}}{{\gamma }^{2-\frac{p}{2}}}{\int }_{\mathbb{R}}{\left(2\pi b\gamma \right)}^{\frac{p}{2}}{\xi }^p|L_A\left\{f\right\}\left(\xi \right){|}^{p}d\xi \right]}^{1/p}\hfill \\ \hfill & \ge \frac{b^{\frac{1}{2}+\frac{1}{p}}}{2}{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}|T_{\phi }^{A}f(\gamma ,·){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}.\hfill \end{array}$

Hence,

(58)$\begin{array}{cc}\hfill {\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\mu }^p|T_{\phi }^{A}f(\gamma ,·){|}^{p}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1/p}& {\left[{\int }_{\mathbb{R}}{\xi }^p|L_A\left\{f\right\}\left(\xi \right){|}^{p}d\xi \right]}^{1/p}\hfill \\ \hfill & \ge b^{1+\frac{1}{p}}{\gamma }^{\frac{2}{p}-1}\sqrt{\frac{\pi C_{\phi ,A}}{2}}{\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2,\hfill \end{array}$

which finishes the proof of the theorem. □

It should be noticed that for $p=2$ in Theorem 6, we obtain Theorem 4. This shows that Theorem 6 is a general form of Theorem 4.

The author in [1] proposed the Heisenberg-type certainty principle for the shearlet transform. The following result is a modification of the principle in the case of the LCWT.

Theorem 7.

Let $\phi \in L^2\left(\mathbb{R}\right)$ and $f\in L^2\left(\mathbb{R}\right)$. For any fixed $q\ge 2$, we have

(59) $\begin{array}{cc}\hfill {\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\mu }^q|T_{\phi }^{A}f(\gamma ,·){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1/q}& {\left[{\int }_{\mathbb{R}}{\xi }^q|L_A\left\{f\right\}{|}^{2}d\xi \right]}^{1/q}\hfill \\ \hfill & \ge b^{1+1/q}{2}^{-1+1/q}{\left(\pi C_{\phi ,A}\right)}^{1/q}{\left({\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2\right)}^{2/q}.\hfill \end{array}$

Proof. 

Applying the Hölder inequality will lead to

(60)$\begin{array}{cc}\hfill & {\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\mu }^q|T_{\phi }^{A}f(\gamma ,·){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{2/q}{\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}|T_{\phi }^{A}f(\gamma ,·){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1-2/q}\hfill \\ \hfill & ={\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\left({\mu }^2|T_{\phi }^{A}f(\gamma ,·){|}^{4/q}\right)}^{q/2}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{2/q}\hfill \\ \hfill & \times {\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\left(|T_{\phi }^{A}f(\gamma ,·){|}^{2-4/q}\right)}^{\frac{1}{1-2/q}}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1-2/q}\hfill \\ \hfill & \ge {\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}\left|{\left({\mu }^2{T}_{\phi }^{A}f(\gamma ,·)\right)}^{4/q}{\left(T_{\phi }^{A}f(\gamma ,·)\right)}^{2-4/q}\right|d\mu \frac{d\gamma }{{\gamma }^2}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\mu }^2|T_{\phi }^{A}f(\gamma ,·){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}.\hfill \end{array}$

Thus, we have

(61)$\begin{array}{c}\hfill {\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\mu }^q|T_{\phi }^{A}f(\gamma ,·){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1/q}\ge \frac{{\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\mu }^2|T_{\phi }^{A}f(\gamma ,·){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1/2}}{{\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}|T_{\phi }^{A}f(\gamma ,·){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1/2-1/q}}.\end{array}$

In a similar fashion, we can deduce that

(62)$\begin{array}{cc}\hfill {\left[{\int }_{\mathbb{R}}{\xi }^q|L_A\left\{f\right\}{|}^{2}d\xi \right]}^{1/q}& \ge \frac{{\left[{\int }_{\mathbb{R}}{\xi }^2|L_A\left\{f\right\}{|}^{2}d\xi \right]}^{1/2}}{{\left[{\int }_{\mathbb{R}}|L_A\left\{f\right\}{|}^{2}d\xi \right]}^{1/2-1/q}}\hfill \\ \hfill & =\frac{{\left[{\int }_{\mathbb{R}}{\xi }^2|L_A\left\{f\right\}{|}^{2}d\xi \right]}^{1/2}}{{\left[2\pi b{C}_{\phi ,A}{\int }_{\mathbb{R}}|L_A\left\{f\right\}{|}^{2}d\xi \right]}^{1/2-1/q}}{\left(2\pi b{C}_{\phi ,A}\right)}^{1/2-1/q}\hfill \\ \hfill & =\frac{{\left[{\int }_{\mathbb{R}}{\xi }^2|L_A\left\{f\right\}{|}^{2}d\xi \right]}^{1/2}}{{\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1/2-1/q}}{\left(2\pi b{C}_{\phi ,A}\right)}^{1/2-1/q}.\hfill \end{array}$

Combining (61) and (62), we obtain

(63)$\begin{array}{cc}\hfill & {\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\mu }^q|T_{\phi }^{A}f(\gamma ,·){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1/q}{\left[{\int }_{\mathbb{R}}{\xi }^q|L_A\left\{f\right\}{|}^{2}d\xi \right]}^{1/q}\hfill \\ \hfill & \ge \frac{{\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\mu }^2|T_{\phi }^{A}f(\gamma ,·){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1/2}{\left[{\int }_{\mathbb{R}}{\xi }^2|L_A\left\{f\right\}{|}^{2}d\xi \right]}^{1/2}}{{\left[{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}|T_{\phi }^{A}f(\gamma ,·){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right]}^{1-2/q}}{\left(2\pi b{C}_{\phi ,A}\right)}^{1/2-1/q}\hfill \\ \hfill & =\frac{b^{3/2}{b}^{-1/2}\sqrt{\pi C_{\phi ,A}}{\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2}{{\left[2\pi b{C}_{\phi ,A}{\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2\right]}^{1-2/q}}{\left(2\pi b{C}_{\phi ,A}\right)}^{1/2-1/q}\hfill \\ \hfill & =b^{1+1/q}{2}^{-1+1/q}{\left(\pi C_{\phi ,A}\right)}^{1/q}{\left({\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2\right)}^{2/q},\hfill \end{array}$

and the proof is complete. □

By setting $q=2$ in Theorem 7, we obtain Theorem 4. This shows that Theorem 4 is a special case of Theorem 7.

A generalization of Pitt’s inequality in the LCWT domain is presented in the following theorem.

Theorem 8

(LCWT Pitt’s Inequality). Let $\phi \in \mathcal{S}\left(\mathbb{R}\right)$ be a linear canonical admissible wavelet. Then, for every $f\in \mathcal{S}\left(\mathbb{R}\right)$ such that $T_{\phi }^{A}f(\gamma ,·)\in \mathcal{S}\left(\mathbb{R}\right)$, it holds that

(64)$\begin{array}{c}\hfill 2\pi b^{\alpha +2}{C}_{\phi ,A}{\int }_{\mathbb{R}}{|\xi |}^{-\alpha }|L_A\left\{f\right\}\left(\xi \right){|}^{2}d\xi \le C_{\alpha }{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{|\mu |}^{\alpha }{|T_{\phi }^{A}f(\gamma ,·)|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}.\end{array}$

Proof. 

It is known that the Pitt’s inequality associated with the FT takes the following form [32]:

(65)${\int }_{\mathbb{R}}{\left|\xi \right|}^{-\alpha }{\left|\mathcal{F}\left\{f\right\}\left(\xi \right)\right|}^{2}d\xi \le C_{\alpha }{\int }_{\mathbb{R}}{\left|v\right|}^{\alpha }{\left|f\left(v\right)\right|}^{2}dv,$

where

$C_{\alpha }={\pi }^{\alpha }{\left[\Gamma \left(\frac{1-\alpha }{4}\right)/\Gamma \left(\frac{1+\alpha }{4}\right)\right]}^2.$

In this case, $\mathcal{S}\left(\mathbb{R}\right)$ is a Schwartz space and $0\le \alpha <1$. Using the same procedure as in the proof of Theorem 3, we can show that Pitt’s inequality associated with the LCT can be written as

(66)$\begin{array}{c}\hfill {\int }_{\mathbb{R}}{\left|\xi \right|}^{-\alpha }|L_A\left\{f\right\}{|}^{2}d\xi \le b^{-\alpha -1}{C}_{\alpha }{\int }_{\mathbb{R}}{\left|v\right|}^{\alpha }{\left|f\left(v\right)\right|}^{2}dv.\end{array}$

Now, replacing f with $T_{\phi }^{A}f(\gamma ,·)$ on both sides of the above equation yields

(67)${\int }_{\mathbb{R}}{\left|\xi \right|}^{-\alpha }|L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}{|}^{2}d\xi \le b^{-\alpha -1}{C}_{\alpha }{\int }_{\mathbb{R}}{\left|\mu \right|}^{\alpha }{\left|T_{\phi }^{A}f(\gamma ,·)\right|}^{2}d\mu .$

By integrating both sides of (67) with respect to $\frac{d\gamma }{{\gamma }^2}$, we obtain

(68)$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\left|\xi \right|}^{-\alpha }|L_A\left\{T_{\phi }^{A}f(\gamma ,·)\right\}{|}^{2}d\xi \frac{d\gamma }{{\gamma }^2}\le b^{-\alpha -1}{C}_{\alpha }{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\left|\mu \right|}^{\alpha }{\left|T_{\phi }^{A}f(\gamma ,·)\right|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}.\end{array}$

Due to Equations (16), (30), (32) and Fubini’s theorem, the left-hand side of (68) can be written as

$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\left|\xi \right|}^{-\alpha }2\pi b|L_A\left\{e^{-i\frac{a}{2b}{(·)}^2}\phi \right\}\left(\gamma \xi \right){|}^2& |L_A\left\{f\right\}\left(\xi \right){|}^{2}d\xi \frac{d\gamma }{\gamma }\hfill \\ \hfill & \le b^{-\alpha -1}{C}_{\alpha }{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\left|\mu \right|}^{\alpha }{\left|T_{\phi }^{A}f(\gamma ,·)\right|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\hfill \\ \hfill 2\pi b{\int }_{{\mathbb{R}}^{+}}|L_A\left\{e^{-i\frac{a}{2b}{(·)}^2}\phi \right\}\left(\gamma \xi \right){|}^2\frac{d\gamma }{\gamma }& {\int }_{\mathbb{R}}{\left|\xi \right|}^{-\alpha }|L_A\left\{f\right\}\left(\xi \right){|}^{2}d\xi \hfill \\ \hfill & \le b^{-\alpha -1}{C}_{\alpha }{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\left|\mu \right|}^{\alpha }{\left|T_{\phi }^{A}f(\gamma ,·)\right|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\hfill \\ \hfill 2\pi b{C}_{\phi ,A}{\int }_{\mathbb{R}}{\left|\xi \right|}^{-\alpha }|L_A\left\{f\right\}\left(\xi \right){|}^{2}d\xi & \le b^{-\alpha -1}{C}_{\alpha }{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\left|\mu \right|}^{\alpha }{\left|T_{\phi }^{A}f(\gamma ,·)\right|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}.\hfill \end{array}$

Simplifying it yields

(69)$2\pi b^{\alpha +2}{C}_{\phi ,A}{\int }_{\mathbb{R}}{\left|\xi \right|}^{-\alpha }|L_A\left\{f\right\}\left(\xi \right){|}^{2}d\xi \le C_{\alpha }{\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\left|\mu \right|}^{\alpha }{\left|T_{\phi }^{A}f(\gamma ,·)\right|}^{2}d\mu \frac{d\gamma }{{\gamma }^2},$

which is the desired result. □

As a special case of Pitt’s inequality related to the LCWT, we easily obtain the following remark.

Remark 3.

It is interesting to note that by taking $\alpha =0$ in Theorem 8, we obtain the following inequality:

(70) $\begin{array}{c}\hfill 2\pi b^2{C}_{\phi ,A}{\int }_{\mathbb{R}}|L_A\left\{f\right\}\left(\xi \right){|}^{2}d\xi \le {\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\left|T_{\phi }^{A}f(\gamma ,·)\right|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}.\end{array}$

It is straightforward to verify that equality holds for $b=1$.

The following interesting results presents the local uncertainty principles associated with the LCWT.

Theorem 9.

Let $E\subset {\mathbb{R}}^{+}\times \mathbb{R}$ such that $0<\frac{m\left(E\right)}{2\pi b{C}_{\phi ,A}}<1$. If $\phi \in L^2\left(\mathbb{R}\right)$ is an admissible wavelet related to the LCT such that ${\parallel \phi \parallel }_{L^2\left(\mathbb{R}\right)}=1$, then for all $f\in L^2\left(\mathbb{R}\right)$, we have

(71) $\begin{array}{c}\hfill {\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}\le \frac{1}{\sqrt{\left(2\pi b{C}_{\phi ,A}-m\left(E\right)\right)}}{\left(\int {\int }_{E^c}|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right)}^{\frac{1}{2}},\end{array}$

where $m\left(E\right)$ represents the Lebesgue measure of E and $E^c={\mathbb{R}}^{+}\times \mathbb{R}\setminus E$.

Proof. 

It follows from identity (23) that

(72)$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^{+}}{\parallel T_{\phi }^{A}f(\gamma ,\mu )\parallel }_{L^2\left(\mathbb{R}\right)}^2\frac{d\gamma }{{\gamma }^2}& ={\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\hfill \\ \hfill & =\int {\int }_E|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}+\int {\int }_{E^c}|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\hfill \\ \hfill & \le {m\left(E\right)\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2{\parallel \phi \parallel }_{L^2\left(\mathbb{R}\right)}^2+\int {\int }_{E^c}|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\hfill \\ \hfill & ={m\left(E\right)\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2+\int {\int }_{E^c}|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}.\hfill \end{array}$

Therefore,

(73)$\begin{array}{c}\hfill {m\left(E\right)\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2+\int {\int }_{E^c}|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\ge 2\pi b{C}_{\phi ,A}{\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2.\end{array}$

Hence,

(74)$\begin{array}{c}\hfill \int {\int }_{E^c}|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\ge \left(2\pi b{C}_{\phi ,A}-m\left(E\right)\right){\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2.\end{array}$

Alternatively,

(75)$\begin{array}{c}\hfill {\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}\le \frac{1}{\sqrt{\left(2\pi b{C}_{\phi ,A}-m\left(E\right)\right)}}{\left(\int {\int }_{E^c}|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right)}^{\frac{1}{2}},\end{array}$

and the proof is complete. □

Theorem 10.

Let $0<r\le 1$ be a real number and $B_r=\{(\mu ,\gamma )\in {\mathbb{R}}^{+}\times \mathbb{R}:\left|(\mu ,\gamma )\right|<r\}$. Fix $0<r_0\le 1$ small enough satisfying $0<\frac{m\left(B_{r_0}\right)}{2\pi b{C}_{\phi ,A}}<1$. Then, for every $l>0$, there exists a constant $C_l$ such that for all $f,\phi \in L^2\left(\mathbb{R}\right)$ and ${\parallel \phi \parallel }_{L^2\left(\mathbb{R}\right)}=1$, we have

(76) $\begin{array}{c}\hfill {\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}\le C_l{\left({\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\left|(\mu ,\gamma )\right|}^{2l}|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right)}^{\frac{1}{2}},\end{array}$

where $C_l=\frac{1}{r_0^l\sqrt{\left(2\pi b{C}_{\phi ,A}-m\left(B_{r_0}\right)\right)}}$.

Proof. 

According to (71), we obtain

(77)$\begin{array}{cc}\hfill {\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}& \le \frac{1}{r_0^l\sqrt{\left(2\pi b{C}_{\phi ,A}-m\left(B_{r_0}\right)\right)}}{\left(\int {\int }_{\left|(\mu ,\gamma )\right|>r_0}{r}_0^{2l}|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right)}^{\frac{1}{2}}\hfill \\ \hfill & \le \frac{1}{r_0^l\sqrt{\left(2\pi b{C}_{\phi ,A}-m\left(B_{r_0}\right)\right)}}{\left(\int {\int }_{\left|(\mu ,\gamma )\right|>r_0}{\left|(\mu ,\gamma )\right|}^{2l}|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right)}^{\frac{1}{2}}\hfill \\ \hfill & \le \frac{1}{r_0^l\sqrt{\left(2\pi b{C}_{\phi ,A}-m\left(B_{r_0}\right)\right)}}{\left({\int }_{{\mathbb{R}}^{+}}{\int }_{\mathbb{R}}{\left|(\mu ,\gamma )\right|}^{2l}|T_{\phi }^{A}f(\gamma ,\mu ){|}^{2}d\mu \frac{d\gamma }{{\gamma }^2}\right)}^{\frac{1}{2}}.\hfill \end{array}$

Thus, we obtained the desired result. □

5. Conclusions

The linear canonical wavelet transform has been introduced. Some of its general properties were investigated. We have provided a detailed explanation of the proof of the uncertainty principles concerning the linear canonical wavelet transform. Future studies will focus on the application of the proposed method in an image interpolation problem that is different than the works in [33,34]. Other potential studies are in the area of scattered data interpolation [35,36,37].

Author Contributions

Conceptualization, M.B.; Formal analysis, S.A.A.K.; Funding acquisition, S.A.A.K.; Investigation, M.B. and S.A.A.K.; Methodology, M.B. and S.A.A.K.; Resources, S.A.A.K.; Validation, M.B. and S.A.A.K.; Writing – original draft, M.B.; Writing – review & editing, M.B. and S.A.A.K. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The first author is supported in part by a grant from Ministry of Education, Culture, Research and Technology, Indonesia under the WCR scheme, and the second author is fully supported by Universiti Malaysia Sabah, Malaysia. The authors are thankful to the anonymous reviewer for the useful comments for the improvement of this paper. Mawardi Bahri thanks Wahyuni Ekasasmita, Andi Tenri Ajeng, and Nasrullah for assistance in the revised version of the manuscript. Special thank you to Research Management Centre and Faculty of Computing and Informatics, Universiti Malaysia Sabah.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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