Windowed linear canonical transform: its relation to windowed Fourier transform and uncertainty principles

Abstract

The windowed linear canonical transform is a natural extension of the classical windowed Fourier transform using the linear canonical transform. In the current work, we first remind the reader about the relation between the windowed linear canonical transform and windowed Fourier transform. It is shown that useful relation enables us to provide different proofs of some properties of the windowed linear canonical transform, such as the orthogonality relation, inversion theorem, and complex conjugation. Lastly, we demonstrate some new results concerning several generalizations of the uncertainty principles associated with this transformation.

Introduction

As it is well known, the classical windowed Fourier transform (WFT) is a useful mathematical tool, which has been broadly studied in quantum physics, signal processing and many other fields of science and engineering. In recent years, a number of efforts have been made with an increasing interest in expanding various types of transformations in the context of the linear canonical transform (LCT), we refer the reader to the papers [1–4]. Some authors [5–7] have introduced an extension of the WFT in the LCT domain, the so-called windowed linear canonical transform (WLCT). The generalized transform is built by including the Fourier kernel with the LCT kernel in the definition of the windowed Fourier transform. They also have investigated its essential properties like linearity, orthogonality relation, inversion theorem, and the inequalities.

In [8], the author has discussed that the fractional Fourier transform is intimately related to the Fourier transformation. According to this idea, some properties of the fractional Fourier transform can be easily obtained using the basic connection between the fractional Fourier transform and Fourier transform. In [9], the authors have investigated the fundamental properties of the continuous shearlet transforms using the direct interaction between the Fourier transform and shearlet transform. In this work, we developed this approach within the framework of the linear canonical transform. We have provided different proofs of the WLCT properties like the orthogonality relation, inversion theorem, and complex conjugation using the direct interaction among the windowed linear canonical transform, the windowed Fourier transform and the Fourier transform, the proofs of which are simpler than those the authors proposed in [7]. As we know, the uncertainty principle is one of the fundamental results of the WLCT, which explains how an original function interacts with its WLCT. Therefore, we have proposed several versions of the uncertainty principles associated with this transformation, which are quite different from those investigated in [5, 7] as well as in [10].

The present work is structured in the following fashion. In Sect. 3, we provided a brief review of the linear canonical transform and basic notations that will be useful later. Section 4 is a part of the core of the article. This section presents the basic relation between the windowed linear canonical transform and windowed Fourier transform. In it, some famous properties for the windowed linear canonical transform are proved using this relation. Section 5 is also a part of the core of the article. This section is devoted to some generalizations of the uncertainty principles related to the windowed linear canonical transform. Lastly, the summary of this work is included in Sect. 6.

Generalities

In this segment, we state the definition of the linear canonical transform (LCT) and its useful properties, as well as the basic notations, which will be used in the derivation of the results of this work. For a detaled information on this transform, we refer to [11–15].

Definition 2.1

Let $B=(a,b,c,d)=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\in {\mathbb{R}}^{2\times 2}$ be a matrix parameter such that $|B| = 1$. The LCT of a function $f \in L^{1}(\mathbb{R})$ is expressed as

$$\begin{aligned} \mathcal{C}_{B}\{ f \} (w) = \textstyle\begin{cases} \int _{\mathbb{R}} f(x) \mathcal{K}_{B}(w,x) \,dx, &b \neq 0, \\ \sqrt{d}e^{i \frac{cd}{2} w^{2}}f(dw), & b = 0, \end{cases}\displaystyle \end{aligned}$$

(1)

where $\mathcal{K}_{B}(w,x)$ is given by

$$\begin{aligned} \mathcal{K}_{B}(w, x) = \frac{1}{\sqrt{2\pi b }} e^{\frac{i}{2} ( \frac{a}{b}x^{2}- \frac{2}{b} xw + \frac{d}{b} w^{2} -\frac{\pi }{2}) }. \end{aligned}$$

(2)

It is evident that relation (2) above fulfills

$$ \mathcal{K}_{B^{-1}}(x, w) = \overline{\mathcal{K}_{B}(w, x)} = \frac{1}{\sqrt{2\pi b }} e^{-\frac{i}{2} ( \frac{a}{b}x^{2}- \frac{2}{b} xw + \frac{d}{b} w^{2}- \frac{\pi }{2} ) }. $$

From equation (1), it can be observed that for $b=0$ the LCT of a signal is a chirp product. Therefore, in the current work, we always consider the case $b >0$.

It is worth noting that for $B=(a,b,c,d)=(0,1,-1,0)$, equation (1) can be expressed as

$$\begin{aligned} F\{f\}(w) = \int _{\mathbb{R}} f(x)e^{-i xw} \,dx, \end{aligned}$$

(3)

leading to the definition of the Fourier transform times $\frac{1}{\sqrt{2\pi } i}$ and providing that the infinite integral exists. The inverse LCT is given by

$$\begin{aligned} f(x) & = \int _{\mathbb{R}} \mathcal{C}_{B}\{ f \} (w) \mathcal{K}_{B^{-1}}(x,w) \,dw \\ & = \int _{\mathbb{R}} \mathcal{C}_{B}\{ f \} (w) \frac{1}{\sqrt{2\pi b }} e^{-\frac{i}{2} ( \frac{a}{b}x^{2}- \frac{2}{b} xw + \frac{d}{b} w^{2}- \frac{\pi }{2} )} \,dw. \end{aligned}$$

(4)

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

$$\begin{aligned} \sqrt{ 2\pi b} e^{i \frac{\pi }{4}} e^{ -\frac{id}{2b} w^{2} } \mathcal{C}_{B}\{ f \} (w) = F \bigl\{ e^{\frac{ia}{2b}x^{2}}f \bigr\} { \biggl( \frac{w}{b} \biggr)}. \end{aligned}$$

(5)

Definition 2.2

Given f a measurable function on $\mathbb{R}$ and $1\leq r < \infty$, define

$$\begin{aligned} &\Vert f \Vert _{L^{r}(\mathbb{R})} = \biggl( \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{r} \,dx \biggr)^{1/r} < \infty, \\ &\Vert f \Vert _{L^{\infty }(\mathbb{R})} = \mathop{\mathrm{ess\, sup}}_{x\in \mathbb{R}} \bigl\vert f(x) \bigr\vert < \infty. \end{aligned}$$

(6)

For $r=2$, we get

$$\begin{aligned} \langle f, g\rangle _{L^{2}(\mathbb{R})} = \int _{\mathbb{R}} f(x) \overline{g(x)} \,dx\quad \text{and}\quad \Vert f \Vert ^{2}_{L^{2}(\mathbb{R})} = \langle f, f\rangle _{L^{2}(\mathbb{R})}. \end{aligned}$$

Based on the above definition we state the following fact, which is known as Parseval’s formula and Planchel’s formula, respectively.

Lemma 2.1

For every $f, g \in L^{2}(\mathbb{R})$, the following relation holds:

$$\begin{aligned} \langle f, g\rangle _{L^{2}(\mathbb{R})} = \bigl\langle \mathcal{C}_{B}\{ f \}, \mathcal{C}_{B}\{ g \} \bigr\rangle _{L^{2}(\mathbb{R})}, \end{aligned}$$

(7)

and

$$\begin{aligned} \Vert f \Vert ^{2}_{L^{2}(\mathbb{R})} = \bigl\Vert \mathcal{C}_{B}\{ f \} \bigr\Vert ^{2}_{L^{2}( \mathbb{R})}. \end{aligned}$$

(8)

The next result will be useful in this paper.

Theorem 2.2

([7])

Let $1\leq r\leq 2$ and s be such that $\frac{1}{r} + \frac{1}{s} =1$. Then for all $g \in L^{r}(\mathbb{R})$, it holds

$$\begin{aligned} \bigl\Vert \mathcal{C}_{B}\{g \} \bigr\Vert _{L^{s}(\mathbb{R})}\leq \Vert g \Vert _{L^{r}( \mathbb{R})}. \end{aligned}$$

(9)

It is straightforward to see that for $r=1$, we get

$$\begin{aligned} \bigl\Vert \mathcal{C}_{B}\{g \} \bigr\Vert _{L^{\infty }(\mathbb{R})}\leq \Vert g \Vert _{L^{1}( \mathbb{R})}. \end{aligned}$$

(10)

Windowed linear canonical transform (WLCT)

Below, we shortly introduce the windowed linear canonical transform (WLCT), which was studied in [5–7, 16].

Definition 3.1

Let $\phi \in L^{2}(\mathbb{R})$ be a non-zero window function. The WLCT of $f\in L^{2}(\mathbb{R})$ with respect to ϕ denoted by $G_{\phi }^{B}$ is given by

$$\begin{aligned} G_{\phi }^{B} f(w,v) & = \frac{1}{\sqrt{2\pi b }} \int _{\mathbb{R}} f(x) \overline{\phi (x -v)} e^{\frac{i}{2} (\frac{a}{b}x^{2}-\frac{2}{b} xw + \frac{d}{b} w^{2} -\frac{\pi }{2})} \,dx \end{aligned}$$

(11)

for $(x,v)\in \mathbb{R}\times \mathbb{R}$.

The relation of the WLCT to the Fourier transform takes the form

$$\begin{aligned} G_{\phi }^{B} f(w,v) & = \frac{e^{-i\frac{\pi }{4}}}{\sqrt{2\pi b }} e^{i \frac{d}{2b}w^{2}} \int _{\mathbb{R}} f(x) \overline{\phi (x -v)} e^{i \frac{a}{2b} x^{2}} e^{-i\frac{xw}{b} } \,dx \\ &= \frac{e^{-i\frac{\pi }{4}}}{\sqrt{2\pi b }} e^{i\frac{d}{2b}w^{2}} F \bigl\{ e^{\frac{a}{b}x^{2}} f T_{v} \bar{\phi } \bigr\} { \biggl(\frac{w}{b} \biggr)}, \end{aligned}$$

(12)

where the shifting operator $T_{v} \bar{\phi }$ is expressed as

$$\begin{aligned} T_{v} \bar{\phi }(x) = \bar{\phi }(x-v). \end{aligned}$$

The relation (12) above is equivalent to

$$\begin{aligned} \sqrt{2\pi b} e^{i\frac{\pi }{4}} e^{-i\frac{d}{2b}w^{2}} G_{\phi }^{B} f(w,v) = F \bigl\{ e^{\frac{a}{b}x^{2}} f T_{v} \bar{\phi } \bigr\} { \biggl(\frac{w}{b} \biggr)}. \end{aligned}$$

(13)

Some useful consequences of the above definition are collected as the following:

• Especially, for $B=(a,b,c,d)=(0,1,-1,0)$, Definition 3.1 changes to the classical WFT definition, namely,

  $$\begin{aligned} G_{\phi }^{B} f(w,v) = {\frac{1}{\sqrt{2\pi i}}} G_{\phi } f(w,v), \end{aligned}$$

  where

  $$\begin{aligned} G_{\phi } f(w,v) = \int _{\mathbb{R}} f(x) \overline{\phi (x -v)} e^{-ix w} \,dx, \end{aligned}$$

  (14)

  which means that

  $$\begin{aligned} G_{\phi } f(w,v) = F \{ f T_{v} \bar{\phi } \} (w ). \end{aligned}$$

  (15)

• If we take the Gaussian signal as ϕ in (11), it is often called the Gabor linear canonical transformation.

• It is straightforward to verify that

  $$\begin{aligned} G_{\phi }^{B} f(w,v) = \mathcal{C}_{B} \{ f T_{v} \bar{\phi } \} (w), \end{aligned}$$

  (16)

  which describes the connection between the windowed linear canonical transform and linear canonical transform.

Essential properties of windowed linear canonical transform

We need the following simple lemma, which will be useful in deriving results in this work. It demonstrates the interaction between the windowed linear canonical transform (WLCT) and windowed Fourier transform.

Lemma 4.1

([7])

The WLCT of a signal $f \in L^{2}(\mathbb{R})$ with $B =(a,b,c,d)$ can be changed to the WFT via

$$\begin{aligned} e^{ -\frac{id}{2b} w^{2}} G_{\phi }^{B} f(w,v) = G_{\phi } \check{f} \biggl( \frac{w}{b},v\biggr), \end{aligned}$$

(17)

where

$$\begin{aligned} \check{f}(x) =\frac{e^{-i \frac{\pi }{4}}}{\sqrt{2\pi b}} e^{ \frac{ia}{2b}x^{2}} f(x). \end{aligned}$$

(18)

Let us now build the orthogonality property and inversion theorem associated with the WLCT by applying the direct connection among the WLCT, WFT, and FT (in comparison with [7]).

Theorem 4.2

Let $\phi,\psi$ be two window functions related to the LCT. For each $f,g \in L^{2} (\mathbb{R})$, one has

$$\begin{aligned} \int _{\mathbb{R}} \int _{\mathbb{R}} G_{\phi }^{B} f(w,v) \overline{G_{\psi }^{B} g(w,v)} \,dw \,dv = \langle \bar{\phi }, \bar{\psi }\rangle _{L^{2}(\mathbb{R})} \langle f,g \rangle _{L^{2}( \mathbb{R})}. \end{aligned}$$

(19)

In particular,

$$\begin{aligned} \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv = \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \Vert f \Vert ^{2}_{L^{2}( \mathbb{R})}. \end{aligned}$$

(20)

Proof

With the help of the orthogonality relation for the WFT (see [17]), we obtain

$$\begin{aligned} \frac{1}{2\pi } \int _{\mathbb{R}} \int _{\mathbb{R}} G_{\phi } f(w,v) \overline{G_{\psi } g(w,v)} \,dw \,dv = \langle \bar{\phi }, \bar{\psi } \rangle _{L^{2}(\mathbb{R})} \langle f,g\rangle _{L^{2}(\mathbb{R})}. \end{aligned}$$

(21)

With (15), the relation (21) may be expressed as

$$\begin{aligned} \frac{1}{2\pi } \int _{\mathbb{R}} \int _{\mathbb{R}} F \{ f T_{v} \bar{\phi } \} (w ) \overline{F \{ g T_{v} \bar{\psi } \} (w )} \,dw \,dv = \langle \bar{ \phi }, \bar{\psi }\rangle _{L^{2}(\mathbb{R})} \langle f,g \rangle _{L^{2}(\mathbb{R})}. \end{aligned}$$

(22)

Equation (22) may be rewritten as

$$\begin{aligned} \frac{1}{2\pi } \int _{\mathbb{R}} \int _{\mathbb{R}} F \bigl\{ e^{ \frac{ia}{2b}x^{2}}f T_{v} \bar{ \phi } \bigr\} (w ) \overline{F \bigl\{ e^{\frac{ia}{2b}x^{2}} g T_{v} \bar{ \psi } \bigr\} (w )} \,dw \,dv = \langle \bar{\phi }, \bar{\psi }\rangle _{L^{2}(\mathbb{R})} \bigl\langle e^{\frac{ia}{2b}x^{2}} f, e^{\frac{ia}{2b}x^{2}}g\bigr\rangle _{L^{2}( \mathbb{R})}. \end{aligned}$$

Putting $w =\frac{w}{b}$, we can write the above identity in the form

$$\begin{aligned} \frac{1}{2\pi b} \int _{\mathbb{R}} \int _{\mathbb{R}} F \bigl\{ e^{ \frac{ia}{2b}x^{2}}f T_{v} \bar{ \phi } \bigr\} \biggl(\frac{w}{b} \biggr) \overline{F \bigl\{ e^{\frac{ia}{2b}x^{2}} g T_{v} \bar{\psi } \bigr\} \biggl(\frac{w}{b} \biggr)} \,dw \,dv = \langle \bar{\phi }, \bar{\psi }\rangle _{L^{2}(\mathbb{R})} \langle f,g\rangle _{L^{2}(\mathbb{R})}. \end{aligned}$$

(23)

By virtue of (13), the left-hand side of (23) takes the form

$$\begin{aligned} \int _{\mathbb{R}} \int _{\mathbb{R}} e^{i\frac{\pi }{4}} e^{-i \frac{d}{2b}w^{2}} G_{\phi }^{B} f(w,v) \overline{e^{i\frac{\pi }{4}} e^{-i\frac{d}{2b}w^{2}} G_{\psi }^{B} g(w,v) } \,dw \,dv = \langle \bar{\phi }, \bar{\psi }\rangle _{L^{2}(\mathbb{R})} \langle f,g \rangle _{L^{2}(\mathbb{R})}. \end{aligned}$$

(24)

Hence,

$$\begin{aligned} \int _{\mathbb{R}} \int _{\mathbb{R}} G_{\phi }^{B} f(w,v) \overline{ G_{\phi }^{B} f(w,v) } \,dw \,dv = \langle \bar{\phi }, \bar{ \psi }\rangle _{L^{2}(\mathbb{R})} \langle f,g \rangle _{L^{2}( \mathbb{R})}, \end{aligned}$$

and the proof is complete. □

Let us implement Lemma 4.1 to derive a basic property of the WLCT.

Theorem 4.3

If a function $f\in L^{2} (\mathbb{R})$ and ϕ is real-valued, then we have

$$\begin{aligned} G_{\phi }^{B} \bar{f}(w,v) =\overline{G_{\phi }^{B^{-1}} f(w,v)}. \end{aligned}$$

(25)

Proof

By including f̌ into the complex conjugate theorem for the windowed Fourier transform (see [17]) defined by (18), we see that

$$\begin{aligned} G_{\phi } \check{\bar{f}}\biggl(\frac{w}{b},v\biggr) = \overline{G_{\phi } \check{f}\biggl(\frac{w}{-b},v\biggr)}. \end{aligned}$$

(26)

In view of (17), the expression on the left of (26) above can be expressed as

$$\begin{aligned} e^{ -\frac{id}{2b} w^{2}} G_{\phi }^{B} \bar{f}(w,v) = G_{\phi } \check{\bar{f}}\biggl(\frac{w}{b},v\biggr), \end{aligned}$$

(27)

and thus, we obtain

$$\begin{aligned} e^{ -\frac{id}{2b} w^{2}} G_{\phi }^{B} \bar{f}(w,v) = \overline{G_{\phi } \check{f}\biggl(\frac{w}{-b},v\biggr)}. \end{aligned}$$

(28)

From equations (14) and (28), we observe that

$$\begin{aligned} G_{\phi }^{B} \bar{f}(w,v) &= e^{\frac{id}{2b} w^{2}} \overline{G_{\phi } \check{f}\biggl(\frac{w}{-b},v\biggr)} \\ &= e^{\frac{id}{2b} w^{2}} \int _{\mathbb{R}} \overline{\frac{e^{-i \frac{\pi }{4}}}{\sqrt{-2\pi b}} e^{-\frac{ia}{2b}x^{2}} f(x) \overline{\phi (x -v)} e^{i\frac{xw}{b} }} \,dx \\ &= e^{\frac{id}{2b} w^{2}} \int _{\mathbb{R}} \overline{\frac{e^{i \frac{\pi }{4}}}{\sqrt{2\pi b}} e^{-\frac{ia}{2b}x^{2}} f(x) \overline{\phi (x -v)} e^{i\frac{xw}{b} }} \,dx \\ &= \int _{\mathbb{R}} \overline{f(x) \overline{\phi (x -v)} \frac{1}{\sqrt{2\pi b }} e^{-\frac{i}{2} (\frac{a}{b}x^{2}-\frac{2}{b} xw + \frac{d}{b} w^{2} -\frac{\pi }{2})}} \,dx \\ &= \overline{G_{\phi }^{B^{-1}} f(w,v)}, \end{aligned}$$

which was to be proved. □

Theorem 4.4

Let $\phi,\psi$ be two window functions related to the LCT. For any $f \in L^{2} (\mathbb{R})$, one has

$$\begin{aligned} f(x) = \frac{1}{\langle \phi, \psi \rangle _{L^{2}(\mathbb{R})}} \int _{\mathbb{R}} \int _{\mathbb{R}} G^{A}_{\phi } f(w,v) K_{B^{-1}}(w,x) \psi (x-v) \,dw \,dv. \end{aligned}$$

(29)

Proof

Due to the inversion theorem for the WFT, we obtain

$$\begin{aligned} f(x) &= \frac{1}{2\pi \langle \phi, \psi \rangle _{L^{2}(\mathbb{R})}} \int _{ \mathbb{R}} \int _{\mathbb{R}} G_{\phi } f(w,v) e^{iw x}\psi (x-v) \,dw \,dv \\ &= \frac{1}{2\pi \langle \phi, \psi \rangle _{L^{2}(\mathbb{R})}} \int _{\mathbb{R}} \int _{\mathbb{R}} F\{f T_{v}\bar{\phi } \} (w) e^{iw x}\psi (x-v) \,dw \,dv. \end{aligned}$$

(30)

According to (13), we see that

$$\begin{aligned} e^{\frac{ia}{2b}x^{2}} f(x) &= \frac{1}{2\pi \langle \phi, \psi \rangle _{L^{2}(\mathbb{R})}} \int _{ \mathbb{R}} \int _{\mathbb{R}} F\bigl\{ e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{ \phi } \bigr\} (w) e^{iw x}\psi (x-v) \,dw \,dv \\ &= \frac{1}{2\pi b\langle \phi, \psi \rangle _{L^{2}(\mathbb{R})}} \int _{\mathbb{R}} \int _{\mathbb{R}} F\bigl\{ e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{ \phi } \bigr\} \biggl(\frac{w}{b}\biggr) e^{i \frac{w x}{b}} \psi (x-v) \,dw \,dv \\ &= \frac{1}{2\pi b\langle \phi, \psi \rangle _{L^{2}(\mathbb{R})}} \int _{\mathbb{R}} \sqrt{2\pi b} e^{i\frac{\pi }{4}} e^{-i \frac{d}{2b}w^{2}} G_{\phi }^{B} f(w,v) e^{i \frac{w x}{b}} \psi (x-v) \,dw \,dv. \end{aligned}$$

(31)

The above relation can be expressed in the form

$$\begin{aligned} f(x) &= \frac{1}{\langle \phi, \psi \rangle _{L^{2}(\mathbb{R})}} \int _{\mathbb{R}} \int _{\mathbb{R}} G^{A}_{\phi } f(w,v) \frac{1}{\sqrt{2\pi b }} e^{-\frac{i}{2} ( \frac{a}{b}x^{2}- \frac{2}{b} xw + \frac{d}{b} w^{2} -\frac{\pi }{2}) } \psi (x-v) \,dw \,dv \\ &= \frac{1}{\langle \phi, \psi \rangle _{L^{2}(\mathbb{R})}} \int _{ \mathbb{R}} \int _{\mathbb{R}} G^{A}_{\phi } f(w,v) K_{B^{-1}}(w,x) \psi (x-v) \,dw \,dv. \end{aligned}$$

(32)

The proof is complete. □

Inequalities for windowed linear canonical transform

We first state the following result, which describes the Hausdorff–Young inequality related to the WLCT.

Theorem 5.1

(WLCT Hausdorff–Young)

For any $1\leq r\leq 2$ such that $\frac{1}{r} + \frac{1}{s} =1$. Suppose that ϕ in $L^{s}(\mathbb{R})$ and f in $L^{r}(\mathbb{R})$, then we have

$$\begin{aligned} \bigl\Vert G_{\phi }^{B} f(w,v) \bigr\Vert _{L^{s}(\mathbb{R})} \leq \Vert \phi \Vert _{L^{s}( \mathbb{R})} \Vert f \Vert _{L^{r}(\mathbb{R})}. \end{aligned}$$

(33)

Proof

We assume that $\| \phi \|_{L^{s}(\mathbb{R})} = 1$. An application of the identity (10) together with the identity (16) will lead to

$$\begin{aligned} \bigl\Vert G_{\phi }^{B} f(w,v) \bigr\Vert _{L^{\infty }(\mathbb{R})} & = \bigl\Vert \mathcal{C}_{B} \{ f T_{v} \bar{\phi } \} \bigr\Vert _{L^{\infty }(\mathbb{R})} \\ &\leq \Vert f T_{v} \bar{\phi } \Vert _{L^{1}(\mathbb{R})} \\ &\leq \Vert f \Vert _{L^{1}(\mathbb{R})} \Vert \phi \Vert _{L^{\infty }(\mathbb{R})} \\ &= \Vert f \Vert _{L^{1}(\mathbb{R})}. \end{aligned}$$

(34)

For $r=2$, we obtain

$$\begin{aligned} \bigl\Vert G_{\phi }^{B} f(w,v) \bigr\Vert _{L^{2}(\mathbb{R})} = \Vert \phi \Vert _{L^{2}( \mathbb{R})} \Vert f \Vert _{L^{2}(\mathbb{R})} \leq \Vert f \Vert _{L^{2}( \mathbb{R})}, \end{aligned}$$

(35)

and applying Riesz–Thorin interpolation theorem yields

$$\begin{aligned} \bigl\Vert G_{\phi }^{B} f(w,v) \bigr\Vert _{L^{s}(\mathbb{R})} \leq \Vert f \Vert _{L^{r}( \mathbb{R})}. \end{aligned}$$

(36)

Let $\phi \in L^{r}(\mathbb{R})$ be a window function and putting $\psi = \frac{\phi }{\|\phi \|_{L^{s}(\mathbb{R})}}$, then we immediately get

$$\begin{aligned} G_{\psi }^{B} f =\frac{1}{ \Vert \phi \Vert _{L^{s}(\mathbb{R})}} G_{\phi }^{B} f. \end{aligned}$$

(37)

From (36), we conclude that

$$\begin{aligned} \bigl\Vert G_{\psi }^{B} f(w,v) \bigr\Vert _{L^{s}(\mathbb{R})} \leq \Vert f \Vert _{L^{r}( \mathbb{R})}. \end{aligned}$$

(38)

This implies that

$$\begin{aligned} \bigl\Vert G_{\phi }^{B} f(w,v) \bigr\Vert _{L^{s}(\mathbb{R})} \leq \Vert \phi \Vert _{L^{s}( \mathbb{R})} \Vert f \Vert _{L^{r}(\mathbb{R})}, \end{aligned}$$

(39)

and the proof is complete. □

We derive the following theorem, which is little different from those proposed in the paper [5].

Theorem 5.2

Let ϕ be a window function belonging to $L^{2}(\mathbb{R})$ such that $\|\phi \|_{L^{2}(\mathbb{R})} =1$. Let $f\in L^{2}(\mathbb{R})$ be a function with $\|f\|_{L^{2}(\mathbb{R})}=1$. If

$$\begin{aligned} \int \int _{V} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \geq 1-\epsilon, \end{aligned}$$

(40)

then for every $\epsilon \geq 0$ one has

$$\begin{aligned} \mu (V) \geq \sqrt{2\pi b} (1-\epsilon ). \end{aligned}$$

(41)

Here $V \subseteq \mathbb{R} \times \mathbb{R}$ is a measurable subset, and $\mu (V)$ is the Lebesgue measure of V.

Proof

Applying the Cauchy–Schwarz inequality results in

$$\begin{aligned} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert &= \biggl\vert \frac{1}{\sqrt{2\pi b }} \int _{ \mathbb{R}} f(x) \overline{\phi (x -v)} e^{\frac{i}{2} (\frac{a}{b}x^{2}- \frac{2}{b} xw + \frac{d}{b} w^{2} -\frac{\pi }{2})} \,dx \biggr\vert \\ & \leq \frac{1}{\sqrt{2\pi b }} \biggl( \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\mathbb{R}} \bigl\vert \phi (x-v) \bigr\vert ^{2} \,dx \biggr)^{\frac{1}{2}} \\ & = \frac{1}{\sqrt{2\pi b }} \Vert f \Vert _{L^{2}(\mathbb{R})} \Vert \phi \Vert _{L^{2}( \mathbb{R})}, \end{aligned}$$

(42)

which implies that

$$\begin{aligned} 1-\epsilon \leq \int \int _{V} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \leq \bigl\Vert G_{\phi }^{B} f \bigr\Vert _{L^{\infty }(\mathbb{R})}^{2} \mu (V) \leq \frac{1}{\sqrt{2\pi b }} \mu (V), \end{aligned}$$

(43)

This is the desired result. □

Theorem 5.3

Let $\phi,\psi$ be two window functions, and $f \in L^{2}(\mathbb{R})$, then for all $r \in [1,\infty )$, we have

$$\begin{aligned} &\biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) G_{\psi }^{B} g(w,v) \bigr\vert ^{r} \,dw \,dv \biggr)^{\frac{1}{r}} \\ &\quad \leq \biggl(\frac{1}{2\pi b} \biggr)^{\frac{r-1}{r}} \Vert f \Vert _{L^{2}( \mathbb{R})} \Vert g \Vert _{L^{2}(\mathbb{R})} \Vert \phi \Vert _{L^{2}(\mathbb{R})} \Vert \psi \Vert _{L^{2}(\mathbb{R})}. \end{aligned}$$

(44)

Proof

According to the Cauchy–Schwarz inequality, we obtain

$$\begin{aligned} &\int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) G_{ \psi }^{B} g(w,v) \bigr\vert \,dw \,dv \\ &\quad\leq \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \biggr)^{\frac{1}{2}} \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\psi }^{B} g(w,v) \bigr\vert ^{2} \,dw \,dv \biggr)^{ \frac{1}{2}}. \end{aligned}$$

(45)

By virtue of (20), we see that

$$\begin{aligned} \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) G_{ \psi }^{B} g(w,v) \bigr\vert \,dw \,dv \leq \Vert f \Vert _{L^{2}(\mathbb{R})} \Vert g \Vert _{L^{2}( \mathbb{R})} \Vert \phi \Vert _{L^{2}(\mathbb{R})} \Vert \psi \Vert _{L^{2}( \mathbb{R})}. \end{aligned}$$

(46)

Thus,

$$\begin{aligned} & \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} |G_{\phi }^{B} f(w,v) \bigl\vert G_{\psi }^{B} g(w,v) \bigr\vert ^{r} \,dw \,dv \biggr)^{\frac{1}{r}} \\ &\quad\leq \bigl\Vert G_{\phi }^{B} f G_{\psi }^{B} g \bigr\Vert ^{\frac{r-1}{r}}_{L^{ \infty }(\mathbb{R}\times \mathbb{R} )} \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) G_{\psi }^{B} g(w,v) \bigr\vert \,dw \,dv \biggr)^{\frac{1}{r}} \\ &\quad\leq \biggl(\frac{1}{2\pi b} \Vert f \Vert _{L^{2}(\mathbb{R})} \Vert \phi \Vert _{L^{2}( \mathbb{R})} \Vert g \Vert _{L^{2}(\mathbb{R})} \Vert \psi \Vert _{L^{2}(\mathbb{R})} \biggr)^{\frac{r-1}{r}} \bigl( \Vert f \Vert _{L^{2}(\mathbb{R})} \Vert g \Vert _{L^{2}( \mathbb{R})} \Vert \phi \Vert _{L^{2}(\mathbb{R})} \Vert \psi \Vert _{L^{2}( \mathbb{R})} \bigr)^{\frac{1}{r}} \\ &\quad= \biggl(\frac{1}{2\pi b} \biggr)^{\frac{r-1}{r}} \Vert f \Vert _{L^{2}( \mathbb{R})} \Vert g \Vert _{L^{2}(\mathbb{R})} \Vert \phi \Vert _{L^{2}(\mathbb{R})} \Vert \psi \Vert _{L^{2}(\mathbb{R})}, \end{aligned}$$

(47)

which completes the proof of Theorem 5.3 as desired. □

As an easy consequence of Theorem 5.3 mentioned above, we get the following result.

Corollary 5.4

Let $\phi, \psi$ be two window functions. Then for every $f \in L^{2}(\mathbb{R})$ with $r\in [2,\infty )$, one has

$$\begin{aligned} \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{r} \,dw \,dv \biggr)^{\frac{1}{r}} \leq \biggl( \frac{1}{2\pi b} \biggr)^{\frac{\frac{r}{2}-1}{r}} \Vert f \Vert _{L^{2}(\mathbb{R})} \Vert \phi \Vert _{L^{2}( \mathbb{R})}. \end{aligned}$$

(48)

Proof

For $r=\infty$, we have

$$\begin{aligned} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert \leq \frac{1}{2\pi b} \Vert f \Vert _{L^{2}(\mathbb{R})} \Vert \phi \Vert _{L^{2}(\mathbb{R})}. \end{aligned}$$

(49)

For all $r \in [1, \infty )$, it holds

$$\begin{aligned} \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2r} \,dw \,dv \biggr)^{\frac{1}{r}} \leq \biggl( \frac{1}{2\pi b} \biggr)^{\frac{r-1}{r}} \Vert f \Vert ^{2}_{L^{2}(\mathbb{R})} \Vert \phi \Vert ^{2}_{L^{2}( \mathbb{R})}. \end{aligned}$$

(50)

Now putting $s=2r\in [2,\infty )$ yields

$$\begin{aligned} \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{s} \,dw \,dv \biggr)^{\frac{2}{s}} \leq \biggl( \frac{1}{2\pi b} \biggr)^{\frac{s-2}{s}} \Vert f \Vert ^{2}_{L^{2}(\mathbb{R})} \Vert \phi \Vert ^{2}_{L^{2}( \mathbb{R})}. \end{aligned}$$

(51)

Hence,

$$\begin{aligned} \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{s} \,dw \,dv \biggr)^{\frac{1}{s}} \leq \biggl( \frac{1}{2\pi b} \biggr)^{\frac{\frac{s}{2}-1}{s}} \Vert f \Vert _{L^{2}(\mathbb{R})} \Vert \phi \Vert _{L^{2}( \mathbb{R})}. \end{aligned}$$

(52)

We thus finish the proof. □

Below, we obtain an immediate generalization of Theorem 5.2 in the following form.

Theorem 5.5

With the notations of Theorem 5.2. For all $r>2$, the following inequality holds

$$\begin{aligned} \mu (V) \geq (1-\epsilon )^{\frac{r}{(r-2)}} (2\pi b). \end{aligned}$$

(53)

Proof

Applying the Hölder inequality, we easily obtain

$$\begin{aligned} 1-\epsilon&\leq \int \int _{V} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \\ &\leq \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2 .\frac{r}{2}} \,dw \,dv \biggr)^{\frac{2}{r}} \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \chi _{V}(w, v)^{\frac{r}{r-2}} \,dw \,dv \biggr)^{ \frac{r-2}{r}}, \end{aligned}$$

(54)

where $\chi _{V}$ denotes the indicator function of the set V. Substituting relation (48) into the first term in the right-hand side of the equation (54), we see that

$$\begin{aligned} 1-\epsilon &\leq \biggl( \biggl(\frac{1}{2\pi b} \biggr)^{ \frac{\frac{r}{2}-1}{r}} \Vert f \Vert _{L^{2}(\mathbb{R})} { \Vert \phi \Vert _{L^{2}(\mathbb{R})}} \biggr)^{2} \bigl(\mu (V) \bigr)^{ \frac{r-2}{r}}. \end{aligned}$$

(55)

For all $r>2$, we get

$$\begin{aligned} 1-\epsilon &\leq \biggl(\frac{1}{2\pi b} \biggr)^{\frac{r-2}{r}} \bigl(\mu (V) \bigr)^{\frac{r-2}{r}}. \end{aligned}$$

(56)

After simplifying the above relation, we get

$$\begin{aligned} \mu (V) \geq (1-\epsilon )^{\frac{r}{(r-2)}} (2\pi b). \end{aligned}$$

(57)

This proves the claim. □

Definition 5.1

A function $f \in L^{2}(\mathbb{R})$ is said to be the ϵ-concentrated on a measurable set $X \subseteq \mathbb{R}$ if for any $\epsilon > 0$, it holds

$$\begin{aligned} \biggl( \int _{\mathbb{R}\setminus X} \bigl\vert f(x) \bigr\vert ^{r} \,dx \biggr)^{1/r} \leq \epsilon \Vert f \Vert _{L^{r}(\mathbb{R})},\quad 0< r\leq \infty. \end{aligned}$$

(58)

Below, based on Definition 5.1 above, we immediately obtain the alternative form of Theorem 5.5.

Theorem 5.6

Let ϕ be a complex window function, and $f \in L^{2}(\mathbb{R})$. If $G_{\phi }^{B} f$ is the ϵ-concentrated on $L^{2}$-norm on measurable set X of $\mathbb{R}$, then for all $r>2$, one has

$$\begin{aligned} \mu (V) \geq \bigl(1-\epsilon ^{2}\bigr)^{\frac{r}{(r-2)}} (2\pi b). \end{aligned}$$

(59)

Proof

For simplicity, we first assume that $\|f \|^{2}_{L^{2}(\mathbb{R})} = \|\phi \|^{2}_{L^{2}(\mathbb{R})} =1$. Invoking (58), we get

$$\begin{aligned} \int _{\mathbb{R}\setminus X} \int _{\mathbb{R}\setminus X} \bigl\vert G_{\phi }^{B} f( \omega,u) \bigr\vert ^{2} \,d\omega \,du \leq \epsilon ^{2} \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f( \omega,u) \bigr\vert ^{2} \,d\omega \,du. \end{aligned}$$

(60)

An application of relation (8) together with relation (16) will lead to

$$\begin{aligned} \int _{\mathbb{R}\setminus X} \int _{\mathbb{R}\setminus X} \bigl\vert G_{\phi }^{B} f( \omega,u) \bigr\vert ^{2} \,d\omega \,du &\leq \epsilon ^{2} \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert \mathcal{C}_{B} \{ f T_{u} \bar{\phi } \} \bigr\vert ^{2} \,d\omega \,du \\ &= \epsilon ^{2} \Vert f \Vert ^{2}_{L^{2}(\mathbb{R})} \Vert \phi \Vert ^{2}_{L^{2}( \mathbb{R})} \\ &= \epsilon ^{2}. \end{aligned}$$

Hence,

$$\begin{aligned} \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f( \omega,u) \bigr\vert ^{2} \,d\omega \,du \leq \int _{X} \int _{X} \bigl\vert G_{\phi }^{A} f( \omega,u) \bigr\vert ^{2} \,d\omega \,du + \epsilon ^{2}. \end{aligned}$$

(61)

Applying equation (20) and the Hölder inequality, we obtain

$$\begin{aligned} 1-\epsilon ^{2} &\leq \int _{X} \int _{X} \bigl\vert G_{\phi }^{B} f( \omega,u) \bigr\vert ^{2} \,d\omega \,du \\ &\leq \biggl( \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert G_{\phi }^{B} f( \omega,u) \bigr\vert ^{2 \frac{r}{2}} \,d\omega \,du \biggr)^{\frac{2}{r}} \bigl( \mu (X) \bigr)^{\frac{r-2}{r}} \\ &\leq \biggl(\frac{1}{2\pi b} \biggr)^{\frac{r-2}{r}} \bigl(\mu (X) \bigr)^{\frac{r-2}{r}}, \end{aligned}$$

which completes the proof of the theorem. □

Next, we present a lemma, which describes the basic concept of Nazarovs’s uncertainty principle for the Fourier transform (see [18]).

Lemma 5.7

Suppose that $X_{1}$ and $X_{2}$ are two finite, measurable subsets of $\mathbb{R}$. Then for every $f \in L^{2}(\mathbb{R})$ there exists a constant $C>0$ such that

$$\begin{aligned} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \leq C e^{C\mu (X_{1}) \mu (X_{2})} \biggl( \int _{\mathbb{R}\setminus X_{1}} \bigl\vert f(x) \bigr\vert ^{2} \,dx + \int _{ \mathbb{R}\setminus X_{2}} \bigl\vert F\{f\}(w) \bigr\vert ^{2} \,dw \biggr). \end{aligned}$$

(62)

We are ready to obtain a straightforward generalization of Nazarovs’s uncertainty principle in the framework of the WLCT.

Theorem 5.8

With the notations of Lemma 5.7above, if $\phi \in L^{2}(\mathbb{R})$, then

$$\begin{aligned} & \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \\ &\quad \leq C e^{C\mu (X_{1}) \mu (X_{2})} \\ &\qquad{}\times \biggl( \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}\setminus X_{1}} \bigl\vert f(x) \bigr\vert ^{2} \,dx + 2\pi \int _{\mathbb{R}} \int _{\mathbb{R}\setminus b X_{2}} \bigl\vert G_{ \phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \biggr). \end{aligned}$$

(63)

Proof

Replacing $f(x)$ by $e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi }(x)$ on both sides of (62), we obtain

$$\begin{aligned} &\int _{\mathbb{R}} \bigl\vert e^{\frac{ia}{2b}x^{2}}f T_{v} \bar{\phi }(x) \bigr\vert ^{2} \,dx \\ &\quad \leq C e^{C\mu (X_{1}) \mu (X_{2})} \\ &\qquad {}\times \biggl( \int _{\mathbb{R}\setminus X_{1}} \bigl\vert e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi }(x) \bigr\vert ^{2} \,dx + \int _{\mathbb{R}\setminus X_{2}} \bigl\vert F \bigl\{ e^{\frac{ia}{2b}x^{2}}f T_{v} \bar{\phi }\bigr\} (w) \bigr\vert ^{2} \,dw \biggr). \end{aligned}$$

(64)

Hence,

$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \\ &\quad \leq C e^{C\mu (X_{1}) \mu (X_{2})} \\ &\qquad{}\times \biggl( \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}\setminus X_{1}} \bigl\vert f(x) \bigr\vert ^{2} \,dx + \int _{\mathbb{R}} \int _{\mathbb{R}\setminus X_{2}} \bigl\vert F \bigl\{ e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi } \bigr\} (w) \bigr\vert ^{2} \,dw \,dv \biggr). \end{aligned}$$

(65)

This further leads to

$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \\ &\quad \leq C e^{C\mu (X_{1}) \mu (X_{2})} \\ &\qquad{}\times \biggl( \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}\setminus X_{1}} \bigl\vert f(x) \bigr\vert ^{2} \,dx + \frac{1}{b} \int _{\mathbb{R}} \int _{\mathbb{R} \setminus b X_{2}} \biggl\vert F\bigl\{ e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi } \bigr\} { \biggl(\frac{w}{b} \biggr)} \biggr\vert ^{2} \,dw \,dv \biggr). \end{aligned}$$

(66)

Applying relation (13), we see that

$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \\ &\quad \leq C e^{C\mu (X_{1}) \mu (X_{2})} \\ &\qquad{}\times \biggl( \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}\setminus X_{1}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \\ &\qquad{}+ \frac{1}{b} \int _{\mathbb{R}} \int _{\mathbb{R} \setminus b X_{2}} \bigl\vert \sqrt{2\pi b} e^{i\frac{\pi }{4}} e^{-i \frac{d}{2b}w^{2}} G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \biggr), \end{aligned}$$

(67)

and the required result follows. □

The following results concern some consequences of Nazarov’s uncertainty principles described by equation (62).

Corollary 5.9

Using the notations as in Lemma 5.7, we have

$$\begin{aligned} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \leq C e^{C\mu (X_{1}) \mu (X_{2})} \biggl( \int _{\mathbb{R}\setminus X_{1}} \bigl\vert f(x) \bigr\vert ^{2} \,dx + 2\pi \int _{ \mathbb{R}\setminus bX_{2}} \bigl\vert \mathcal{C}_{B}\{ f \} (w) \bigr\vert ^{2} \,dw \biggr), \end{aligned}$$

(68)

which is Nazarovs’s uncertainty principle in the context of the LCT.

Proof

Including $f(x)$ as $e^{\frac{ia}{2b}x^{2}}f(x)$ on both sides of (62) and then implementing (5) will lead to the desired result. □

The above corollary is consistent with the result studied in [19], where, in this case, the LCT is a particular case of the offset linear canonical transform.

Corollary 5.10

Under the assumptions of Lemma 5.7, if $\phi \in L^{2}(\mathbb{R})$, then one has

$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \\ &\quad \leq C e^{C\mu (X_{1}) \mu (X_{2})} \\ &\qquad {}\times\biggl( \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}\setminus X_{1}} \bigl\vert f(x) \bigr\vert ^{2} \,dx + 2\pi \int _{\mathbb{R}} \int _{\mathbb{R}\setminus X_{2}} \bigl\vert G_{\phi } f(w,v) \bigr\vert ^{2} \,dw \,dv \biggr), \end{aligned}$$

(69)

which is Nazarovs’s uncertainty principle associated with the WFT defined by (14).

Proof

By (15), the proof is straightforward. □

Theorem 5.11

(WLCT local uncertainty principle)

Let $\phi \in L^{2}(\mathbb{R})$ be a window function. If $0 <\alpha <\frac{1}{2}$, then there exists some constant $C_{\alpha }$ such that for any $f\in L^{2}(\mathbb{R})$ and $X\subseteq \mathbb{R}$ measurable, one has

$$\begin{aligned} \int _{\mathbb{R}} \int _{bX} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \leq \frac{C_{\alpha } \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} }{2\pi } \bigl[\mu (X) \bigr]^{2 \alpha } \int _{\mathbb{R}} \vert x \vert ^{2\alpha } \bigl\vert f(x) \bigr\vert ^{2} \,dx, \end{aligned}$$

(70)

and for $\alpha > \frac{1}{2}$, it holds

$$\begin{aligned} 2\pi \int _{\mathbb{R}} \int _{bX} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \leq{}& C_{\alpha } \mu (X) \Vert \phi \Vert _{L^{2}(\mathbb{R})}^{ \frac{2\alpha -1}{\alpha }} \biggl( \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx \biggr)^{\frac{2\alpha -1}{2\alpha }} \\ & {}\times \Vert \phi \Vert _{L^{2}(\mathbb{R})}^{\frac{1}{\alpha }} \biggl( \int _{ \mathbb{R}} \bigl\vert \vert x \vert ^{\alpha } f(x) \bigr\vert ^{2} \,dx \biggr)^{ \frac{1}{2\alpha }}, \end{aligned}$$

(71)

Proof

Let a set $X\subseteq \mathbb{R}$ with finite measure. The local uncertainty principle for the Fourier transform is expressed as (see [20])

$$\begin{aligned} \int _{X} \bigl\vert F\{f\}(w) \bigr\vert ^{2} \,dw \leq C_{\alpha } \bigl[\mu (X)\bigr]^{2\alpha } \bigl\Vert \vert x \vert ^{ \alpha } f \bigr\Vert ^{2}_{L^{2}(\mathbb{R})}. \end{aligned}$$

(72)

In fact, by inserting $f(x)$ as $f T_{v} \bar{\phi }(x)$ on both sides of the relation (72) above, we get

$$\begin{aligned} \int _{X} \bigl\vert F\{f T_{v} \bar{\phi }\}(w) \bigr\vert ^{2} \,dw \leq C_{\alpha } \bigl[ \mu (X) \bigr]^{2\alpha } \bigl\Vert \vert x \vert ^{\alpha } f T_{v} \bar{\phi } \bigr\Vert ^{2}_{L^{2}( \mathbb{R})}. \end{aligned}$$

(73)

Furthermore,

$$\begin{aligned} &\int _{\mathbb{R}} \int _{X} \bigl\vert F\bigl\{ e^{\frac{ia}{2b}x^{2}}f T_{v} \bar{\phi }\bigr\} (w) \bigr\vert ^{2} \,dw \,dv \\ &\quad \leq C_{\alpha } \bigl[\mu (X)\bigr]^{2\alpha } \int _{ \mathbb{R}} \int _{\mathbb{R}} \bigl\vert \vert x \vert ^{\alpha } e^{\frac{ia}{2b}x^{2}} f(x) \overline{\phi (x -v)} \bigr\vert ^{2} \,dx \,dv. \end{aligned}$$

(74)

Setting $w = \frac{w}{b}$, we obtain

$$\begin{aligned} &\frac{1}{b} \int _{\mathbb{R}} \int _{bX} \biggl\vert F\bigl\{ e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi }\bigr\} \biggl(\frac{w}{b}\biggr) \biggr\vert ^{2} \,dw \,dv \\ &\quad \leq C_{\alpha } \bigl[\mu (X) \bigr]^{2 \alpha } \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert \vert x \vert ^{\alpha } f(x) \overline{\phi (x -v)} \bigr\vert ^{2} \,dx \,dv. \end{aligned}$$

(75)

It follows from the expression (13) that

$$\begin{aligned} 2\pi \int _{\mathbb{R}} \int _{bX} \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \leq C_{\alpha } \bigl[\mu (X) \bigr]^{2\alpha } \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \vert x \vert ^{2\alpha } \bigl\vert f(x) \bigr\vert ^{2} \,dx, \end{aligned}$$

(76)

which gives the proof of (70). Based on the local uncertainty principle for the Fourier transform

$$\begin{aligned} \int _{X} \bigl\vert F\{f\}(w) \bigr\vert ^{2} \,dw \leq C_{\alpha } \mu (X) \Vert f \Vert _{L^{2}( \mathbb{R})}^{2-\frac{1}{\alpha }} \bigl\Vert \vert x \vert ^{\alpha } f \bigr\Vert _{L^{2}( \mathbb{R})}^{\frac{1}{\alpha }}, \end{aligned}$$

(77)

we apply the same arguments to get equation (71). □

Recently, Kubo et al. [21] have proposed the logarithmic Sobolev-type uncertainty principle for the Fourier transform. Below, we search this uncertainty principle in the setting of the WLCT. For this purpose, we present the following.

Definition 5.2

For $1 \leq r < \infty$ and $s>0$ define the weighted Lebesgue space as

$$\begin{aligned} \mathcal{W}_{s}^{r}(\mathbb{R}) = \bigl\{ f\in L^{r}(\mathbb{R}): \langle x\rangle ^{s} \in L^{r}(\mathbb{R}) \bigr\} , \end{aligned}$$

(78)

where $\langle x\rangle = (1 +x^{2})^{\frac{1}{2}}$ is the weight function.

We have the following.

Theorem 5.12

Let $\phi \in L^{2}(\mathbb{R})$ be a window function. For every $f\in \mathcal{S}(\mathbb{R})\cap \mathcal{W}_{1}^{2}(\mathbb{R})$, we have

$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \ln \biggl(\frac{1+ \vert x \vert ^{2}}{2} \biggr) \,dx + 2\pi \int _{\mathbb{R}} \int _{ \mathbb{R}} \ln \vert w \vert \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \\ &\quad \geq \biggl( \frac{\Gamma ^{\prime } ( 1/2 )}{\Gamma ( 1/2 )} + 2 \pi \ln b \biggr) \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx. \end{aligned}$$

(79)

In this case, $\mathcal{S}(\mathbb{R})$ is the Sobolev space on $\mathbb{R}$ defined by

$$\begin{aligned} \mathcal{S}(\mathbb{R}) = \bigl\{ f\in L^{2}(\mathbb{R}): Df \in L^{2}( \mathbb{R}) \bigr\} , \end{aligned}$$

where D stands for the differential operator, and Γ indicates the Gamma function.

Proof

From the logarithmic Sobolev-type uncertainty principle for the Fourier transform, we obtain that (see [21])

$$\begin{aligned} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \ln \biggl(\frac{1+ \vert x \vert ^{2}}{2} \biggr) \,dx + \int _{\mathbb{R}} \ln \vert w \vert \bigl\vert F\{f\}(w) \bigr\vert ^{2} \,dw \geq \biggl( \frac{\Gamma ^{\prime } ( 1/2 )}{\Gamma ( 1/2 )} \biggr) \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx. \end{aligned}$$

(80)

Now replacing $f(x)$ by $e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi }(x)$ on both sides of identity (80) yields

$$\begin{aligned} &\int _{\mathbb{R}} \bigl\vert e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi }(x) \bigr\vert ^{2} \ln \biggl(\frac{1+ \vert x \vert ^{2}}{2} \biggr) \,dx + \int _{\mathbb{R}} \ln \vert w \vert \bigl\vert F\bigl\{ e^{\frac{ia}{2b}x^{2}} f T_{v} \bar{\phi } \bigr\} (w) \bigr\vert ^{2} \,dw \\ &\quad\geq \biggl( \frac{\Gamma ^{\prime } ( 1/2 )}{\Gamma ( 1/2 )} \biggr) \int _{\mathbb{R}} \bigl\vert e^{\frac{ia}{2b}x^{2}}f T_{v} \bar{\phi }(x) \bigr\vert ^{2} \,dx. \end{aligned}$$

(81)

Hence,

$$\begin{aligned} &\int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert f T_{v} \bar{\phi }(x) \bigr\vert ^{2} \ln \biggl(\frac{1+ \vert x \vert ^{2}}{2} \biggr) \,dx \,dv + \int _{\mathbb{R}} \int _{\mathbb{R}} \ln \vert w \vert \bigl\vert F\bigl\{ e^{\frac{ia}{2b}x^{2}}f T_{v} \bar{\phi } \bigr\} (w) \bigr\vert ^{2} \,dw \,dv \\ &\quad \geq \biggl( \frac{\Gamma ^{\prime } ( 1/2 )}{\Gamma ( 1/2 )} \biggr) \int _{\mathbb{R}} \int _{\mathbb{R}} \bigl\vert f T_{v} \bar{\phi }(x) \bigr\vert ^{2} \,dx\,dv. \end{aligned}$$

(82)

This implies that

$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \ln \biggl(\frac{1+ \vert x \vert ^{2}}{2} \biggr) \,dx + \frac{1}{b} \int _{\mathbb{R}} \int _{\mathbb{R}} \ln \biggl\vert \frac{w}{b} \biggr\vert \biggl\vert F\bigl\{ e^{ \frac{ia}{2b}x^{2}}f T_{v} \bar{\phi } \bigr\} \biggl(\frac{w}{b}\biggr) \biggr\vert ^{2} \,dw \,dv \\ &\quad \geq \biggl( \frac{\Gamma ^{\prime } ( 1/2 )}{\Gamma ( 1/2 )} \biggr) \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx. \end{aligned}$$

(83)

Applying (13) gives

$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \ln \biggl(\frac{1+ \vert x \vert ^{2}}{2} \biggr) \,dx + 2\pi \int _{\mathbb{R}} \int _{ \mathbb{R}} \ln \biggl\vert \frac{w}{b} \biggr\vert \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \\ & \quad\geq \biggl( \frac{\Gamma ^{\prime } ( 1/2 )}{\Gamma ( 1/2 )} \biggr) \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx. \end{aligned}$$

(84)

According to (20), we deduce that

$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \ln \biggl(\frac{1+ \vert x \vert ^{2}}{2} \biggr) \,dx + 2\pi \int _{\mathbb{R}} \int _{ \mathbb{R}} \ln \vert w \vert \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \\ &\quad \geq \biggl( \frac{\Gamma ^{\prime } ( 1/2 )}{\Gamma ( 1/2 )} + 2 \pi \ln b \biggr) \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx, \end{aligned}$$

and the proof is complete. □

As an immediate consequence of the above theorem, we obtain the following (see [7]).

Corollary 5.13

Let $\phi \in L^{2}(\mathbb{R})$ be a window function. For every $f\in \mathcal{S}(\mathbb{R})$, we have

$$\begin{aligned} &\Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{\mathbb{R}} \ln \vert x \vert \bigl\vert f(x) \bigr\vert ^{2} \,dx + \int _{\mathbb{R}} \int _{\mathbb{R}} \ln \vert w \vert \bigl\vert G_{\phi }^{B} f(w,v) \bigr\vert ^{2} \,dw \,dv \\ & \quad\geq (D + \ln b ) \Vert \phi \Vert ^{2}_{L^{2}(\mathbb{R})} \int _{ \mathbb{R}} \bigl\vert f(x) \bigr\vert ^{2} \,dx, \end{aligned}$$

(85)

where $D = \Phi (\frac{1}{2})-\ln \pi, \Phi (x) = \frac{d}{dx} \ln [ \Gamma (x)]$.

Conclusion

In this paper, we have investigated some properties of the windowed linear canonical transform like the orthogonality relation, inversion theorem, and conjugate function by utilizing the direct interaction among the windowed linear canonical transform, windowed Fourier transform and the Fourier transform. We have also discussed several generalizations of the uncertainty principles in the setting of the windowed linear canonical transform.