Quantitative Uncertainty Principles Associated with the Deformed Gabor Transform

Mejjaoli, Hatem; Shah, Firdous A.

doi:https://doi.org/10.1155/2022/8658689

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Research Article | Open Access

Volume 2022 | Article ID 8658689 | https://doi.org/10.1155/2022/8658689

Quantitative Uncertainty Principles Associated with the Deformed Gabor Transform

Hatem Mejjaoli¹and Firdous A. Shah²

Academic Editor: Serkan Araci

Received10 Jan 2022

Accepted15 Feb 2022

Published23 May 2022

1. Introduction

The nonstationery signals constitute a wider class of signals arising in natural or artificial communication systems. As such, the mathematical representation of such signals is one of the core areas of interest among researchers working in diverse aspects of harmonic analysis. Indeed, the nonstationery signals require frequency analysis that is local in time, resulting in the notion of time-frequency analysis. The utmost development in the context of time-frequency analysis came in the form of the well-known Gabor transform [1], which deals with the decomposition of nontransient signals in terms of time- and frequency-shifted basis functions, known as Gabor window functions. With the aid of these window functions, one can analyze the spectral contents of nontransient signals in localized neighbourhoods of time [2]. Mathematically, the Gabor transformation of any evaluated at the location in the time-frequency plane is defined by [3]where is an arbitrary window function. Keeping in view the fact that Gabor transform (1) relies on the family of analyzing functions determined by the translation and modulation operators acting on the window function , Mejjaoli [4] introduced the notion of deformed Gabor transform by revamping the classical family of analyzing elements aswhere and denote the generalized translation and modulation operators acting on the window function aswhere , , , denotes the well-known deformed Hankel transform. For any , the deformed Gabor transform with respect to is given aswhere is given by (2).

On the flip side, the notion of uncertainty principles is central in harmonic analysis and with the advent time-frequency analysis, the investigation of the uncertainty inequalities received considerable heed and such inequalities have already been extensively studied for diverse integral transforms ranging from the classical Fourier to the recently introduced quadratic-phase Fourier transforms [5]. The pioneering Heisenberg’s uncertainty principle asserts that it is impossible for any ideal function to attain compact support in both the time and frequency domains. In literature, many amendments of the usual Heisenberg’s uncertainty principle have been carried out, with the most notable ones being the Beckner-type uncertainty principles [6], Benedick’s uncertainty principles [7], entropy-based uncertainty inequalities [8], Pitt’s inequality [9], weighted-type inequalities [10], Nazarov’s uncertainty principles, local uncertainty principles, and several others [11]. These uncertainty principles are broadly classified into qualitative and quantitative inequalities. In the present article, our primary goal is to formulate certain quantitative uncertainty principles pertaining to the deformed Gabor transform (4). Nevertheless, we shall also present certain prerequisite developments regarding the notion of the DGT.

The highlights of the article are pointed out as follows:(1)To present the notion of generalized Gabor transform operator in the setting of the deformed Gabor transform(2)To obtain some Heisenberg-type uncertainty principles by following diverse strategies, such as the principle of nonexclusive entropy, the contraction semigroup method, and so on(3)Formulation of Pitt’s and Beckner’s uncertainty principles associated with DGT(4)To study some weighted uncertainty inequalities pertaining to the DGT(5)To obtain few other uncertainty inequalities for the deformed Gabor transform based on concentration over sets, such as the Benedick–Amrein–Berthier and the local-type uncertainty principles

The remainder of this paper is organized into four sections: Section 2 deals with preliminaries including the fundamental results about the deformed Hankel transform and the generalized translation operators. In addition, the notions of deformed Gabor transform abreast to the fundamental properties are also studied. In Section 3, we formulate certain Heisenberg-type uncertainty principles, whereas the Beckner uncertainty principle and some other weighted uncertainty inequalities for the deformed Gabor transform are presented in Section 4. Finally, in Section 5, we obtain some other uncertainty inequalities for the DGT based on concentration over sets, such as the Benedick–Amrein–Berthier and the local-type uncertainty principles.

2. Deformed Hankel and Gabor Transforms

The aim of this section is to present the prerequisites concerning the deformed Hankel and Gabor transforms which shall be frequently used in formulating the main results. The main references are [12–15].

2.1. The Deformed Hankel Transform

Here, we shall take a survey of the deformed Hankel transform together with the fundamental properties. To facilitate the narrative, we set some notations as follows:(1) the space of bounded continuous functions on .(2)The space C_b,e of even bounded continuous functions on .(3)For , the conjugate exponent shall be denoted by .(4), , , and .(5), , denotes the space of measurable functions on satisfying

In case , the inner product on the space is given by

We are now in a position to recall the notion of deformed Hankel transform. In this direction, we have the following definition.

Definition 2.1. For , and , the integral operatoris termed as the deformed Hankel transform, where denotes the kerneland are the normalized Bessel function of index :Deformed Hankel kernel (8) satisfies the following properties:(i)For , we have Moreover, for all , we have .(ii)There exists a finite positive constant depending on and such thatIn what follows, we shall replace by the rescaled version but continue to use the same symbol . As such,Remarks are as follows: (i)We note that the authors in [9] conjectured (12) when .(ii)The deformed Hankel transform is bounded on the space and(iii)The deformed Hankel transform provides a natural generalization of the classical Hankel transform. Indeed, if we set(iv)then the deformed Hankel transform of an even function on the real line yields a Hankel-type transform on . In fact, in case and is also even over , thenIn the upcoming theorem, we collect certain elementary properties of the deformed Hankel transform. The proof of the theorem can be acquired from [12].

Theorem 2.1. For any pair of functions , the following assertions are true:(i)The deformed Hankel transform satisfies(ii)The deformed Hankel transform is an isometric isomorphism on , satisfying(iii)The deformed Hankel transform is an involution unitary operator on ; that is,

Proposition 2.1. Let be in and . Then, belongs to andIn the following, we present some useful results concerning the notion of generalized translation operator on .

Definition 2.2 (see [16]). For any , the generalized translation operator on is defined byIt is imperative to mention that relation (20) holds pointwise, provided is a member of the generalized Weiner space .Next, we shall present some useful properties of generalized translation operator (20).

Proposition 2.2 (see [14, 15]). If is the generalized translation operator on , then the following statements are true:(i)For any , we have(ii)If , then(iii)For all and , we have(iv)For all in and , we haveRecently, the authors in [13] have obtained some important results for generalized translation operator (20). Keeping the notations as in [13], we present the next theorem.

Theorem 2.2. Let and let . For , the generalized translation operator is expressible aswithwhere denotes the kernel, which is supported on the set . Moreover, operator (26) satisfies the following norm inequality:

Apart from the integral representation of the generalized translation operator given in Theorem 2.2, another useful “trigonometric” form has been obtained in [14, 15]. In continuation to this, we present the next theorem.

Theorem 2.3 (see [14, 15]). Suppose is such that , where is an even function and is an odd function. Then, the generalized translation operator can be expressed aswhere are the degenerate polynomials and

Corollary 2.1 (see [14, 15]). For all in , we have

Besides the aforementioned trigonometric form, the authors in [14, 15] have also studied certain important properties of the generalized translation operator, which play a crucial role in the subsequent developments on the subject. Here, we recall some of the needful properties, whose proof can be found in [14, 15].

Proposition 2.3. Suppose that is nonnegative, is even, and belongs to generalized Wigner space . Then, for any , we have(i).(ii) and

Theorem 2.4. Let be the space of even functions in . Then, we have(i) and(ii)The generalized translation operator defined on can be extended to , . Moreover, for any , we have(iii)For every , we have

2.2. The Deformed Gabor Transform

In this section, we shall take a tour of the deformed Gabor transform introduced in [4]. Primarily, we fix some notations which shall be frequently used while formulating the main results. For , we denote as the space of measurable functions on satisfyingwhere .

Definition 2.3. For , the generalized modulation acting on is given bywhere is the usual generalized translation operator.
Remark: by virtue of positivity of the generalized translation operator as mentioned in Theorem 2.4, we infer that (37) is well defined. Moreover, by virtue of (17) and (32), we haveBased on the generalized translation and modulation operations defined in (20) and (37), respectively, we consider the family of functions , asThen, it can be easily verified thatWith the aid of (39), we are in a position to present the formal definition of the deformed Gabor Fourier transform.

Definition 2.4. Given , the deformed Gabor transform with respect to the window is denoted by and is defined aswhere is given by (39).
For any and , it can be easily verified thatwithIn the next theorem, some basic properties of deformed Gabor Fourier transform (41) are assembled.

Theorem 2.5 (see [4]). For any and , the following statements are true:(i)The deformed Gabor Fourier transform is a bounded operator and(ii)The deformed Gabor transform (41) satisfies the following energy preserving relation:(iii)For any given pair of functions , the following orthogonality relation holds:(iv)For , we haveThe following lemma follows by a straightforward calculation.

Lemma 2.1. Let ; then, for any , we have

Theorem 2.6. Suppose that a function satisfies . Then, the functionbelongs to and satisfies

3. Heisenberg-Type Inequalities for the Deformed Gabor Transform

Heisenberg’s uncertainty principle is surely the stepping stone for harmonic analysis, which is in fact an analogy of the prominent Heisenberg’s uncertainty principle in quantum mechanics asserting that it is impossible to ascertain both the position and momentum of particles simultaneously [5]. The harmonic analysis variant of the uncertainty principle is also referred to as the duration-bandwidth theorem, due to the fact that the principle states that the widths of a signal in the time domain (duration) and in the frequency domain (band-width) are constrained and cannot be made arbitrarily narrow. In this section, we shall establish certain Heisenberg-type uncertainty inequalities in the context of the deformed Gabor transform by choosing the window function as a nontrivial even function in the space .

3.1. Generalized Heisenberg’s Uncertainty Principle

In order to facilitate the formulation of new variants of Heisenberg’s principle for the deformed Gabor transform (41), we ought to recall a fundamental uncertainty inequality in the context of deformed Hankel transform .

Proposition 3.1 (see [12, 10]). For , there exists a positive constant , such that for every in , we havewhere .

Theorem 3.1. Let be the deformed Gabor transform of any . Then, for , we havewhere is the same constant as in Proposition 3.1.

Proof. We shall take into consideration the ideal case, assuming that the integrals appearing in (52) are finite. For an arbitrary , inequality (51) yieldsIntegrating under the measure followed by the implication of Cauchy–Schwarz’s inequality yieldsUsing the fact thatwe deduce thatHence, the proof of Theorem 3.1 is complete.

Proposition 3.2 (Nash’s inequality for ). Given , we can always find a constant such thatfor all .

Proof. Clearly inequality (57) is true in case . For a given , we consider ; then, it follows from Plancherel’s formula (45) thatwhereOn the other hand, (46) implies thatMoreover, we haveTherefore, it follows thatAfter minimizing over , the RHS of the above inequality implies thatThe intended outcome occurs fairly from (63).

3.2. Heisenberg’s Uncertainty Principle via -Entropy

The -entropy pertaining to a given probability density function over is given via the integral expression [17]:where

The main goal of the present section is to investigate upon the local characteristics of the -entropy associated with deformed Gabor transform (41).

Proposition 3.3. For all , we have

Proof. Assume that . Then, by virtue of (44), we obtainIn particular, . We now relax the above assumption and considerThen, it is quite evident that and .
Hence, . Indeed, we havewhich further implies thatInvoking , it follows thatHence, the proof of Proposition 3.3 is complete.
Employing the -entropy of deformed Gabor transform (41), we can have another variant of Heisenberg’s principle for .

Theorem 3.2. For every and , we havewhere

Proof. For every , we considerAfter doing some elementary computations, we observe thatwhere is the probability measure on . Since is convex on , Jensen’s inequality implies thatwhich further implies thatAssume that . Then, Proposition 3.3 implies thatHowever, the RHS of inequality (78) attains its upper bound atand consequently,whereTherefore, for every and satisfying , we haveIt is evident that for , the dilates and belong to . Therefore, after substituting with and with and noting , the above inequality yieldsUsing (42), we obtainIn particular, the inequality holds at the pointso thatwhereHence, the desired result is obtained after replacing with and with . □
Remark: for , we have

3.3. -Heisenberg’s Uncertainty Principle

In this section, we shall establish a unified form of -Heisenberg’s inequality for deformed Gabor transform (41). Our strategy of the proof is motivated by [18], wherein the authors have studied the -Heisenberg’s uncertainty inequality in the context of Lie groups. To facilitate the narrative, we set the following notation:

It is quite straightforward to verify that for every , there exist with

Lemma 3.1. Let be the deformed Gabor transform of any and . Then, there exists a positive constant such that

Proof. Trivially, inequality (91) holds for .
Assume that . Moreover, for , we consider and . Then, we haveSubsequently, by virtue of (47), we obtainOn the flip side, relation (44) and Hölder’s inequality implies thatIt is quite straightforward to demonstrate that there exists a positive constant such thatConsequently, we obtainChoosing and using (90), we obtain the desired inequality.

Theorem 3.3. Let be the deformed Gabor transform of any arbitrary function . Then, for and , the following inequality holds:

Proof. Trivially, inequality (97) holds, whenever . Assume that . Using Lemma 3.1, it follows that for and for all , we haveHowever, on the flip side, we haveTherefore, we haveas is bounded for and . By optimizing the above inequality over , we can obtain inequality (97) for and .
Next, we shall consider the case, when . For and , it is easy to see that , which is for becomesTherefore, we haveUpon optimizing over , we shall obtain a positive constant such thatCombining (97) and (103), we get the desired inequality.

Corollary 3.1. For and , we can always find some satisfyingfor all .

Proof. The result follows immediately by applying Theorem 3.3 with and Plancherel’s formula (45).

4. Weighted-Type Inequalities for the Deformed Gabor Transform

Pitt’s inequality has a fundamental importance in the deformed Hankel setting because it describes the variance between a sufficiently smooth function and the corresponding deformed Hankel transform. Recently, Gorbachev et al. [9] have proposed a sharp form of Pitt’s and Beckner-type inequalities for the deformed Hankel transform. Explicitly, for any , they formulated thatwhere

Here, our primary goal is to formulate a new variant of Pitt’s inequality (105) pertaining to the DGT given in (41).

Theorem 4.1. Let be the deformed Gabor transform corresponding to ; then,where is given by (106) and .

Proof. By virtue of (105), we obtainwhich upon integration under Haar measure implies thatImplementing Lemma 2.1, expression (109) takes the following form:Or,Using the hypothesis on , relation (32) becomeswhich establishes Pitt’s inequality for deformed Gabor transform (41).

Remark. Plugging , we find that in (107), equality holds, which is exactly in accordance with (48). Next, we present the deformed Hankel–Beckner’s inequality, which asserts that for [9],The above inequality is intimately intertwined with Heisenberg’s inequality, owing to that it is sometimes called the logarithmic variant of the uncertainty inequality. In the recent literature, many novel ramifications of such an inequality have been witnessed from time to time [10]. Our next motive is to obtain an associate of Beckner-type inequality (113) in the context of DGT defined in (41).

Theorem 4.2. Let ; then, we have

Proof. By replacing in (113) with , we obtainIntegrating (115) under implies thatBy virtue of (45), we obtainIn order to derive a useful computation for the later integral in (117), we invoke Lemma 2.1 together with (32), so thatSubstituting (118) in (117), we obtain the result for DGT (41).
Next, we shall give another proof of Theorem 4.2, which primarily relies upon (107).

Proof. (alternate proof of Theorem 4.2). Given , considerDifferentiating (119) with respect to , we obtainwhereFor , relation (121) yieldsUsing (107), we getandTherefore, we deduce thatEquivalently,As a consequence of Plancherel’s formula (45) and (122), it follows thatAlternatively,which establishes the result.

Corollary 4.1. Let be the deformed Gabor transform of any arbitrary function with respect to the window function with . Then, we have

Proof. Noting and then using the well-known Jensen’s inequality in (114), we getwhich upon simplification yields desired result (129).
Remarks are as follows:(i)By virtue of the identity [19] we infer that that is precisely the constant as appearing in Theorem 3.1.(ii)In a manner similar to (113), we get(iii)By virtue of (131), it follows that the constant in the RHS of (133) isthat is precisely the constant as appearing in Proposition 3.1.

5. Concentration-Based Inequalities for the Deformed Gabor Transform

This section is devoted for the formulation of some other uncertainty inequalities based on concentration over sets. More precisely, we shall obtain the Benedick–Amrein–Berthier and the local-type uncertainty inequalities for deformed Gabor transform (41).

5.1. Benedick–Amrein–Berthier’s Uncertainty Principle

In order to formulate the Benedick–Amrein–Berthier’s inequality for the deformed Gabor transforms, we ought to recall the fundamental inequality [10]:where and are the subsets of with finite measure and is a constant.

Theorem 5.1. Let be the deformed Gabor transform of any . Then, we havewhere is given by (135).

Proof. For any , it follows that , provided . Therefore, changing to in (135) implies thatwhich after integration under yields the following inequality:Invoking Lemma 2.1 in association with Plancherel’s formula (45), we getso thatUsing Lemma 2.1 and relation (32) and keeping in view that , we obtainwhich is the desired Benedick–Amrein–Berthier’s uncertainty principle for deformed Gabor transform (41).
As a consequence of Theorem 5.1, we obtain the following generalized Heisenberg-type uncertainty inequality for DGT (41).

Corollary 5.1. Let be the deformed Gabor transform of any arbitrary function . Then, for , there exist satisfying

Proof. Take . Then, for any and , (136) implies thatwhere . Hence, it follows thatReplacing by and by , relation (42) implies thatTherefore, we haveAs a result, optimizing the right side of the aforementioned inequality for yields desired inequality (142).

5.2. Local-Type Uncertainty Principles

In this section, we shall formulate certain local uncertainty inequalities pertaining to the deformed Gabor transform (41) by employing the following inequality of the deformed Hankel transforms [8].

Proposition 5.1 (see [8]). For satisfying , there exists with , so thatfor every .

Theorem 5.2. Let ; then, for satisfying and any , the following is true:where is as mentioned in Proposition 5.1.

Proof. Since , whenever , we can change to in (147), so thatIntegrating the above inequality under the measure , we getFurthermore, using Lemma 2.1 yieldsAs such, inequality (151) becomesOr equivalently, for any , we haveHence, the proof of Theorem 5.2 is complete. □
Given a subset of , the Paley–Wiener space is defined byIn view of Plancherel’s formula (17), we are led to the following inequality.

Corollary 5.2. For with and , the following is true:where and the constant is the same as given in Proposition 5.1.
Swapping the functions and in Proposition 5.1 yields the below mentioned inequality.

Corollary 5.3. For with and , we havewhere is the same constant as mentioned in Proposition 5.1.
Adopting the strategy as in Theorem 5.2 and implementing Corollary 5.3, the following inequality is obtained.

Corollary 5.4. For with and , for all , we havewhere is the same constant as mentioned in Proposition 5.1.
For each subset of , we consider the following generalization of Paley–Wiener spaces:Applications of (45), Corollary 5.4, and the definition of generalized Paley–Wiener spaces yield the following inequality.

Corollary 5.5. Given are two subsets satisfying . Also, let ; then,(i)If , we have(ii)For any , we haveWe now formulate yet another variant of Heisenberg-type uncertainty principle pertaining to the DGT defined in (41).

Theorem 5.3. Let be the deformed Gabor transform of any and and . Then,where

Proof. Let and . Then, for , we haveUsing Theorem 5.2 and some simple calculation, we obtainBesides, it also follows thatCombining relations (163)–(165), we getWe chooseyielding the inequality as desired. □
To wind up the ongoing discourse, we have the following local uncertainty inequality for deformed Gabor transform (41).

Theorem 5.4 (Faris–Price’s inequality). Let satisfy and . For every measurable subset with and , there exists with

Proof. Consider the normalized case . For each , we obtainwhere denotes the ball on with radius . However, for every , relation (43) and Hölder’s inequality implies thatOn the flip side, we also haveTherefore,Using the inequality due to Hölder together with (44), it follows thatTherefore, for every , we haveIn particular, the inequality holds forand hence,Or equivalently,

Remark: for , we define the modulation of by as follows [20]:

Also, the generalized Gabor transform is given by

It is clear that

Therefore, by virtue of Plancherel’s formula (17), we obtain that the two integral transforms are equivalent. As such, all results proved for one are valuables for the second. Hence, we reclaim that all results proved in [4] and in this paper for the deformed Gabor transform are valuables for the integral transform and it is suffice to replace by to derive the analogues results.

6. Conclusion

In the present article, we have accomplished two major objectives regarding the uncertainty inequalities pertaining to the deformed Gabor transform (DGT). Firstly, we obtained Heisenberg’s and Beckner’s uncertainty principles for the deformed Gabor transform. Besides, we also obtained certain weighted uncertainty inequalities for DGT. Secondly, we formulated a few concentration-based inequalities, such as the Benedick–Amrein–Berthier and the local-type uncertainty principles for the deformed Gabor transform.

Data Availability

Data sharing not applicable to this article as no data sets were generated or analysed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this article and approved the final manuscript.

Acknowledgments

The first author dedicates this paper to the Emeritus Professor Khalifa Trimèche.

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Copyright © 2022 Hatem Mejjaoli and Firdous A. Shah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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