Research Quantitative Uncertainty Principles Associated with the Deformed Gabor Transform

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Introduction
e nonstationery signals constitute a wider class of signals arising in natural or arti cial 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. e 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 f ∈ L 2 (R) evaluated at the location (y, ]) in the time-frequency plane is de ned by [3] G h (f)(y, ]) where h ∈ L 2 (R) is an arbitrary window function. Keeping in view the fact that Gabor transform (1) relies on the family of analyzing functions h(t − y)e iy] determined by the translation and modulation operators acting on the window function h, Mejjaoli [4] introduced the notion of deformed Gabor transform by revamping the classical family of analyzing elements as h ],y (x) τ k,n (− 1) n y h ] (x), x ∈ R, where τ k,n and h ] denote the generalized translation and modulation operators acting on the window function h as h ] ≔ F k,n τ k,n ] |h| 2 , F k,n τ k,n x h B k,n (., x)F k,n (h), where F k,n , n ∈ N, k ≥ (n − 1)/n, denotes the well-known deformed Hankel transform. For any f ∈ L 2 k,n (R), the deformed Gabor transform with respect to h ∈ L 2 k,n,e (R) is given as where h ],y (x) is given by (2). On the ip side, the notion of uncertainty principles is central in harmonic analysis and with the advent timefrequency 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]. e 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], weightedtype inequalities [10], Nazarov's uncertainty principles, local uncertainty principles, and several others [11]. ese 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. e 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 e 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.

Deformed Hankel and Gabor Transforms
e 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. e main references are [12][13][14][15].

e 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) C b (R) the space of bounded continuous functions on R. (2) e space C b,e R of even bounded continuous functions on R.
In case p � 2, the inner product on the space L p k,n (R) 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.
is termed as the deformed Hankel transform, where B k,n (λ, x) denotes the kernel B k,n (λ, x) � J nk− n/2 n|λx| 1/n +(− i) n n 2 n · Γ(nk − n/2 + 1) Γ(nk + n/2 + 1) λxJ nk+n/2 n|λx| 1/n , and J α (u) are the normalized Bessel function of index α: Deformed Hankel kernel (8) satisfies the following properties: Moreover, for all λ ∈ R, we have B k,n (λz, t) � B k,n (z, λt). (ii) ere exists a finite positive constant C depending on n and k such that 2 Journal of Mathematics In what follows, we shall replace B k,n by the rescaled version B k,n /C but continue to use the same symbol B k,n . As such, Remarks are as follows: (i) We note that the authors in [9] conjectured (12) when k ≥ n − 1/n. (ii) e deformed Hankel transform is bounded on the space L 1 k,n (R) and (iii) e deformed Hankel transform F k,n provides a natural generalization of the classical Hankel transform. Indeed, if we set (iv) then the deformed Hankel transform F k,n of an even function f on the real line yields a Hankel-type transform on k,n (R) and is also even over R, then · n(r|ξ|) 1/n r 2/n(2nk+2− n/2)− 1 dr, ∀ξ ∈ R. (15) In the upcoming theorem, we collect certain elementary properties of the deformed Hankel transform. e proof of the theorem can be acquired from [12]. Theorem 2.1. For any pair of functions f, g ∈ L 2 k,n (R), the following assertions are true: (iii) e deformed Hankel transform is an involution unitary operator on L 1 k,n (R); that is, Proposition 2.1. Let f be in L p k,n (R) and p ∈ [1,2]. en, F k,n (f) belongs to L p′ k,n (R) and In the following, we present some useful results concerning the notion of generalized translation operator on L 2 k,n (R).
Definition 2.2 (see [16]). For any x ∈ R, the generalized translation operator f↦τ k,n It is imperative to mention that relation (20) holds pointwise, provided f is a member of the generalized Weiner space W k,n (R).
Next, we shall present some useful properties of generalized translation operator (20). [14,15]). If τ k,n x is the generalized translation operator on L 2 k,n (R), then the following statements are true:

Proposition 2.2 (see
(ii) If f ∈ W k,n (R), then (iii) For all x, y ∈ R and f ∈ W k,n (R), we have (iv) For all f in W k,n (R) and g ∈ L 1 k,n (R) ∩ L ∞ k,n (R), we have Recently, 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 x ∈ R and let f ∈ C b (R). For k ≥ n − 1/n, the generalized translation operator τ k,n x is expressible as with Journal of Mathematics dζ k,n x,y (z) � K k,n (x, y, z)dc k,n (z), if xy ≠ 0, where K k,n (x, y, .) denotes the kernel, which is supported on the set z ∈ R: ||x| 1/n − |y| 1/n | < |z| 1/n < |x| 1/n + |y| 1/n . Moreover, operator (26) satisfies the following norm inequality: Apart from the integral representation of the generalized translation operator given in eorem 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]).
where f e is an even function and f o is an odd function. en, the generalized translation operator τ k,n x can be expressed as where C nk− n/2 n are the degenerate polynomials and 《x, y》 ϕ,n ≔ |x| 2/n +|y| 2/n − 2|xy| 1/n cos ϕ Corollary 2.1 (see [14,15]). For all f in C b,e (R), 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 f is nonnegative, is even, and belongs to generalized Wigner space W k,n (R). en, for any x ∈ R, we have (33) (ii) e generalized translation operator τ k,n x defined on

e 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 1⩽p⩽∞, we denote L p μ k,n (R 2 ) as the space of measurable functions f on R 2 satisfying where dμ k,n (x, y) ≔ dc k,n (x)dc k,n (y).

Definition 2.3.
For ] ∈ R, the generalized modulation acting on h ∈ L 2 k,n,e (R) is given by where τ k,n ] is the usual generalized translation operator. Remark: by virtue of positivity of the generalized translation operator as mentioned in eorem 2.4, we infer that (37) is well defined. Moreover, by virtue of (17) and (32), Based on the generalized translation and modulation operations defined in (20) and (37), respectively, we consider the family of functions h ],y , ], y ∈ R as en, it can be easily verified that With the aid of (39), we are in a position to present the formal definition of the deformed Gabor Fourier transform.
k,n (R), the deformed Gabor transform with respect to the window h ∈ L 2 k,n,e (R) is denoted by G k,n h (f) and is defined as where h ],y (x) is given by (39). For any λ > 0 and (y, ]) ∈ R 2 , it can be easily verified that with In the next theorem, some basic properties of deformed Gabor Fourier transform (41) are assembled. Theorem 2.5 (see [4]). For any f ∈ L 2 k,n (R) and h ∈ L 2 k,n,e (R), the following statements are true: (ii) e deformed Gabor transform (41) satisfies the following energy preserving relation: (iii) For any given pair of functions f, g ∈ L 2 k,n (R), the following orthogonality relation holds: The following lemma follows by a straightforward calculation.
belongs to L 2 k,n (R) and satisfies lim

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]. e harmonic analysis variant of the uncertainty principle is also referred to as the durationbandwidth 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 G k,n h by choosing the window function h as a nontrivial even function in the space L 2 k,n (R).

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 F k,n .

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Proposition 3.1 (see [12,10]). For s, t > 0, there exists a positive constant C k,n (s, t), such that for every f in L 2 k,n (R), we have where C k,n (s, t) 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) yields Integrating under the measure dc k,n (]) followed by the implication of Cauchy-Schwarz's inequality yields Using the fact that we deduce that Hence, the proof of eorem 3.1 is complete.
□ Proposition 3.2 (Nash's inequality for G k,n h ). Given s > 0, we can always find a constant C(k, n, s) > 0 such that for all f ∈ L 2 k,n (R).
□ 6 Journal of Mathematics
e main goal of the present section is to investigate upon the local characteristics of the (k, n)-entropy associated with deformed Gabor transform (41).
Employing the (k, n)-entropy of deformed Gabor transform (41), we can have another variant of Heisenberg's principle for G k,n h .

□ Theorem 3.2.
For every f ∈ L 2 k,n (R) and p, q > 0, we have where Proof. For every t, p, q ∈ R + , we consider After doing some elementary computations, we observe that R 2 η k,n t,p,q (y, ])dμ k,n (y, ]) � 1, where dσ k,n t,p,q (y, ]) ≔ η k,n t,p,q (y, ])dμ k,n (y, ]) is the probability measure on R 2 . Since φ(t) � t log(t) is convex on (0, ∞), Jensen's inequality implies that which further implies that Journal of Mathematics 7 Assume that ‖f‖ L 2 k,n (R) ‖h‖ L 2 k,n (R) � 1. en, Proposition 3.3 implies that However, the RHS of inequality (78) attains its upper bound at and consequently, where erefore, for every f ∈ L 2 k,n (R) and h ∈ L 2 k,n,e (R) satisfying ‖f‖ L 2 k,n (R) ‖h‖ L 2 k,n (R) � 1, we have It is evident that for λ > 0, the dilates f λ and h 1/λ belong to L 2 k,n (R). erefore, after substituting f with f λ and h with h 1/λ and noting ‖f λ ‖ L 2 k,n (R) ‖h 1/λ ‖ L 2 k,n (R) � ‖f‖ L 2 k,n (R) ‖h‖ L 2 k,n (R) � 1, the above inequality yields (84) In particular, the inequality holds at the point so that where C p,q (k, n) � C p,q p p/p+q q q/p+q p + q Hence, the desired result is obtained after replacing f with f/ ‖f‖ L 2 k,n (R) and h with h/‖h‖ L 2 k,n (R) . □ Remark: for p � q � 2, we have

L p -Heisenberg's Uncertainty Principle.
In this section, we shall establish a unified form of L p -Heisenberg's inequality for deformed Gabor transform (41). Our strategy of the proof is motivated by [18], wherein the authors have studied the L 2 -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 1 ≤ q < ∞, there exist C > 0 with Lemma 3.1. Let G k,n h (f) be the deformed Gabor transform of any f ∈ L 2 k,n (R) and 1 < p ≤ 2, 0 < s < (2k − 1)n + 2/2np ′ . en, there exists a positive constant C such that Proof. Trivially, inequality (91) holds for ‖|y| s f‖ L 2 k,n (R) + ‖|y| s f‖ L 2p k,n (R) � ∞. Assume that ‖|y| s f‖ L 2 k,n (R) + ‖|y| s f‖ L 2p k,n (R) < ∞. Moreover, for r > 0, we consider f r � 1 (− r,r) f and f r � f − f r .
Next, we shall consider the case, when t ≤ 1/2. For u ≥ 0 and t ′ ≤ 1/2 < t, it is easy to see that u 4t′ ≤ 1 + u 4t , which is for Upon optimizing over ε, we shall obtain a positive constant C such that Combining (97) and (103), we get the desired inequality. □ Corollary 3.1. For 0 < s < (2k − 1)n + 2/4n and t > 0, we can always find some C > 0 satisfying for all f ∈ L 2 k,n (R).

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 Becknertype inequalities for the deformed Hankel transform. Explicitly, for any f ∈ S(R)⊆L 2 k,n (R), they formulated that where Here, our primary goal is to formulate a new variant of Pitt's inequality (105) pertaining to the DGT given in (41).
Proof. By virtue of (105), we obtain which upon integration under Haar measure dc k,n (]) implies that dμ k,n (y, ]).
(113) e 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).

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Proof. By replacing f in (113) with G k,n h (f) (., ]), we obtain log|y| G k,n h (f)(y, ]) 2 dc k,n (t) Integrating (115) under dc k,n (]) implies that By virtue of (45), we obtain In order to derive a useful computation for the later integral in (117), we invoke Lemma 2.1 together with (32), so that Substituting (118) in (117), we obtain the result for DGT (41).

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Journal of Mathematics − 2‖h‖ 2 2 dμ k,n (y, ]) ≤ 0. (126) As a consequence of Plancherel's formula (45) and (122), it follows that − 2‖h‖ 2 Alternatively, which establishes the result. □ Corollary 4.1. Let G k,n h (f) be the deformed Gabor transform of any arbitrary function f ∈ S(R) with respect to the window function h ∈ L 2 k,n,e (R) ∩ L ∞ k,n (R) with ‖h‖ L 2 k,n (R) � 1. en, we have Proof. Noting ‖h‖ L 2 k,n (R) � 1 and then using the well-known Jensen's inequality in (114), we get which upon simplification yields desired result (129). Remarks are as follows: (i) By virtue of the identity [19] Γ ′ (z) we infer that that is precisely the constant as appearing in eorem 3.1. (ii) In a manner similar to (113), we get Journal of Mathematics 13 R |t| 2 |f(t)| 2 dc k,n (t) (133) (iii) By virtue of (131), it follows that the constant in the RHS of (133) is that is precisely the constant as appearing in Proposition 3.1.

Concentration-Based Inequalities for the Deformed Gabor Transform
is 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).

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 E 1 and E 2 are the subsets of R with finite measure and C k,n (E 1 , E 2 ) is a constant.
Proof. For any ] ∈ R, it follows that G k,n h (f)(., ]) ∈ L 2 k,n (R), provided f ∈ L 2 k,n (R). erefore, changing f to G k,n h (f)(., ]) in (135) implies that which after integration under dc k,n (]) yields the following inequality: Invoking Lemma 2.1 in association with Plancherel's formula (45), we get so that Using Lemma 2.1 and relation (32) and keeping in view that h ∈ L 2 k,n,e (R) ∩ L ∞ k,n (R), we obtain which is the desired Benedick-Amrein-Berthier's uncertainty principle for deformed Gabor transform (41). As a consequence of eorem 5.1, we obtain the following generalized Heisenberg-type uncertainty inequality for DGT (41). □ Corollary 5.1. Let G k,n h (f) be the deformed Gabor transform of any arbitrary function f ∈ L 2 k,n (R). en, for p, q > 0, there exist C k,n (p, q) > 0 satisfying Proof. Take E 1 � E 2 � (− 1, 1). en, for any f ∈ L 2 k,n (R) and p, q > 0, (136) implies that where C(k, n) ≔ C k,n (E 1 , E 2 ). Hence, it follows that Replacing f by f λ and h by h 1/λ , relation (42) implies that erefore, we have As a result, optimizing the right side of the aforementioned inequality for λ > 0 yields desired inequality (142).

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].

Corollary 5.2.
For E ⊂ R with 0 < c k,n (E) < ∞ and 0 < s < (2k − 1)n + 2/2n, the following is true: where f ∈ PW k,n (E) and the constant C(k, n, s) is the same as given in Proposition 5.1.
Swapping the functions f and F k,n (f) in Proposition 5.1 yields the below mentioned inequality.
Adopting the strategy as in Theorem 5.2 and implementing Corollary 5.3, the following inequality is obtained.

Corollary 5.4.
For F ⊂ R with 0 < c k,n (F) < ∞ and 0 < t < (2k − 1)n + 2/2n, for all f ∈ L 2 k,n (R), we have where C(k, n, t) is the same constant as mentioned in Proposition 5.1. For each subset F of R, we consider the following generalization of Paley-Wiener spaces: Applications of (45), Corollary 5.4, and the definition of generalized Paley-Wiener spaces GPW k,n (F) yield the following inequality.