# Quantitative Uncertainty Principles Associated with the Deformed Gabor Transform

Authors: Hatem Mejjaoli; Firdous A. Shah
Journal: Journal of Mathematics
 (2022)

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## Body

## 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 f∈L2ℝ evaluated at the location y,ν in the time-frequency plane is defined by [3](1)Ghfy,ν=∫ℝftht−y¯e−iyνdt,where h∈L2ℝ is an arbitrary window function. Keeping in view the fact that Gabor transform (1) relies on the family of analyzing functions ht−yeiyν 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(2)hν,yx=τ−1nyk,nhνx,x∈ℝ,where τk,n and hν denote the generalized translation and modulation operators acting on the window function h as(3)hν≔ℱk,nτνk,nh2,ℱk,nτxk,nh=Bk,n.,x¯ℱk,nh,where ℱk,n, n∈ℕ, k≥n−1/n, denotes the well-known deformed Hankel transform. For any f∈Lk,n2ℝ, the deformed Gabor transform with respect to h∈Lk,n,e2ℝ is given as(4)Ghk,nfy,ν≔∫ℝfxhν,yx¯dγk,nx,where hν,yx 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 principlesThe remainder of this paper is organized into four sections: Section2 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)
Cbℝ the space of bounded continuous functions on ℝ.(2)
The spaceCb,eℝ of even bounded continuous functions on ℝ.(3)
Forp∈1,∞, the conjugate exponent shall be denoted by p′.(4)
Mk,n≔nn2k−1/2/2n2k−1+2/2Γn2k−1+2/2, dγk,nx≔Mk,nx2k−2n+2/ndx, k≥n−1/n, and n∈ℕ.(5)
Lk,npℝ, 1≤p≤∞, denotes the space of measurable functions f on ℝ satisfying(5)fLk,npℝ≔∫ℝfxpdγk,nx1/p<∞,if1≤p<∞,fLk,n∞ℝ≔esssupx∈ℝfx<∞.In casep=2, the inner product on the space Lk,npℝ is given by(6)f,gLk,n2ℝ≔∫ℝfxgx¯dγk,nx.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.
Fork≥n−1/n, and f∈Lk,n1ℝ, the integral operator(7)ℱk,nfλ=∫ℝfxBk,nλ,xdγk,nx,for allλ∈ℝ,is termed as the deformed Hankel transform, where Bk,nλ,x denotes the kernel(8)Bk,nλ,x=Jnk−n/2nλx1/n+−inn2nΓnk−n/2+1Γnk+n/2+1λxJnk+n/2nλx1/n,and Jαu are the normalized Bessel function of index α:(9)Jαu≔Γα+1u2−αJαu=Γα+1∑m=0∞−1mm!Γα+m+1u22m.
Deformed Hankel kernel (8) satisfies the following properties:(i)
Forz,t∈ℝ, we have(10)Bk,nz,t=Bk,nt,z,Bk,nz,0=1,Bk,nz,t¯=Bk,n−1nz,t.Moreover, for allλ∈ℝ, we have Bk,nλz,t=Bk,nz,λt.(ii)
There exists a finite positive constantC depending on n and k such that(11)Bk,nx,y≤C,∀x,y∈ℝ.
In what follows, we shall replaceBk,n by the rescaled version Bk,n/C but continue to use the same symbol Bk,n. As such,(12)Bk,nx,y≤1,∀x,y∈ℝ.
Remarks are as follows:(i)
We note that the authors in [9] conjectured (12) when k≥n−1/n.(ii)
The deformed Hankel transform is bounded on the spaceLk,n1ℝ and(13)ℱk,nfLk,n∞ℝ≤fLk,n1ℝ,∀f∈Lk,n1ℝ.(iii)
The deformed Hankel transformℱk,n provides a natural generalization of the classical Hankel transform. Indeed, if we set(14)Bk,nevenx,y=12Bk,nx,y+Bk,nx,−y=jnk−n/2nxy1/n,(iv)
then the deformed Hankel transformℱk,n of an even function f on the real line yields a Hankel-type transform on ℝ+. In fact, in case fx=Fx∈Lk,n1ℝ and is also even over ℝ, then(15)ℱk,nfξ=n/22nk−n/2Γ2nk+2−n/2∫0∞Frj2nk−n/2nrξ1/nr2/n2nk+2−n/2−1dr,∀ξ∈ℝ.
In 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 functionsf,g∈Lk,n2ℝ, the following assertions are true:(i)
The deformed Hankel transformℱk,n satisfies(16)∫ℝfxgx¯dγk,nx=∫ℝℱk,nλℱk,ngλ¯dγk,nλ.(ii)
The deformed Hankel transformf↦ℱk,nf is an isometric isomorphism on Lk,n2ℝ, satisfying(17)∫ℝfx2dγk,nx=∫ℝℱk,nfλ2dγk,nλ.(iii)
The deformed Hankel transform is an involution unitary operator onLk,n1ℝ; that is,(18)ℱk,n−1fx=ℱk,nf−1nxx∈ℝ.Proposition 2.1.
Letf be in Lk,npℝ and p∈1,2. Then, ℱk,nf belongs to Lk,np′ℝ and(19)ℱk,nfLk,np′ℝ≤fLk,npℝ.
In the following, we present some useful results concerning the notion of generalized translation operator onLk,n2ℝ.Definition 2.2 (see [16]).
For anyx∈ℝ, the generalized translation operator f↦τxk,nf on Lk,n2ℝ is defined by(20)ℱk,nτxk,nf=Bk,n.,x¯ℱk,nf.
It is imperative to mention that relation (20) holds pointwise, provided f is a member of the generalized Weiner space Wk,nℝ.(21)Wk,nℝ≔f∈Lk,n1ℝ:ℱk,nf∈Lk,n1ℝ.
Next, we shall present some useful properties of generalized translation operator (20).Proposition 2.2 (see [14, 15]).
Ifτxk,n is the generalized translation operator on Lk,n2ℝ, then the following statements are true:(i)
For anyf∈Lk,n2ℝ, we have(22)τxk,nfLk,n2ℝ≤fLk,n2ℝ,∀x∈ℝ.(ii)
Iff∈Wk,nℝ, then(23)τxk,nfy=∫ℝBk,n−1nx,ξBk,n−1ny,ξℱk,nfξdγk,nξ,∀x,y∈ℝ.(iii)
For allx,y∈ℝ and f∈Wk,nℝ, we have(24)τxk,nfy=τyk,nfx.(iv)
For allf in Wk,nℝ and g∈Lk,n1ℝ∩Lk,n∞ℝ, we have(25)∫ℝτxk,nfygydγk,ny=∫ℝfyτ−1nxk,ngydγk,ny,∀x∈ℝ.
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.
Letx∈ℝ and let f∈Cbℝ. For k≥n−1/n, the generalized translation operator τxk,n is expressible as(26)τxk,nfy=∫ℝfzdζx,yk,nz,with(27)dζx,yk,nz=Kk,nx,y,zdγk,nz,ifxy≠0,dδxz,ify=0,dδyz,ifx=0,,where Kk,nx,y,. denotes the kernel, which is supported on the set z∈ℝ:x1/n−y1/n<z1/n<x1/n+y1/n. Moreover, operator (26) satisfies the following norm inequality:(28)τxk,nfLk,npℝ≤4fLk,npℝ,∀x∈ℝ.Apart from the integral representation of the generalized translation operator given in Theorem2.2, 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]).
Supposef∈Cbℝ is such that f=fe+fo, where fe is an even function and fo is an odd function. Then, the generalized translation operator τxk,n can be expressed as(29)τxk,nfy=Mk,n2n∫0πfe《x,y》ϕ,n1+−1nn!sgnxy2kn−nnCnnk−n/2cosϕsinϕ2nk−ndϕ+∫0πfo《x,y》ϕ,nn!sgnx2kn−nnCnnk−n/2x1/n−y1/ncosϕ《x,y》ϕ,n1/n+n!sgny2kn−nnCnnk−n/2y1/n−x1/ncosϕ《x,y》ϕ,n1/nsinϕ2nk−ndϕ,where Cnnk−n/2 are the degenerate polynomials and(30)《x,y》ϕ,n≔x2/n+y2/n−2xy1/ncosϕn/2.Corollary 2.1 (see [14, 15]).
For allf in Cb,eℝ, we have(31)τxk,nfy=Mk,n2n∫0πF《x,y》ϕ,n1+−1nn!sgnxy2kn−nnCnnk−n/2cosϕsinϕ2nk−ndϕ.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 thatf is nonnegative, is even, and belongs to generalized Wigner space Wk,nℝ. Then, for any x∈ℝ, we have(i)
τxk,nf≥0.(ii)
τxk,nf∈Lk,n1ℝ and(32)∫ℝτxk,nfydγk,ny=∫ℝfydγk,ny.Theorem 2.4.
LetLk,n,epℝ be the space of even functions in Lk,npℝ. Then, we have(i)
τxk,nf≥0,τxk,nf∈Lk,n1ℝ and(33)∫ℝτxk,nfydγk,ny=∫ℝfydγk,ny,∀x∈ℝ.(ii)
The generalized translation operatorτxk,n defined on Lk,n,e1ℝ∩Lk,n∞ℝ can be extended to Lk,n,epℝ, 1⩽p⩽∞. Moreover, for any f∈Lk,n,epℝ, we have(34)τxk,nfLk,npℝ≤fLk,npℝ,∀x∈ℝ.(iii)
For everyf∈Lk,n1ℝ, we have(35)∫ℝτxk,nfydγk,ny=∫ℝfydγk,ny.


### 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 1⩽p⩽∞, we denote Lμk,npℝ2 as the space of measurable functions f on ℝ2 satisfying(36)fLμk,npℝ2≔∫ℝ2fx,ypdμk,nx,y1/p<∞,1≤p<∞,esssupx,y∈ℝ2fx,y<∞,p=∞,where dμk,nx,y≔dγk,nxdγk,ny.Definition 2.3.
Forν∈ℝ, the generalized modulation acting on h∈Lk,n,e2ℝ is given by(37)hν≔ℱk,nτνk,nh2,where τνk,n is the usual generalized translation operator.
Remark: by virtue of positivity of the generalized translation operator as mentioned in Theorem2.4, we infer that (37) is well defined. Moreover, by virtue of (17) and (32), we have(38)hνLk,n2ℝ=hLk,n2ℝ,∀h∈Lk,n,e2ℝ.
Based on the generalized translation and modulation operations defined in (20) and (37), respectively, we consider the family of functions hν,y, ν,y∈ℝ as(39)hν,yx=τ−1nyk,nhνx,x∈ℝ.
Then, it can be easily verified that(40)hy,νLk,n2ℝ≤hLk,n2ℝ,∀y,ν∈ℝ.
With the aid of (39), we are in a position to present the formal definition of the deformed Gabor Fourier transform.Definition 2.4.
Givenf∈Lk,n2ℝ, the deformed Gabor transform with respect to the window h∈Lk,n,e2ℝ is denoted by Ghk,nf and is defined as(41)Ghk,nfy,ν≔∫ℝfxhν,yx¯dγk,nx,where hν,yx is given by (39).
For anyλ>0 and y,ν∈ℝ2, it can be easily verified that(42)Gh1/λk,nfλy,ν=Ghk,nfyλ,λν,with(43)gλx≔1λ2k−1n+2/ngxλ,∀x∈ℝ.
In the next theorem, some basic properties of deformed Gabor Fourier transform (41) are assembled.Theorem 2.5 (see [4]).
For anyf∈Lk,n2ℝ and h∈Lk,n,e2ℝ, the following statements are true:(i)
The deformed Gabor Fourier transformGhk,nf is a bounded operator and(44)Ghk,nfLμk,n∞ℝ2≤fLk,n2ℝhLk,n2ℝ.(ii)
The deformed Gabor transform (41) satisfies the following energy preserving relation:(45)Ghk,nfLμk,n2ℝ2=hLk,n2ℝfLk,n2ℝ.(iii)
For any given pair of functionsf,g∈Lk,n2ℝ, the following orthogonality relation holds:(46)∫ℝ2Ghk,nfy,νGhk,ngy,ν¯dμk,ny,ν=hLk,n2ℝ2∫ℝfxgx¯dγk,nx.(iv)
Forp∈2,∞, we have(47)Ghk,nfLμk,npℝ2≤fLk,n2ℝhLk,n2ℝ.
The following lemma follows by a straightforward calculation.Lemma 2.1.
Leth∈Lk,n,e2ℝ∩Lk,n∞ℝ; then, for any f∈Lk,n2ℝ, we have(48)ℱk,nGhk,nf.,νξ=ℱk,nfξτνk,nh2−1nξ.Theorem 2.6.
Suppose that a functionh∈Lk,n,e2ℝ∩Lk,n∞ℝ satisfies hLk,n2ℝ=1. Then, the function(49)fjx=∫−jj∫ℝGhk,nfy,νhy,νxdμk,nν,y,belongs to Lk,n2ℝ and satisfies(50)limj⟶∞f−fjLk,n2ℝ=0,wheneverf∈Lk,n2ℝ.


## 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)
Cbℝ the space of bounded continuous functions on ℝ.(2)
The spaceCb,eℝ of even bounded continuous functions on ℝ.(3)
Forp∈1,∞, the conjugate exponent shall be denoted by p′.(4)
Mk,n≔nn2k−1/2/2n2k−1+2/2Γn2k−1+2/2, dγk,nx≔Mk,nx2k−2n+2/ndx, k≥n−1/n, and n∈ℕ.(5)
Lk,npℝ, 1≤p≤∞, denotes the space of measurable functions f on ℝ satisfying(5)fLk,npℝ≔∫ℝfxpdγk,nx1/p<∞,if1≤p<∞,fLk,n∞ℝ≔esssupx∈ℝfx<∞.In casep=2, the inner product on the space Lk,npℝ is given by(6)f,gLk,n2ℝ≔∫ℝfxgx¯dγk,nx.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.
Fork≥n−1/n, and f∈Lk,n1ℝ, the integral operator(7)ℱk,nfλ=∫ℝfxBk,nλ,xdγk,nx,for allλ∈ℝ,is termed as the deformed Hankel transform, where Bk,nλ,x denotes the kernel(8)Bk,nλ,x=Jnk−n/2nλx1/n+−inn2nΓnk−n/2+1Γnk+n/2+1λxJnk+n/2nλx1/n,and Jαu are the normalized Bessel function of index α:(9)Jαu≔Γα+1u2−αJαu=Γα+1∑m=0∞−1mm!Γα+m+1u22m.
Deformed Hankel kernel (8) satisfies the following properties:(i)
Forz,t∈ℝ, we have(10)Bk,nz,t=Bk,nt,z,Bk,nz,0=1,Bk,nz,t¯=Bk,n−1nz,t.Moreover, for allλ∈ℝ, we have Bk,nλz,t=Bk,nz,λt.(ii)
There exists a finite positive constantC depending on n and k such that(11)Bk,nx,y≤C,∀x,y∈ℝ.
In what follows, we shall replaceBk,n by the rescaled version Bk,n/C but continue to use the same symbol Bk,n. As such,(12)Bk,nx,y≤1,∀x,y∈ℝ.
Remarks are as follows:(i)
We note that the authors in [9] conjectured (12) when k≥n−1/n.(ii)
The deformed Hankel transform is bounded on the spaceLk,n1ℝ and(13)ℱk,nfLk,n∞ℝ≤fLk,n1ℝ,∀f∈Lk,n1ℝ.(iii)
The deformed Hankel transformℱk,n provides a natural generalization of the classical Hankel transform. Indeed, if we set(14)Bk,nevenx,y=12Bk,nx,y+Bk,nx,−y=jnk−n/2nxy1/n,(iv)
then the deformed Hankel transformℱk,n of an even function f on the real line yields a Hankel-type transform on ℝ+. In fact, in case fx=Fx∈Lk,n1ℝ and is also even over ℝ, then(15)ℱk,nfξ=n/22nk−n/2Γ2nk+2−n/2∫0∞Frj2nk−n/2nrξ1/nr2/n2nk+2−n/2−1dr,∀ξ∈ℝ.
In 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 functionsf,g∈Lk,n2ℝ, the following assertions are true:(i)
The deformed Hankel transformℱk,n satisfies(16)∫ℝfxgx¯dγk,nx=∫ℝℱk,nλℱk,ngλ¯dγk,nλ.(ii)
The deformed Hankel transformf↦ℱk,nf is an isometric isomorphism on Lk,n2ℝ, satisfying(17)∫ℝfx2dγk,nx=∫ℝℱk,nfλ2dγk,nλ.(iii)
The deformed Hankel transform is an involution unitary operator onLk,n1ℝ; that is,(18)ℱk,n−1fx=ℱk,nf−1nxx∈ℝ.Proposition 2.1.
Letf be in Lk,npℝ and p∈1,2. Then, ℱk,nf belongs to Lk,np′ℝ and(19)ℱk,nfLk,np′ℝ≤fLk,npℝ.
In the following, we present some useful results concerning the notion of generalized translation operator onLk,n2ℝ.Definition 2.2 (see [16]).
For anyx∈ℝ, the generalized translation operator f↦τxk,nf on Lk,n2ℝ is defined by(20)ℱk,nτxk,nf=Bk,n.,x¯ℱk,nf.
It is imperative to mention that relation (20) holds pointwise, provided f is a member of the generalized Weiner space Wk,nℝ.(21)Wk,nℝ≔f∈Lk,n1ℝ:ℱk,nf∈Lk,n1ℝ.
Next, we shall present some useful properties of generalized translation operator (20).Proposition 2.2 (see [14, 15]).
Ifτxk,n is the generalized translation operator on Lk,n2ℝ, then the following statements are true:(i)
For anyf∈Lk,n2ℝ, we have(22)τxk,nfLk,n2ℝ≤fLk,n2ℝ,∀x∈ℝ.(ii)
Iff∈Wk,nℝ, then(23)τxk,nfy=∫ℝBk,n−1nx,ξBk,n−1ny,ξℱk,nfξdγk,nξ,∀x,y∈ℝ.(iii)
For allx,y∈ℝ and f∈Wk,nℝ, we have(24)τxk,nfy=τyk,nfx.(iv)
For allf in Wk,nℝ and g∈Lk,n1ℝ∩Lk,n∞ℝ, we have(25)∫ℝτxk,nfygydγk,ny=∫ℝfyτ−1nxk,ngydγk,ny,∀x∈ℝ.
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.
Letx∈ℝ and let f∈Cbℝ. For k≥n−1/n, the generalized translation operator τxk,n is expressible as(26)τxk,nfy=∫ℝfzdζx,yk,nz,with(27)dζx,yk,nz=Kk,nx,y,zdγk,nz,ifxy≠0,dδxz,ify=0,dδyz,ifx=0,,where Kk,nx,y,. denotes the kernel, which is supported on the set z∈ℝ:x1/n−y1/n<z1/n<x1/n+y1/n. Moreover, operator (26) satisfies the following norm inequality:(28)τxk,nfLk,npℝ≤4fLk,npℝ,∀x∈ℝ.Apart from the integral representation of the generalized translation operator given in Theorem2.2, 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]).
Supposef∈Cbℝ is such that f=fe+fo, where fe is an even function and fo is an odd function. Then, the generalized translation operator τxk,n can be expressed as(29)τxk,nfy=Mk,n2n∫0πfe《x,y》ϕ,n1+−1nn!sgnxy2kn−nnCnnk−n/2cosϕsinϕ2nk−ndϕ+∫0πfo《x,y》ϕ,nn!sgnx2kn−nnCnnk−n/2x1/n−y1/ncosϕ《x,y》ϕ,n1/n+n!sgny2kn−nnCnnk−n/2y1/n−x1/ncosϕ《x,y》ϕ,n1/nsinϕ2nk−ndϕ,where Cnnk−n/2 are the degenerate polynomials and(30)《x,y》ϕ,n≔x2/n+y2/n−2xy1/ncosϕn/2.Corollary 2.1 (see [14, 15]).
For allf in Cb,eℝ, we have(31)τxk,nfy=Mk,n2n∫0πF《x,y》ϕ,n1+−1nn!sgnxy2kn−nnCnnk−n/2cosϕsinϕ2nk−ndϕ.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 thatf is nonnegative, is even, and belongs to generalized Wigner space Wk,nℝ. Then, for any x∈ℝ, we have(i)
τxk,nf≥0.(ii)
τxk,nf∈Lk,n1ℝ and(32)∫ℝτxk,nfydγk,ny=∫ℝfydγk,ny.Theorem 2.4.
LetLk,n,epℝ be the space of even functions in Lk,npℝ. Then, we have(i)
τxk,nf≥0,τxk,nf∈Lk,n1ℝ and(33)∫ℝτxk,nfydγk,ny=∫ℝfydγk,ny,∀x∈ℝ.(ii)
The generalized translation operatorτxk,n defined on Lk,n,e1ℝ∩Lk,n∞ℝ can be extended to Lk,n,epℝ, 1⩽p⩽∞. Moreover, for any f∈Lk,n,epℝ, we have(34)τxk,nfLk,npℝ≤fLk,npℝ,∀x∈ℝ.(iii)
For everyf∈Lk,n1ℝ, we have(35)∫ℝτxk,nfydγk,ny=∫ℝfydγk,ny.


## 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 1⩽p⩽∞, we denote Lμk,npℝ2 as the space of measurable functions f on ℝ2 satisfying(36)fLμk,npℝ2≔∫ℝ2fx,ypdμk,nx,y1/p<∞,1≤p<∞,esssupx,y∈ℝ2fx,y<∞,p=∞,where dμk,nx,y≔dγk,nxdγk,ny.Definition 2.3.
Forν∈ℝ, the generalized modulation acting on h∈Lk,n,e2ℝ is given by(37)hν≔ℱk,nτνk,nh2,where τνk,n is the usual generalized translation operator.
Remark: by virtue of positivity of the generalized translation operator as mentioned in Theorem2.4, we infer that (37) is well defined. Moreover, by virtue of (17) and (32), we have(38)hνLk,n2ℝ=hLk,n2ℝ,∀h∈Lk,n,e2ℝ.
Based on the generalized translation and modulation operations defined in (20) and (37), respectively, we consider the family of functions hν,y, ν,y∈ℝ as(39)hν,yx=τ−1nyk,nhνx,x∈ℝ.
Then, it can be easily verified that(40)hy,νLk,n2ℝ≤hLk,n2ℝ,∀y,ν∈ℝ.
With the aid of (39), we are in a position to present the formal definition of the deformed Gabor Fourier transform.Definition 2.4.
Givenf∈Lk,n2ℝ, the deformed Gabor transform with respect to the window h∈Lk,n,e2ℝ is denoted by Ghk,nf and is defined as(41)Ghk,nfy,ν≔∫ℝfxhν,yx¯dγk,nx,where hν,yx is given by (39).
For anyλ>0 and y,ν∈ℝ2, it can be easily verified that(42)Gh1/λk,nfλy,ν=Ghk,nfyλ,λν,with(43)gλx≔1λ2k−1n+2/ngxλ,∀x∈ℝ.
In the next theorem, some basic properties of deformed Gabor Fourier transform (41) are assembled.Theorem 2.5 (see [4]).
For anyf∈Lk,n2ℝ and h∈Lk,n,e2ℝ, the following statements are true:(i)
The deformed Gabor Fourier transformGhk,nf is a bounded operator and(44)Ghk,nfLμk,n∞ℝ2≤fLk,n2ℝhLk,n2ℝ.(ii)
The deformed Gabor transform (41) satisfies the following energy preserving relation:(45)Ghk,nfLμk,n2ℝ2=hLk,n2ℝfLk,n2ℝ.(iii)
For any given pair of functionsf,g∈Lk,n2ℝ, the following orthogonality relation holds:(46)∫ℝ2Ghk,nfy,νGhk,ngy,ν¯dμk,ny,ν=hLk,n2ℝ2∫ℝfxgx¯dγk,nx.(iv)
Forp∈2,∞, we have(47)Ghk,nfLμk,npℝ2≤fLk,n2ℝhLk,n2ℝ.
The following lemma follows by a straightforward calculation.Lemma 2.1.
Leth∈Lk,n,e2ℝ∩Lk,n∞ℝ; then, for any f∈Lk,n2ℝ, we have(48)ℱk,nGhk,nf.,νξ=ℱk,nfξτνk,nh2−1nξ.Theorem 2.6.
Suppose that a functionh∈Lk,n,e2ℝ∩Lk,n∞ℝ satisfies hLk,n2ℝ=1. Then, the function(49)fjx=∫−jj∫ℝGhk,nfy,νhy,νxdμk,nν,y,belongs to Lk,n2ℝ and satisfies(50)limj⟶∞f−fjLk,n2ℝ=0,wheneverf∈Lk,n2ℝ.


## 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 Ghk,n by choosing the window function h as a nontrivial even function in the space Lk,n2ℝ.


### 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 ℱk,n.Proposition 3.1 (see [12, 10]).
Fors,t>0, there exists a positive constant Ck,ns,t, such that for every f in Lk,n2ℝ, we have(51)ξsℱk,nfξLk,n2ℝt/s+txtfxLk,n2ℝs/s+t≥Ck,ns,tfLk,n2ℝ,where Ck,ns,t=2k−1n+2/2nnst/s+t,s,t≥1/n.Theorem 3.1.
LetGhk,nf be the deformed Gabor transform of any f∈Lk,n2ℝ. Then, for s,t>0, we have(52)∫ℝ2y2tGhk,nfy,ν2dμk,ny,νs/s+t∫ℝξ2sℱk,nfξ2dγk,nξt/s+t≥Ck,ns,t2hLk,n2ℝ2s/s+tfLk,n2ℝ2,where Ck,ns,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 ν, inequality (51) yields(53)∫ℝξ2sℱk,nGhk,nf.,νξ2dγk,nξt/s+t∫ℝy2tGhk,nfy,ν2dγk,nys/s+t≥Ck,ns,t2∫ℝGhk,nfy,ν2dγk,ny.
Integrating under the measuredγk,nν followed by the implication of Cauchy–Schwarz’s inequality yields(54)∫ℝ2ξ2sℱk,nGhk,nf.,νξ2dμk,nξ,νt/s+t∫ℝ2y2tGhk,nfy,ν2dμk,ny,νs/s+t≥Ck,ns,t2∫ℝ2Ghk,nfy,ν2dμk,ny,ν.
Using the fact that(55)∫ℝ2ξ2sℱk,nGhk,nf.,νξ2dμk,nξ,ν=hLk,n2ℝ2∫ℝξ2sℱk,nfξ2dγk,nξ,we deduce that(56)hLk,n2ℝ2t/s+t∫ℝξ2sℱk,nfξ2dμk,nξt/t+s∫ℝy2tGhk,nfy,ν2dμk,ny,νs/s+t≥Ck,ns,t2∫ℝ2Ghfy,ν2dμk,ny,ν=Ck,ns,t2hLk,n2ℝ2fLk,n2ℝ2.
Hence, the proof of Theorem3.1 is complete.Proposition 3.2 (Nash’s inequality forGhk,n).
Givens>0, we can always find a constant Ck,n,s>0 such that(57)hLk,n2ℝfLk,n2ℝ≤Ck,n,sy,νsGhk,nfLμk,n2ℝ2,for all f∈Lk,n2ℝ.Proof.
Clearly inequality (57) is true in case f=0. For a given R>0, we consider 0≠f∈Lk,n2ℝ; then, it follows from Plancherel’s formula (45) that(58)hLk,n2ℝ2fLk,n2ℝ2=Ghk,nfLk,n2ℝ2=1B20,RGhk,nfLμk,n2ℝ22+1−1B20,RGhk,nfLμk,n2ℝ22,where(59)B20,R≔y,ν∈ℝ2:y,ν≤R.
On the other hand, (46) implies that(60)1B20,RGhk,nfLμk,n2ℝ22≤hLk,n2ℝ2fLk,n2ℝ2∫ℝ21B20,Rdμk,ny,ν≤CR22k−1n+2/nhLk,n2ℝ2fLk,n2ℝ2.
Moreover, we have(61)1−1B20,RGhk,nfLμk,n2ℝ22≤R−2s1−1B20,Ry,νsGhk,nfLμk,n2ℝ22≤R−2sy,νsGhk,nfLμk,n2ℝ22.
Therefore, it follows that(62)hLk,n2ℝ2fLk,n2ℝ2≤CR22k−1n+2/nhLk,n2ℝ2fLk,n2ℝ2+R−2sy,νsGhk,nfLμk,n2ℝ22.
After minimizing overR>0, the RHS of the above inequality implies that(63)hLk,n2ℝ2fLk,n2ℝ2≤Ck,n,shLk,n2ℝ2ns/2k−1n+2+nsfLk,n2ℝ2ns/2k−1n+2+nsy,νsGhk,nfLμk,n2ℝ222k−1n+2/2k−1n+2+ns.
The intended outcome occurs fairly from (63).


### 3.2. Heisenberg’s Uncertainty Principle viak,n-Entropy

Thek,n-entropy pertaining to a given probability density function ρ over ℝ2 is given via the integral expression [17]:(64)Ek,nρ≔−∫ℝ2logρy,νρy,νdμk,ny,ν,where(65)∫ℝ2ρy,νdμk,ny,ν=1.The main goal of the present section is to investigate upon the local characteristics of thek,n-entropy associated with deformed Gabor transform (41).Proposition 3.3.
For allf∈Lk,n2ℝ, we have(66)Ek,nGhk,nf2≥−2fLk,n2ℝ2hLk,n2ℝ2lnfLk,n2ℝhLk,n2ℝ.Proof.
Assume thatfLk,n2ℝhLk,n2ℝ=1. Then, by virtue of (44), we obtain(67)Ghk,nfy,ν≤fLk,n2ℝhLk,n2ℝ=1.
In particular,Ek,nGhk,nf2⩾0. We now relax the above assumption and consider(68)ϕ≔ffLk,n2ℝ,ψ≔hhLk,n2ℝ.
Then, it is quite evident thatϕ,ψ∈Lk,n2ℝ and ϕLk,n2ℝψLk,n2ℝ=1.
Hence,Ek,nGψk,nϕ2⩾0. Indeed, we have(69)Gψk,nϕ=1fLk,n2ℝhLk,n2ℝGhk,nf,which further implies that(70)Ek,nGψk,nϕ2=1fLk,n2ℝ2hLk,n2ℝ2Ek,nGhk,nf2+2lnfLk,n2ℝhLk,n2ℝ.
InvokingEk,nGψk,nϕ2⩾0, it follows that(71)Ek,nGhk,nf2⩾−2fLk,n2ℝ2hLk,n2ℝ2lnfLk,n2ℝhLk,n2ℝ.
Hence, the proof of Proposition3.3 is complete.
Employing thek,n-entropy of deformed Gabor transform (41), we can have another variant of Heisenberg’s principle for Ghk,n.Theorem 3.2.
For everyf∈Lk,n2ℝ and p,q>0, we have(72)∫ℝ2ypGhk,nfy,ν2dμk,ny,νq/p+q∫ℝ2νqGhk,nfy,ν2dμk,ny,νp/p+q≥Cp,qk,nfLk,n2ℝ2hLk,n2ℝ2,where(73)Cp,qk,n=2k−1n+2/npq/p+qqp/p+qexppq2k−1n+2/np+qlnpq4Mk,n2Γ2k−1n+2/npΓ2k−1n+2/nq−1.Proof.
For everyt,p,q∈ℝ+, we consider(74)ηt,p,qk,ny,ν≔pq4Mk,n2Γ2k−1n+2/nqΓ2k−1n+2/nqexp−yp+νq/tt2k−1n+2/np+q/pq.
After doing some elementary computations, we observe that(75)∫ℝ2ηt,p,qk,ny,νdμk,ny,ν=1,where dσt,p,qk,ny,ν≔ηt,p,qk,ny,νdμk,ny,ν is the probability measure on ℝ2. Since φt=tlogt is convex on 0,∞, Jensen’s inequality implies that(76)∫ℝ2Ghk,nfy,νηt,p,qk,ny,ν2lnGhk,nfy,ν2ηt,p,qk,ny,νdσt,p,qk,ny,ν≥0,which further implies that(77)Ek,nGhk,nf2+lnpq4Mk,n2Γ2k−1n+2/npΓ2k−1n+2/npfLk,n2ℝ2hLk,n2ℝ2≤lnt2k−1n+2/np+q/pqfLk,n2ℝ2hLk,n2ℝ2+1t∫ℝ2yp+νqGhk,nfy,ν2dμk,ny,ν.
Assume thatfLk,n2ℝhLk,n2ℝ=1. Then, Proposition 3.3 implies that(78)∫ℝ2yp+vqGhk,nfy,ν2dμk,ny,ν≥tlnpq4Mk,n2Γ2k−1n+2/npΓ2k−1n+2/np−lnt2k−1n+2/np+q/pq.
However, the RHS of inequality (78) attains its upper bound at(79)t0=exppq2k−1n+2/np+qlnpq4Mk,n2Γ2k−1n+2/np+qΓ2k−1n+2/nqp+q−1,and consequently,(80)∫ℝ2yp+vqGhk,nfy,ν2dμk,ny,ν≥Cp,q,where(81)Cp,q=2k−1n+2/np+qpq×exppq2k−1n+2/np+qlnpq4Mk,n2Γ2k−1n+2/nΓ2k−1n+2/nq−1.
Therefore, for everyf∈Lk,n2ℝ and h∈Lk,n,e2ℝ satisfying fLk,n2ℝhLk,n2ℝ=1, we have(82)∫ℝ2ypGhk,nfy,ν2dμk,ny,ν+∫ℝ2vqGhk,nfy,ν2dμk,ny,ν≥Cp,q.
It is evident that forλ>0, the dilates fλ and h1/λ belong to Lk,n2ℝ. Therefore, after substituting f with fλ and h with h1/λ and noting fλLk,n2ℝh1/λLk,n2ℝ=fLk,n2ℝhLk,n2ℝ=1, the above inequality yields(83)∫ℝ2ypGh1λk,nfλy,ν2dμk,ny,ν+∫ℝ2νqGh1/λk,nfλy,ν2dμk,ny,ν≥Cp,q.
Using (42), we obtain(84)λ−p∫ℝ2ypGhk,nfy,ν2dμk,ny,ν+λq∫ℝ2νqGhk,nfy,ν2dμk,ny,ν≥Cp,q.
In particular, the inequality holds at the point(85)λ=p∫ℝ2ypGhk,nfy,ν2dμk,ny,νq∫ℝ2νqGhk,nfy,ν2dμk,ny,ν1/p+q,so that(86)∫ℝ2ypGhk,nfy,ν2dμk,ny,νq/p+q∫ℝ2νqGhk,nfy,ν2dμk,ny,νp/p+q≥Cp,qk,n,where(87)Cp,qk,n=Cp,qpp/p+qqq/p+qp+q=2k−1n+2/npq/p+qqp/p+qexppq2k−1n+2/np+qlnpq4Mk,n2Γ2k−1n+2/npΓ2k−1n+2/np−1.
Hence, the desired result is obtained after replacingf with f/fLk,n2ℝ and h with h/hLk,n2ℝ. □
Remark: forp=q=2, we have(88)yGhk,nfLμk,n2ℝ2νGhk,nfLμk,n2ℝ2≥2k−1n+2/n2eMk,nΓ2k−1n+2/2n2/2k−1n+2/nfLk,n2ℝ2hLk,n2ℝ2.


### 3.3.Lp-Heisenberg’s Uncertainty Principle

In this section, we shall establish a unified form ofLp-Heisenberg’s inequality for deformed Gabor transform (41). Our strategy of the proof is motivated by [18], wherein the authors have studied the L2-Heisenberg’s uncertainty inequality in the context of Lie groups. To facilitate the narrative, we set the following notation:(89)Γλy,ν≔e−λy,ν2,y,ν∈ℝ2,λ>0.It is quite straightforward to verify that for every1≤q<∞, there exist C>0 with(90)ΓλLμk,nqℝ2=Cλ−2k−1n+2/nq.Lemma 3.1.
LetGhk,nf be the deformed Gabor transform of any f∈Lk,n2ℝ and 1<p≤2,0<s<2k−1n+2/2np′. Then, there exists a positive constant C such that(91)ΓλGhk,nfLμk,np′ℝ2≤ChLk,n2ℝλ−2sysfLk,n2ℝ+ysfLk,n2pℝ,λ>0.Proof.
Trivially, inequality (91) holds for ysfLk,n2ℝ+ysfLk,n2pℝ=∞.
Assume thatysfLk,n2ℝ+ysfLk,n2pℝ<∞. Moreover, for r>0, we consider fr=1−r,rf and fr=f−fr. Then, we have(92)fry≤r−sysfy.
Subsequently, by virtue of (47), we obtain(93)ΓλGhk,n1ℝ/−r,rfLμk,np′ℝ2≤ΓλLμk,n∞ℝ2Ghk,n1ℝ/−r,rfLμk,np′ℝ2≤hLk,n2ℝ1ℝ/−r,rfLk,n2ℝ≤r−shLk,n2ℝysfLk,n2ℝ.
On the flip side, relation (44) and Hölder’s inequality implies that(94)ΓλGhk,n1−r,rfLμk,np′ℝ2≤ΓλLμk,np′ℝ2Ghk,n1−r,rfLk,n∞ℝ≤hLk,n2ℝΓλLμk,np′ℝ21−r,rfLk,n2ℝ≤hLk,n2ℝΓλLμk,np′ℝ2y−s1−r,rLk,n2p′ℝysfLk,n2pℝ.
It is quite straightforward to demonstrate that there exists a positive constantC such that(95)y−s1−r,rLk,n2p′ℝ=Cr−s+2k−1n+2/2np′.
Consequently, we obtain(96)ΓλGhk,nfLμk,np′ℝ2≤ΓλGhk,nfsLμk,np′ℝ2+ΓλGhk,nfsLμk,np′ℝ2≤Cr−shLk,n2ℝysfLk,n2ℝ+r2k−1/n+2/2np′ΓλLμk,np′ℝ2ysfLk,n2pℝ.
Choosingr=λ2 and using (90), we obtain the desired inequality.Theorem 3.3.
LetGhk,nf be the deformed Gabor transform of any arbitrary function f∈Lk,n2ℝ. Then, for 1<p⩽2,0<s<2k−1n+2/2np′ and t>0, the following inequality holds:(97)Ghk,nfLμk,np′ℝ2≤ChLk,n2ℝt/s+tysfLk,n2ℝ+ysfLk,n2pℝt/s+ty,ν4tGhk,nfLμk,np′ℝ2s/s+t.Proof.
Trivially, inequality (97) holds, whenever Ghk,nf=0. Assume that Ghk,nf≠0. Using Lemma 3.1, it follows that for t≤1/2 and for all λ>0, we have(98)Ghk,nfLμk,np′ℝ2≤ΓλGhk,nfLμk,np′ℝ2+1−ΓλGhk,nfLμk,np′ℝ2≤ChLk,n2ℝλ−2sysfLk,n2ℝ+ysfLk,n2pℝ+1−ΓλGhk,nfLμk,np′ℝ2.
However, on the flip side, we have(99)1−ΓλGhk,nfLμk,np′ℝ2=λ2tλy,ν2−2t1−Γλy,ν4tGhk,nfLμk,np′ℝ2.
Therefore, we have(100)Ghk,nfLμk,np′ℝ2≤ChLk,n2ℝλ−2sysfLk,n2ℝ+ysfLk,n2pℝ+Cλ2ty,ν4tGhk,nfLμk,np′ℝ2,as 1−e−uu−2t is bounded for u≥0 and t≤1/2. By optimizing the above inequality over λ, we can obtain inequality (97) for 0<s<2k−1n+2/2np′ and t≤1/2.
Next, we shall consider the case, whent≤1/2. For u≥0 and t′≤1/2<t, it is easy to see that u4t′≤1+u4t, which is for u=y,ν/ε becomes(101)y,νε4t′<1+y,νε4t,for allε>0.
Therefore, we have(102)y,ν4t′Ghk,nfLμk,np′ℝ2≤ε4t′Ghk,nfLμk,np′ℝ2+ε4t′−ty,ν4tGhk,nfLμk,np′ℝ2.
Upon optimizing overε, we shall obtain a positive constant C such that(103)y,ν4t′Ghk,nfLμk,np′ℝ2≤CGhk,nfLμk,np′ℝ2t−t′/ty,ν4tGhk,nfLμk,np′ℝ2t′/t.
Combining (97) and (103), we get the desired inequality.Corollary 3.1.
For0<s<2k−1n+2/4n and t>0, we can always find some C>0 satisfying(104)fLk,n2ℝ⩽ChLk,n2ℝ−s/s+tysfLk,n2ℝ+ysfLk,n4ℝt/s+ty,ν4tGhk,nfLμk,n2ℝ2s/s+t,for all f∈Lk,n2ℝ.Proof.
The result follows immediately by applying Theorem3.3 with p=2 and Plancherel’s formula (45).


## 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 ℱk,n.Proposition 3.1 (see [12, 10]).
Fors,t>0, there exists a positive constant Ck,ns,t, such that for every f in Lk,n2ℝ, we have(51)ξsℱk,nfξLk,n2ℝt/s+txtfxLk,n2ℝs/s+t≥Ck,ns,tfLk,n2ℝ,where Ck,ns,t=2k−1n+2/2nnst/s+t,s,t≥1/n.Theorem 3.1.
LetGhk,nf be the deformed Gabor transform of any f∈Lk,n2ℝ. Then, for s,t>0, we have(52)∫ℝ2y2tGhk,nfy,ν2dμk,ny,νs/s+t∫ℝξ2sℱk,nfξ2dγk,nξt/s+t≥Ck,ns,t2hLk,n2ℝ2s/s+tfLk,n2ℝ2,where Ck,ns,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 ν, inequality (51) yields(53)∫ℝξ2sℱk,nGhk,nf.,νξ2dγk,nξt/s+t∫ℝy2tGhk,nfy,ν2dγk,nys/s+t≥Ck,ns,t2∫ℝGhk,nfy,ν2dγk,ny.
Integrating under the measuredγk,nν followed by the implication of Cauchy–Schwarz’s inequality yields(54)∫ℝ2ξ2sℱk,nGhk,nf.,νξ2dμk,nξ,νt/s+t∫ℝ2y2tGhk,nfy,ν2dμk,ny,νs/s+t≥Ck,ns,t2∫ℝ2Ghk,nfy,ν2dμk,ny,ν.
Using the fact that(55)∫ℝ2ξ2sℱk,nGhk,nf.,νξ2dμk,nξ,ν=hLk,n2ℝ2∫ℝξ2sℱk,nfξ2dγk,nξ,we deduce that(56)hLk,n2ℝ2t/s+t∫ℝξ2sℱk,nfξ2dμk,nξt/t+s∫ℝy2tGhk,nfy,ν2dμk,ny,νs/s+t≥Ck,ns,t2∫ℝ2Ghfy,ν2dμk,ny,ν=Ck,ns,t2hLk,n2ℝ2fLk,n2ℝ2.
Hence, the proof of Theorem3.1 is complete.Proposition 3.2 (Nash’s inequality forGhk,n).
Givens>0, we can always find a constant Ck,n,s>0 such that(57)hLk,n2ℝfLk,n2ℝ≤Ck,n,sy,νsGhk,nfLμk,n2ℝ2,for all f∈Lk,n2ℝ.Proof.
Clearly inequality (57) is true in case f=0. For a given R>0, we consider 0≠f∈Lk,n2ℝ; then, it follows from Plancherel’s formula (45) that(58)hLk,n2ℝ2fLk,n2ℝ2=Ghk,nfLk,n2ℝ2=1B20,RGhk,nfLμk,n2ℝ22+1−1B20,RGhk,nfLμk,n2ℝ22,where(59)B20,R≔y,ν∈ℝ2:y,ν≤R.
On the other hand, (46) implies that(60)1B20,RGhk,nfLμk,n2ℝ22≤hLk,n2ℝ2fLk,n2ℝ2∫ℝ21B20,Rdμk,ny,ν≤CR22k−1n+2/nhLk,n2ℝ2fLk,n2ℝ2.
Moreover, we have(61)1−1B20,RGhk,nfLμk,n2ℝ22≤R−2s1−1B20,Ry,νsGhk,nfLμk,n2ℝ22≤R−2sy,νsGhk,nfLμk,n2ℝ22.
Therefore, it follows that(62)hLk,n2ℝ2fLk,n2ℝ2≤CR22k−1n+2/nhLk,n2ℝ2fLk,n2ℝ2+R−2sy,νsGhk,nfLμk,n2ℝ22.
After minimizing overR>0, the RHS of the above inequality implies that(63)hLk,n2ℝ2fLk,n2ℝ2≤Ck,n,shLk,n2ℝ2ns/2k−1n+2+nsfLk,n2ℝ2ns/2k−1n+2+nsy,νsGhk,nfLμk,n2ℝ222k−1n+2/2k−1n+2+ns.
The intended outcome occurs fairly from (63).


## 3.2. Heisenberg’s Uncertainty Principle viak,n-Entropy

Thek,n-entropy pertaining to a given probability density function ρ over ℝ2 is given via the integral expression [17]:(64)Ek,nρ≔−∫ℝ2logρy,νρy,νdμk,ny,ν,where(65)∫ℝ2ρy,νdμk,ny,ν=1.The main goal of the present section is to investigate upon the local characteristics of thek,n-entropy associated with deformed Gabor transform (41).Proposition 3.3.
For allf∈Lk,n2ℝ, we have(66)Ek,nGhk,nf2≥−2fLk,n2ℝ2hLk,n2ℝ2lnfLk,n2ℝhLk,n2ℝ.Proof.
Assume thatfLk,n2ℝhLk,n2ℝ=1. Then, by virtue of (44), we obtain(67)Ghk,nfy,ν≤fLk,n2ℝhLk,n2ℝ=1.
In particular,Ek,nGhk,nf2⩾0. We now relax the above assumption and consider(68)ϕ≔ffLk,n2ℝ,ψ≔hhLk,n2ℝ.
Then, it is quite evident thatϕ,ψ∈Lk,n2ℝ and ϕLk,n2ℝψLk,n2ℝ=1.
Hence,Ek,nGψk,nϕ2⩾0. Indeed, we have(69)Gψk,nϕ=1fLk,n2ℝhLk,n2ℝGhk,nf,which further implies that(70)Ek,nGψk,nϕ2=1fLk,n2ℝ2hLk,n2ℝ2Ek,nGhk,nf2+2lnfLk,n2ℝhLk,n2ℝ.
InvokingEk,nGψk,nϕ2⩾0, it follows that(71)Ek,nGhk,nf2⩾−2fLk,n2ℝ2hLk,n2ℝ2lnfLk,n2ℝhLk,n2ℝ.
Hence, the proof of Proposition3.3 is complete.
Employing thek,n-entropy of deformed Gabor transform (41), we can have another variant of Heisenberg’s principle for Ghk,n.Theorem 3.2.
For everyf∈Lk,n2ℝ and p,q>0, we have(72)∫ℝ2ypGhk,nfy,ν2dμk,ny,νq/p+q∫ℝ2νqGhk,nfy,ν2dμk,ny,νp/p+q≥Cp,qk,nfLk,n2ℝ2hLk,n2ℝ2,where(73)Cp,qk,n=2k−1n+2/npq/p+qqp/p+qexppq2k−1n+2/np+qlnpq4Mk,n2Γ2k−1n+2/npΓ2k−1n+2/nq−1.Proof.
For everyt,p,q∈ℝ+, we consider(74)ηt,p,qk,ny,ν≔pq4Mk,n2Γ2k−1n+2/nqΓ2k−1n+2/nqexp−yp+νq/tt2k−1n+2/np+q/pq.
After doing some elementary computations, we observe that(75)∫ℝ2ηt,p,qk,ny,νdμk,ny,ν=1,where dσt,p,qk,ny,ν≔ηt,p,qk,ny,νdμk,ny,ν is the probability measure on ℝ2. Since φt=tlogt is convex on 0,∞, Jensen’s inequality implies that(76)∫ℝ2Ghk,nfy,νηt,p,qk,ny,ν2lnGhk,nfy,ν2ηt,p,qk,ny,νdσt,p,qk,ny,ν≥0,which further implies that(77)Ek,nGhk,nf2+lnpq4Mk,n2Γ2k−1n+2/npΓ2k−1n+2/npfLk,n2ℝ2hLk,n2ℝ2≤lnt2k−1n+2/np+q/pqfLk,n2ℝ2hLk,n2ℝ2+1t∫ℝ2yp+νqGhk,nfy,ν2dμk,ny,ν.
Assume thatfLk,n2ℝhLk,n2ℝ=1. Then, Proposition 3.3 implies that(78)∫ℝ2yp+vqGhk,nfy,ν2dμk,ny,ν≥tlnpq4Mk,n2Γ2k−1n+2/npΓ2k−1n+2/np−lnt2k−1n+2/np+q/pq.
However, the RHS of inequality (78) attains its upper bound at(79)t0=exppq2k−1n+2/np+qlnpq4Mk,n2Γ2k−1n+2/np+qΓ2k−1n+2/nqp+q−1,and consequently,(80)∫ℝ2yp+vqGhk,nfy,ν2dμk,ny,ν≥Cp,q,where(81)Cp,q=2k−1n+2/np+qpq×exppq2k−1n+2/np+qlnpq4Mk,n2Γ2k−1n+2/nΓ2k−1n+2/nq−1.
Therefore, for everyf∈Lk,n2ℝ and h∈Lk,n,e2ℝ satisfying fLk,n2ℝhLk,n2ℝ=1, we have(82)∫ℝ2ypGhk,nfy,ν2dμk,ny,ν+∫ℝ2vqGhk,nfy,ν2dμk,ny,ν≥Cp,q.
It is evident that forλ>0, the dilates fλ and h1/λ belong to Lk,n2ℝ. Therefore, after substituting f with fλ and h with h1/λ and noting fλLk,n2ℝh1/λLk,n2ℝ=fLk,n2ℝhLk,n2ℝ=1, the above inequality yields(83)∫ℝ2ypGh1λk,nfλy,ν2dμk,ny,ν+∫ℝ2νqGh1/λk,nfλy,ν2dμk,ny,ν≥Cp,q.
Using (42), we obtain(84)λ−p∫ℝ2ypGhk,nfy,ν2dμk,ny,ν+λq∫ℝ2νqGhk,nfy,ν2dμk,ny,ν≥Cp,q.
In particular, the inequality holds at the point(85)λ=p∫ℝ2ypGhk,nfy,ν2dμk,ny,νq∫ℝ2νqGhk,nfy,ν2dμk,ny,ν1/p+q,so that(86)∫ℝ2ypGhk,nfy,ν2dμk,ny,νq/p+q∫ℝ2νqGhk,nfy,ν2dμk,ny,νp/p+q≥Cp,qk,n,where(87)Cp,qk,n=Cp,qpp/p+qqq/p+qp+q=2k−1n+2/npq/p+qqp/p+qexppq2k−1n+2/np+qlnpq4Mk,n2Γ2k−1n+2/npΓ2k−1n+2/np−1.
Hence, the desired result is obtained after replacingf with f/fLk,n2ℝ and h with h/hLk,n2ℝ. □
Remark: forp=q=2, we have(88)yGhk,nfLμk,n2ℝ2νGhk,nfLμk,n2ℝ2≥2k−1n+2/n2eMk,nΓ2k−1n+2/2n2/2k−1n+2/nfLk,n2ℝ2hLk,n2ℝ2.


## 3.3.Lp-Heisenberg’s Uncertainty Principle

In this section, we shall establish a unified form ofLp-Heisenberg’s inequality for deformed Gabor transform (41). Our strategy of the proof is motivated by [18], wherein the authors have studied the L2-Heisenberg’s uncertainty inequality in the context of Lie groups. To facilitate the narrative, we set the following notation:(89)Γλy,ν≔e−λy,ν2,y,ν∈ℝ2,λ>0.It is quite straightforward to verify that for every1≤q<∞, there exist C>0 with(90)ΓλLμk,nqℝ2=Cλ−2k−1n+2/nq.Lemma 3.1.
LetGhk,nf be the deformed Gabor transform of any f∈Lk,n2ℝ and 1<p≤2,0<s<2k−1n+2/2np′. Then, there exists a positive constant C such that(91)ΓλGhk,nfLμk,np′ℝ2≤ChLk,n2ℝλ−2sysfLk,n2ℝ+ysfLk,n2pℝ,λ>0.Proof.
Trivially, inequality (91) holds for ysfLk,n2ℝ+ysfLk,n2pℝ=∞.
Assume thatysfLk,n2ℝ+ysfLk,n2pℝ<∞. Moreover, for r>0, we consider fr=1−r,rf and fr=f−fr. Then, we have(92)fry≤r−sysfy.
Subsequently, by virtue of (47), we obtain(93)ΓλGhk,n1ℝ/−r,rfLμk,np′ℝ2≤ΓλLμk,n∞ℝ2Ghk,n1ℝ/−r,rfLμk,np′ℝ2≤hLk,n2ℝ1ℝ/−r,rfLk,n2ℝ≤r−shLk,n2ℝysfLk,n2ℝ.
On the flip side, relation (44) and Hölder’s inequality implies that(94)ΓλGhk,n1−r,rfLμk,np′ℝ2≤ΓλLμk,np′ℝ2Ghk,n1−r,rfLk,n∞ℝ≤hLk,n2ℝΓλLμk,np′ℝ21−r,rfLk,n2ℝ≤hLk,n2ℝΓλLμk,np′ℝ2y−s1−r,rLk,n2p′ℝysfLk,n2pℝ.
It is quite straightforward to demonstrate that there exists a positive constantC such that(95)y−s1−r,rLk,n2p′ℝ=Cr−s+2k−1n+2/2np′.
Consequently, we obtain(96)ΓλGhk,nfLμk,np′ℝ2≤ΓλGhk,nfsLμk,np′ℝ2+ΓλGhk,nfsLμk,np′ℝ2≤Cr−shLk,n2ℝysfLk,n2ℝ+r2k−1/n+2/2np′ΓλLμk,np′ℝ2ysfLk,n2pℝ.
Choosingr=λ2 and using (90), we obtain the desired inequality.Theorem 3.3.
LetGhk,nf be the deformed Gabor transform of any arbitrary function f∈Lk,n2ℝ. Then, for 1<p⩽2,0<s<2k−1n+2/2np′ and t>0, the following inequality holds:(97)Ghk,nfLμk,np′ℝ2≤ChLk,n2ℝt/s+tysfLk,n2ℝ+ysfLk,n2pℝt/s+ty,ν4tGhk,nfLμk,np′ℝ2s/s+t.Proof.
Trivially, inequality (97) holds, whenever Ghk,nf=0. Assume that Ghk,nf≠0. Using Lemma 3.1, it follows that for t≤1/2 and for all λ>0, we have(98)Ghk,nfLμk,np′ℝ2≤ΓλGhk,nfLμk,np′ℝ2+1−ΓλGhk,nfLμk,np′ℝ2≤ChLk,n2ℝλ−2sysfLk,n2ℝ+ysfLk,n2pℝ+1−ΓλGhk,nfLμk,np′ℝ2.
However, on the flip side, we have(99)1−ΓλGhk,nfLμk,np′ℝ2=λ2tλy,ν2−2t1−Γλy,ν4tGhk,nfLμk,np′ℝ2.
Therefore, we have(100)Ghk,nfLμk,np′ℝ2≤ChLk,n2ℝλ−2sysfLk,n2ℝ+ysfLk,n2pℝ+Cλ2ty,ν4tGhk,nfLμk,np′ℝ2,as 1−e−uu−2t is bounded for u≥0 and t≤1/2. By optimizing the above inequality over λ, we can obtain inequality (97) for 0<s<2k−1n+2/2np′ and t≤1/2.
Next, we shall consider the case, whent≤1/2. For u≥0 and t′≤1/2<t, it is easy to see that u4t′≤1+u4t, which is for u=y,ν/ε becomes(101)y,νε4t′<1+y,νε4t,for allε>0.
Therefore, we have(102)y,ν4t′Ghk,nfLμk,np′ℝ2≤ε4t′Ghk,nfLμk,np′ℝ2+ε4t′−ty,ν4tGhk,nfLμk,np′ℝ2.
Upon optimizing overε, we shall obtain a positive constant C such that(103)y,ν4t′Ghk,nfLμk,np′ℝ2≤CGhk,nfLμk,np′ℝ2t−t′/ty,ν4tGhk,nfLμk,np′ℝ2t′/t.
Combining (97) and (103), we get the desired inequality.Corollary 3.1.
For0<s<2k−1n+2/4n and t>0, we can always find some C>0 satisfying(104)fLk,n2ℝ⩽ChLk,n2ℝ−s/s+tysfLk,n2ℝ+ysfLk,n4ℝt/s+ty,ν4tGhk,nfLμk,n2ℝ2s/s+t,for all f∈Lk,n2ℝ.Proof.
The result follows immediately by applying Theorem3.3 with p=2 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 f∈Sℝ⊆Lk,n2ℝ, they formulated that(105)∫ℝξ−2λℱk,nfξ2dγk,nξ≤Ck,nλ∫ℝx2λfx2dγk,nx,where(106)Ck,nλ≔n22nλΓ2k−1−2λn+2/4nΓ2k−1+2λn+2/4n2,0≤λ<2k−1n+22n.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.
LetGhk,nf be the deformed Gabor transform corresponding to f∈Sℝ⊆Lk,n2ℝ; then,(107)hLk,n2ℝ2∫ℝ2ξ−λℱk,nfξ2dγk,nξ≤Ck,nλ∫ℝ2yλGhk,nfy,ν2dμk,ny,ν,where Ck,nλ is given by (106) and 0≤λ<2k−1n+2/2n.Proof.
By virtue of (105), we obtain(108)∫ℝξ−2λℱk,nGhk,nf.,νξ2dγk,nξ≤Ck,nλ∫ℝy2λGhk,nfy,ν2dγk,nt,for allν∈ℝ,which upon integration under Haar measure dγk,nν implies that(109)∫ℝ2ξ−2λℱk,nGhk,nf.,νξ2dμk,nξ,ν≤Ck,nλ∫ℝ2y2λGhk,nfy,ν2dμk,ny,ν.
Implementing Lemma2.1, expression (109) takes the following form:(110)∫ℝ2ξ−2λℱk,nfξ2τνk,nh2−1nξdμk,nξ,ν≤Ck,nλ∫ℝ2y2λGhk,nfy,ν2dμk,ny,ν.
Or,(111)∫ℝξ−2λℱk,nfξ2∫ℝτνk,nh2−1nξdγk,nνdγk,nξ≤Ck,nλ∫ℝ2y2λGhk,nfy,ν2dμk,ny,ν.
Using the hypothesis onh, relation (32) becomes(112)hLk,n2ℝ2∫ℝξ−2λℱk,nfξ2dγk,nξ≤Ck,nλ∫ℝ2y2λGhk,nfy,ν2dμk,ny,ν,which establishes Pitt’s inequality for deformed Gabor transform (41).Remark.
Pluggingλ=0, 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 f∈Sℝ [9],(113)∫ℝlogyft2dγk,nt+∫ℝlogξℱk,nfξ2dγk,nξ≥nΓ′2k−1n+2/4Γ2k−1n+2/4+log2n∫ℝft2dγk,nt.
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.
Letf∈Sℝ; then, we have(114)∫ℝ2logyGhk,nfy,ν2dμk,ny,ν+hLk,n2ℝ2∫ℝlogξℱk,nfξ2dγk,nξ≥nΓ′2k−1n+2/4Γ2k−1n+2/4+log2nhLk,n2ℝ2fLk,n2ℝ2.Proof.
By replacingf in (113) with Ghk,nf.,ν, we obtain(115)∫ℝlogyGhk,nfy,ν2dγk,nt+∫ℝlogξℱk,nGhk,nf.,νξ2dγk,nξ≥nΓ′2k−1n+2/4Γ2k−1n+2/4+log2n∫ℝGhk,nfy,ν2dγk,nt,for allν∈ℝ.
Integrating (115) under dγk,nν implies that(116)∫ℝ2logyGhk,nfy,ν2dμk,ny,ν+∫ℝ∫ℝlogξℱk,nGhk,nf.,νξ2dμk,nξ,ν≥nΓ′2k−1n+2/4Γ2k−1n+2/4+log2n∫ℝ2Ghk,nfy,ν2dμk,ny,ν.
By virtue of (45), we obtain(117)∫ℝ2logyGhk,nfy,ν2dμk,ny,ν+∫ℝ2logξℱk,nGhk,nf.,νξ2dμk,nξ,ν≥nΓ′2k−1n+2/4Γ2k−1n+2/4+log2nhLk,n2ℝ2fLk,n2ℝ2.
In order to derive a useful computation for the later integral in (117), we invoke Lemma 2.1 together with (32), so that(118)∫ℝ2logξℱk,nGhk,nf.,νξ2dμk,ny,ν=∫ℝ∫ℝlogξℱk,nGhk,nf.,νξ2dγk,nξdγk,nν=∫ℝlogξℱk,nfξ2dγk,nξhLk,n2ℝ2.
Substituting (118) in (117), we obtain the result for DGT (41).
Next, we shall give another proof of Theorem4.2, which primarily relies upon (107).Proof.
(alternate proof of Theorem4.2). Given 0≤λ<2k−1n+2/2n, consider(119)Sλ=hLk,n2ℝ2∫ℝξ−2λℱk,nfξ2dγk,nξ−Ck,nλ∫ℝ2y2λGhk,nfy,ν2dμk,ny,ν.
Differentiating (119) with respect to λ, we obtain(120)S′λ=−2hLk,n2ℝ2∫ℝξ−2λlogξℱk,nfξ2dγk,nξ−2Ck,nλ∫ℝy2λlogyGhk,nfy,ν2dμk,ny,ν−Ck,n′λ∫ℝ2y2λGhk,nfy,ν2dμk,ny,ν,where(121)Ck,n′λ=−nCk,nλΓ′2k−1−2λn+2/4Γ2k−1−2λn+2/4+Γ′2k−1+2λn+2/4Γ2k−1+2λn+2/4+2log2n.
Forλ=0, relation (121) yields(122)Ck,n′0=−2nΓ′2k−1n+2/4Γ2k−1n+2/4+log2n.
Using (107), we get(123)Sλ≤0,∀λ∈0,2k−1n+2/n+2λ2,and(124)S0=hLk,n2ℝ2∫ℝℱk,nfξ2dγk,nξ−Ck,n0∫ℝ2Ghk,nfy,ν2dμk,ny,ν=hLk,n2ℝ2fLk,n2ℝ2−hLk,n2ℝ2fLk,n2ℝ2=0.Therefore, we deduce that(125)S′0+≔limλ⟶0+Sλλ≤0.
Equivalently,(126)−2hLk,n2ℝ2∫ℝlogξℱk,nfξ2dγk,nξ−2Ck,n0∫ℝ2logyGhk,nfy,ν2dμk,ny,ν−Ck,n′0∫ℝ2Ghk,nfy,ν2dμk,ny,ν≤0.
As a consequence of Plancherel’s formula (45) and (122), it follows that(127)−2hLk,n2ℝ2∫ℝlogξℱk,nfξ2dγk,nξ−2∫ℝ2logyGhk,nfy,ν2dμk,ny,ν+2nΓ′2k−1n+2/4Γ2k−1n+2/4+log2nhLk,n2ℝ2fLk,n2ℝ2≤0.
Alternatively,(128)∫ℝ2logyGhk,nfy,ν2dμk,ny,ν+hLk,n2ℝ2∫ℝlogξℱk,nfξ2dγk,nξ≥nΓ′2k−1n+2/4Γ2k−1n+2/4+log2nhLk,n2ℝ2fLk,n2ℝ2,which establishes the result.Corollary 4.1.
LetGhk,nf be the deformed Gabor transform of any arbitrary function f∈Sℝ with respect to the window function h∈Lk,n,e2ℝ∩Lk,n∞ℝ with hLk,n2ℝ=1. Then, we have(129)∫ℝ2y2Ghk,nfy,ν2dμk,ny,ν1/2∫ℝξℱk,nfξ2dγk,nξ1/2≥expnΓ′2k−1n+2/4Γ2k−1n+2/4+log2nfLk,n2ℝ2.Proof.
NotinghLk,n2ℝ=1 and then using the well-known Jensen’s inequality in (114), we get(130)log∫ℝ2y2Ghk,nfy,ν2fLk,n2ℝ2dμk,ny,ν∫ℝξ2ℱk,nfξ2fLk,n2ℝ2dγk,nξ1/2=log∫ℝ2y2Ghk,nfy,ν2fLk,n2ℝ2dμk,ny,ν1/2+log∫ℝξ2ℱk,nfξ2fLk,n2ℝ2dγk,nξ1/2≥∫ℝ2logyGhk,nfy,ν2fLk,n2ℝ2dμk,ny,ν+∫ℝlogξℱk,nfξ2fLk,n2ℝ2dγk,nξ≥nΓ′2k−1n+2/4Γ2k−1n+2/4+log2n,which upon simplification yields desired result (129).
Remarks are as follows:(i)
By virtue of the identity [19](131)Γ′zΓz=logz−12z−2∫0∞tt2+z2e2πt−1dt,we infer that(132)expnΓ′2k−1n+2/4Γ2k−1n+2/4+log2n≈2k−1n+22nn,for2k−1n+22n≫1,that is precisely the constant as appearing in Theorem3.1.(ii)
In a manner similar to (113), we get(133)∫ℝt2ft2dγk,nt1/2∫ℝξ2ℱk,nfξ2dγk,nξ1/2≥expnΓ′2k−1n+2/4Γ2k−1n+2/4+log2n∫ℝft2dγk,nt.(iii)
By virtue of (131), it follows that the constant in the RHS of (133) is(134)expnΓ′2k−1n+2/4Γ2k−1n+2/4+log2n≈2k−1n+22nn,for2k−1n+22n≫1,that 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]:(135)∫ℝft2dγk,nt≤Ck,nE1,E2∫ℝ/E1ft2dγk,nt+∫ℝ/E2ℱk,nfξ2dγk,nξ,where E1 and E2 are the subsets of ℝ with finite measure and Ck,nE1,E2 is a constant.Theorem 5.1.
LetGhk,nf be the deformed Gabor transform of any f∈Lk,n2ℝ. Then, we have(136)∫ℝ/E1∫ℝGhk,nfy,ν2dμk,ny,ν+hLk,n2ℝ2∫ℝ/E2ℱk,nfξ2dγk,nξ≥hLk,n2ℝ2fLk,n2ℝ2Ck,nE1,E2,where Ck,nE1,E2 is given by (135).Proof.
For anyν∈ℝ, it follows that Ghk,nf.,ν∈Lk,n2ℝ, provided f∈Lk,n2ℝ. Therefore, changing f to Ghk,nf.,ν in (135) implies that(137)∫ℝGhk,nfy,ν2dγk,nt≤Ck,nE1,E2∫ℝ/E1Ghk,nfy,ν2dγk,nt+∫ℝ/E2ℱk,nGhk,nf.,νξ2dγk,nξ,which after integration under dγk,nν yields the following inequality:(138)∫ℝ2Ghk,nfy,ν2dμk,ny,ν≤Ck,nE1,E2∫ℝ/E1∫ℝGhk,nfy,ν2dμk,ny,ν+∫ℝ/E2∫ℝℱk,nGhk,nf.,νξ2dμk,nξ,ν.
Invoking Lemma2.1 in association with Plancherel’s formula (45), we get(139)∫ℝ/E1∫ℝGhk,nfy,ν2dμk,ny,ν+∫ℝ/E2∫ℝℱk,nfξτνk,nh2−1nξ2dμk,nξ,ν≥hLk,n2ℝ2fLk,n2ℝ2Ck,nE1,E2,so that(140)∫ℝ/E1∫ℝGhk,nfy,ν2dμk,ny,ν+∫ℝ\E2ℱk,nfξ2∫ℝτνk,nh2−1nξdγk,nνdγk,nξ≥hLk,n2ℝ2fLk,n2ℝ2Ck,nE1,E2.
Using Lemma2.1 and relation (32) and keeping in view that h∈Lk,n,e2ℝ∩Lk,n∞ℝ, we obtain(141)∫ℝ/E1∫ℝGhk,nfy,ν2dμk,ny,ν+hLk,n2ℝ2∫ℝ/E2ℱk,nfξ2dγk,nξ≥hLk,n2ℝ2fLk,n2ℝ2Ck,nE1,E2,which is the desired Benedick–Amrein–Berthier’s uncertainty principle for deformed Gabor transform (41).
As a consequence of Theorem5.1, we obtain the following generalized Heisenberg-type uncertainty inequality for DGT (41).Corollary 5.1.
LetGhk,nf be the deformed Gabor transform of any arbitrary function f∈Lk,n2ℝ. Then, for p,q>0, there exist Ck,np,q>0 satisfying(142)∫ℝ2y2pGhk,nfy,ν2dμk,ny,νq/2∫ℝξ2qℱk,nfξ2dγk,nξp/2≥Ck,np,qhLk,n2ℝqfLk,n2ℝp+q.Proof.
TakeE1=E2=−1,1. Then, for any f∈Lk,n2ℝ and p,q>0, (136) implies that(143)∫ℝ/−1,1∫ℝGhk,nfy,v2dμk,ny,v+hLk,n2ℝ2∫ℝ/−1,1ℱk,nfξ2dγk,nξ≥hLk,n2ℝ2hLk,n2ℝ2Ck,n,where Ck,n≔Ck,nE1,E2. Hence, it follows that(144)∫ℝ2y2pGhk,nfy,ν2dμk,ny,ν+hLk,n2ℝ2∫ℝξ2qℱk,nfξ2dγk,nξ≥hLk,n2ℝ2fLk,n2ℝ2Ck,n.
Replacingf by fλ and h by h1/λ, relation (42) implies that(145)∫ℝ2y2pGhk,nfyλ,λν2dμk,ny,ν+λ2k−1n+2/nhLk,n2ℝ2∫ℝξ2qℱk,nfλξ2dγk,nξ≥hLk,n2ℝ2fLk,n2ℝ2Ck,n.
Therefore, we have(146)λ2p∫ℝ2y2pGhk,nfy,ν2dμk,ny,ν+λ−2qhLk,n2ℝ2∫ℝξ2qℱk,nfξ2dγk,nξ≥hLk,n2ℝ2fLk,n2ℝ2Ck,n.
As a result, optimizing the right side of the aforementioned inequality forλ>0 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]).
ForE⊂ℝ satisfying 0<γk,nE≔∫Edγk,nx<∞, there exists ℭk,n,s>0 with 0<s<2k−1n+2/2n, so that(147)∫Eℱk,nfξ2dγk,nξ≤ℭk,n,sγk,nE2ns/2k−1n+2xsfLk,n2ℝ2,for every f∈Lk,n2ℝ.Theorem 5.2.
Let0<s<2k−1n+2/2n; then, for E⊂ℝ satisfying 0<γk,nE<∞ and any f∈Lk,n2ℝ, the following is true:(148)∫Eℱk,nfξ2dγk,nξ≤ℭk,n,sγk,nE2ns/2k−1n+2hLk,n2ℝ2∫ℝ2y2sGhk,nfy,ν2dμk,ny,ν,where ℭk,n,s is as mentioned in Proposition 5.1.Proof.
SinceGhk,nf.,ν∈Lk,n2ℝ, whenever f∈Lk,n2ℝ, we can change f to Ghk,nf.,ν in (147), so that(149)∫Eℱk,nGhk,nf.,νξ2dγk,nξ≤ℭk,n,sγk,nE2ns/2k−1n+2ysGhk,nf.,νLk,n2ℝ2,∀ν∈ℝ.
Integrating the above inequality under the measuredγk,nν, we get(150)∫E∫ℝℱk,nGhk,nf.,νξ2dμk,nξ,ν≤ℭk,n,sγk,nE2ns/2k−1n+2∫ℝ2y2sGhk,nfy,ν2dμk,ny,ν.
Furthermore, using Lemma2.1 yields(151)∫E∫ℝℱk,nfξ2τνh2−1nξdγk,nξdγk,nν≤ℭk,n,sγk,nE2ns/2k−1n+2∫ℝ2y2sGhk,nfy,ν2dμk,ny,ν.
As such, inequality (151) becomes(152)hLk,n2ℝ2∫Eℱk,nfξ2dγk,nξ≤ℭk,n,sγk,nE2ns/2k−1n+2∫ℝ2y2sGhk,nfy,ν2dμk,ny,ν.
Or equivalently, for anys∈0,2k−1n+2/2n, we have(153)∫ℝ2y2sGhk,nfy,ν2dμk,ny,ν≥hLk,n2ℝ2ℭk,n,sγk,nE2ns/2k−1n+2∫Eℱk,nfξ2dγk,nξ.
Hence, the proof of Theorem5.2 is complete. □
Given a subsetE of ℝ, the Paley–Wiener space PWk,nE is defined by(154)PWk,nE≔f∈Lk,n2ℝ:suppℱk,nf⊂E.
In view of Plancherel’s formula (17), we are led to the following inequality.Corollary 5.2.
ForE⊂ℝ with 0<γk,nE<∞ and 0<s<2k−1n+2/2n, the following is true:(155)fLk,n2ℝd2≤ℭk,n,sγk,nE2ns/2k−1n+2hLk,n2ℝ2∫ℝ2y2sGhk,nfy,ν2dμk,ny,ν,where f∈PWk,nE and the constant ℭk,n,s is the same as given in Proposition 5.1.
Swapping the functionsf and ℱk,nf in Proposition 5.1 yields the below mentioned inequality.Corollary 5.3.
ForF⊂ℝ with 0<γk,nF<∞ and 0<t<2k−1n+2/2n, we have(156)∫Ffy2dγk,ny≤ℭk,n,tγk,nF2nt/2k−1n+2ξtℱk,nfLk,n2ℝ2,∀f∈Lk,n2ℝ,where ℭk,n,t is the same constant as mentioned in Proposition 5.1.
Adopting the strategy as in Theorem5.2 and implementing Corollary 5.3, the following inequality is obtained.Corollary 5.4.
ForF⊂ℝ with 0<γk,nF<∞ and 0<t<2k−1n+2/2n, for all f∈Lk,n2ℝ, we have(157)∫ℝ∫FGhk,nfy,ν2dμk,ny,ν≤ℭk,n,tγk,nF2nt/2k−1n+2hLk,n2ℝ2∫ℝξ2tℱk,nfξ2dγk,nξ,where ℭk,n,t is the same constant as mentioned in Proposition 5.1.
For each subsetF of ℝ, we consider the following generalization of Paley–Wiener spaces:(158)GPWk,nF≔f∈Lk,n2ℝ:∀ν∈ℝ,suppGhk,nf.,ν⊂F.
Applications of (45), Corollary 5.4, and the definition of generalized Paley–Wiener spaces GPWk,nF yield the following inequality.Corollary 5.5.
Given are two subsetsE,F⊂ℝ satisfying 0<γk,nE,γk,nF<∞. Also, let 0<s,t<2k−1n+2/2n; then,(i)
Iff∈GPWk,nF, we have(159)fLk,n2ℝd2≤ℭk,n,tγk,nF2nt/2k−1n+2∫ℝξ2tℱk,nfξ2dγk,nξ.(ii)
For anyf∈PWk,nE∩GPWk,nF, we have(160)fLk,n2ℝds+t≤ℭk,n,ts/2ℭk,n,st/2γk,nEγk,nF2ts/2k−1n+2/n∫ℝξ2tℱk,nfξ2dγk,nξs/2∫ℝ2y2sGhk,nfy,ν2dμk,ny,νt/2.
We now formulate yet another variant of Heisenberg-type uncertainty principle pertaining to the DGT defined in (41).Theorem 5.3.
LetGhk,nf be the deformed Gabor transform of any f∈Lk,n2ℝ and 0<p<2k−1n+2/2n and q>0. Then,(161)fLk,n2ℝd2≤Ck,n,p,qypGhk,nfLμk,n2ℝ22q/p+qξqℱk,nfLk,n2ℝ2p/p+q,where(162)Ck,n,p,q=2nMk,n/2k−1n+22np/2k−1n+2ℭk,n,phLk,n2ℝ2q/p+qpqq/p+q+qpq/p+q.Proof.
Letq>0 and r>0. Then, for 0<p<2k−1n+2/2n, we have(163)fLk,n2ℝd2=ℱk,nfLk,n2ℝd2=∫−rrℱk,nfξ2dγk,nξ+∫ℝ/−r,rℱk,nfξ2dγk,nξ.
Using Theorem5.2 and some simple calculation, we obtain(164)∫−rrℱk,nfξ2dγk,nξ≤2nMk,n/2k−1n+22np/2k−1n+2ℭk,n,phLk,n2ℝ2r2p∫ℝ2y2pGhk,nfy,ν2dμk,ny,ν.
Besides, it also follows that(165)∫ℝ/−r,rℱk,nfξ2dγk,nξ≤r−2q∫ℝξ2qℱk,nfξ2dγk,nξ.
Combining relations (163)–(165), we get(166)fLk,n2ℝd2≤2nMk,n/2k−1n+22np/2k−1n+2ℭk,n,phLk,n2ℝ2r2pypGhk,nfLμk,n2ℝ22+r−2qξqℱk,nfLk,n2ℝ2.
We choose(167)r=qhLk,n2ℝ2p2nMk,n/2k−1n+22np/2k−1n+2ℭk,n,p1/2p+2qξqℱk,nfLk,n2ℝypGhk,nfLμk,n2ℝ21/p+q,yielding 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).
Letη,p∈ℝ satisfy 0<η<2k−1n+2/n and p⩾1. For every measurable subset T⊂ℝ2 with 0<μk,nT<∞ and f∈Lk,n2ℝd, there exists Ck,nη,p>0 with(168)∫TGhk,nfy,νpdμk,ny,ν1/p≤Ck,nη,pμk,nT1/pp+1y,νηGhk,nfLμk,n2ℝ222k−1n+2/2k−1+ηn+2p+1fLk,n2ℝhLk,n2ℝ2k−1+ηn+2p+1−22k−1n+2/2k−1+ηn+2p+1.Proof.
Consider the normalized casefLk,n2ℝ=hLk,n2ℝ=1. For each r>1, we obtain(169)Ghk,nfLμk,npT≤Ghk,nf1B20,rLμk,npT+Ghk,nf1B2c0,rLμk,npT,where B20,r≔y,ν∈ℝ2:y,ν≤r denotes the ball on ℝ2 with radius r. However, for every η∈0,2k−1n+2/n, relation (43) and Hölder’s inequality implies that(170)Ghk,nf1B20,rLμk,npT=∫ℝ2Ghk,nfy,νp1B20,ry,ν1Ty,νdμk,ny,ν1/p≤Ghk,nfLμk,n∞ℝ2p/p+1∫ℝ2Ghk,nfy,νp/p+11B20,ry,ν1Ty,νdμk,ny,ν1/p≤μk,nT1/pp+1Ghk,nf1B20,rLμk,n1ℝ21/p+1≤μk,nT1/pp+1y,νηGhk,nfLμk,n2ℝ21/p+1y,ν−η1B20,rLμk,n2ℝ21/p+1.
On the flip side, we also have(171)y,ν−η1B20,rLμk,n2ℝ2≤nMk,nΓ2k−1n+2/2n22k−1−ηn+2Γ2k−1n+2/nr2k−1−ηn+2/n.
Therefore,(172)Ghk,nf1B20,rLμk,npT≤μk,nT1/pp+1nMk,nΓ2k−1n+2/2n22k−1−ηn+2Γ2k−1n+2/n1/2p+1r2k−1−ηn+2/np+1y,νηGhk,nfLμk,n2ℝ21/p+1.
Using the inequality due to Hölder together with (44), it follows that(173)Ghk,nf1B2c0,rLμk,npT≤Ghk,nfLμk,n∞ℝ2p−1/p+1∫ℝ2Ghk,nfy,ν2p/p+11B2c0,ry,ν1Ty,νdμk,ny,ν1/p≤μk,nT1/pp+1∫ℝ2Ghk,nfy,ν21B2c0,ry,νdμk,ny,ν1/p+1≤μk,nT1/pp+1y,νηGhk,nfLμk,n2ℝ22/p+1r−2η/p+1.
Therefore, for everyη∈0,2k−1n+2/n, we have(174)∫TGhk,nfy,νpdμk,ny,ν1/p≤μk,nT1/pp+1y,νηGhk,nfLμk,n2ℝ21/p+1nMk,nΓ2k−1n+2/2n22k−1−ηn+2Γ2k−1n+2/n1/2p+1r2k−1−ηn+2/np+1+y,νηGhk,nfLμk,n2ℝ21/p+1r−2η/p+1.
In particular, the inequality holds for(175)r0=2k−1−ηn+2Γ2k−1n+2/nnMk,nΓ2k−1n+2/2n2n/2k−1+ηn+22nη2k−1−ηn+2np+1/2k−1+ηn+2y,νηGhk,nfLμk,n2ℝ2n/2k−1+ηn+2,and hence,(176)∫TGhk,nfy,νpdμk,ny,ν1/p≤μk,nT1/pp+12k−1−ηn+2Γ2k−1n+2/nnMk,nΓ2k−1n+2/2n22nη/2k−1+ηn+2p+1y,νηGhk,nfLμk,n2ℝ222k−1n+2/2k−1+ηn+2p+12k−1−ηn+22nη2nη/2k−1+ηn+22k−1+ηn+22k−1−ηn+2.
Or equivalently,(177)∫TGhk,nfy,νpdμk,ny,ν1/p≤Ck,nη,pμk,nT1/pp+1y,νηGhk,nfLμk,n2ℝ222k−1n+2/2k−1+ηn+2p+1fLk,n2ℝhLk,n2ℝ22k−1n+2/2k−1+ηn+2p+1.Remark: forh∈Lk,n,e2ℝ, we define the modulation of h by ν as follows [20]:(178)ℳνh≔ℱk,nτνk,nℱk,nh2.Also, the generalized Gabor transformVhk,n is given by(179)∀y,ν∈ℝ2,Vhk,nfy,ν≔∫ℝfxτ−1nyk,nℳνhy¯dγkx.It is clear that(180)Vhk,n=Gℱk,nhk,n.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 Ghk,n are valuables for the integral transform Vhk,n and it is suffice to replace h by ℱk,nh to derive the analogues results.


## 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]:(135)∫ℝft2dγk,nt≤Ck,nE1,E2∫ℝ/E1ft2dγk,nt+∫ℝ/E2ℱk,nfξ2dγk,nξ,where E1 and E2 are the subsets of ℝ with finite measure and Ck,nE1,E2 is a constant.Theorem 5.1.
LetGhk,nf be the deformed Gabor transform of any f∈Lk,n2ℝ. Then, we have(136)∫ℝ/E1∫ℝGhk,nfy,ν2dμk,ny,ν+hLk,n2ℝ2∫ℝ/E2ℱk,nfξ2dγk,nξ≥hLk,n2ℝ2fLk,n2ℝ2Ck,nE1,E2,where Ck,nE1,E2 is given by (135).Proof.
For anyν∈ℝ, it follows that Ghk,nf.,ν∈Lk,n2ℝ, provided f∈Lk,n2ℝ. Therefore, changing f to Ghk,nf.,ν in (135) implies that(137)∫ℝGhk,nfy,ν2dγk,nt≤Ck,nE1,E2∫ℝ/E1Ghk,nfy,ν2dγk,nt+∫ℝ/E2ℱk,nGhk,nf.,νξ2dγk,nξ,which after integration under dγk,nν yields the following inequality:(138)∫ℝ2Ghk,nfy,ν2dμk,ny,ν≤Ck,nE1,E2∫ℝ/E1∫ℝGhk,nfy,ν2dμk,ny,ν+∫ℝ/E2∫ℝℱk,nGhk,nf.,νξ2dμk,nξ,ν.
Invoking Lemma2.1 in association with Plancherel’s formula (45), we get(139)∫ℝ/E1∫ℝGhk,nfy,ν2dμk,ny,ν+∫ℝ/E2∫ℝℱk,nfξτνk,nh2−1nξ2dμk,nξ,ν≥hLk,n2ℝ2fLk,n2ℝ2Ck,nE1,E2,so that(140)∫ℝ/E1∫ℝGhk,nfy,ν2dμk,ny,ν+∫ℝ\E2ℱk,nfξ2∫ℝτνk,nh2−1nξdγk,nνdγk,nξ≥hLk,n2ℝ2fLk,n2ℝ2Ck,nE1,E2.
Using Lemma2.1 and relation (32) and keeping in view that h∈Lk,n,e2ℝ∩Lk,n∞ℝ, we obtain(141)∫ℝ/E1∫ℝGhk,nfy,ν2dμk,ny,ν+hLk,n2ℝ2∫ℝ/E2ℱk,nfξ2dγk,nξ≥hLk,n2ℝ2fLk,n2ℝ2Ck,nE1,E2,which is the desired Benedick–Amrein–Berthier’s uncertainty principle for deformed Gabor transform (41).
As a consequence of Theorem5.1, we obtain the following generalized Heisenberg-type uncertainty inequality for DGT (41).Corollary 5.1.
LetGhk,nf be the deformed Gabor transform of any arbitrary function f∈Lk,n2ℝ. Then, for p,q>0, there exist Ck,np,q>0 satisfying(142)∫ℝ2y2pGhk,nfy,ν2dμk,ny,νq/2∫ℝξ2qℱk,nfξ2dγk,nξp/2≥Ck,np,qhLk,n2ℝqfLk,n2ℝp+q.Proof.
TakeE1=E2=−1,1. Then, for any f∈Lk,n2ℝ and p,q>0, (136) implies that(143)∫ℝ/−1,1∫ℝGhk,nfy,v2dμk,ny,v+hLk,n2ℝ2∫ℝ/−1,1ℱk,nfξ2dγk,nξ≥hLk,n2ℝ2hLk,n2ℝ2Ck,n,where Ck,n≔Ck,nE1,E2. Hence, it follows that(144)∫ℝ2y2pGhk,nfy,ν2dμk,ny,ν+hLk,n2ℝ2∫ℝξ2qℱk,nfξ2dγk,nξ≥hLk,n2ℝ2fLk,n2ℝ2Ck,n.
Replacingf by fλ and h by h1/λ, relation (42) implies that(145)∫ℝ2y2pGhk,nfyλ,λν2dμk,ny,ν+λ2k−1n+2/nhLk,n2ℝ2∫ℝξ2qℱk,nfλξ2dγk,nξ≥hLk,n2ℝ2fLk,n2ℝ2Ck,n.
Therefore, we have(146)λ2p∫ℝ2y2pGhk,nfy,ν2dμk,ny,ν+λ−2qhLk,n2ℝ2∫ℝξ2qℱk,nfξ2dγk,nξ≥hLk,n2ℝ2fLk,n2ℝ2Ck,n.
As a result, optimizing the right side of the aforementioned inequality forλ>0 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]).
ForE⊂ℝ satisfying 0<γk,nE≔∫Edγk,nx<∞, there exists ℭk,n,s>0 with 0<s<2k−1n+2/2n, so that(147)∫Eℱk,nfξ2dγk,nξ≤ℭk,n,sγk,nE2ns/2k−1n+2xsfLk,n2ℝ2,for every f∈Lk,n2ℝ.Theorem 5.2.
Let0<s<2k−1n+2/2n; then, for E⊂ℝ satisfying 0<γk,nE<∞ and any f∈Lk,n2ℝ, the following is true:(148)∫Eℱk,nfξ2dγk,nξ≤ℭk,n,sγk,nE2ns/2k−1n+2hLk,n2ℝ2∫ℝ2y2sGhk,nfy,ν2dμk,ny,ν,where ℭk,n,s is as mentioned in Proposition 5.1.Proof.
SinceGhk,nf.,ν∈Lk,n2ℝ, whenever f∈Lk,n2ℝ, we can change f to Ghk,nf.,ν in (147), so that(149)∫Eℱk,nGhk,nf.,νξ2dγk,nξ≤ℭk,n,sγk,nE2ns/2k−1n+2ysGhk,nf.,νLk,n2ℝ2,∀ν∈ℝ.
Integrating the above inequality under the measuredγk,nν, we get(150)∫E∫ℝℱk,nGhk,nf.,νξ2dμk,nξ,ν≤ℭk,n,sγk,nE2ns/2k−1n+2∫ℝ2y2sGhk,nfy,ν2dμk,ny,ν.
Furthermore, using Lemma2.1 yields(151)∫E∫ℝℱk,nfξ2τνh2−1nξdγk,nξdγk,nν≤ℭk,n,sγk,nE2ns/2k−1n+2∫ℝ2y2sGhk,nfy,ν2dμk,ny,ν.
As such, inequality (151) becomes(152)hLk,n2ℝ2∫Eℱk,nfξ2dγk,nξ≤ℭk,n,sγk,nE2ns/2k−1n+2∫ℝ2y2sGhk,nfy,ν2dμk,ny,ν.
Or equivalently, for anys∈0,2k−1n+2/2n, we have(153)∫ℝ2y2sGhk,nfy,ν2dμk,ny,ν≥hLk,n2ℝ2ℭk,n,sγk,nE2ns/2k−1n+2∫Eℱk,nfξ2dγk,nξ.
Hence, the proof of Theorem5.2 is complete. □
Given a subsetE of ℝ, the Paley–Wiener space PWk,nE is defined by(154)PWk,nE≔f∈Lk,n2ℝ:suppℱk,nf⊂E.
In view of Plancherel’s formula (17), we are led to the following inequality.Corollary 5.2.
ForE⊂ℝ with 0<γk,nE<∞ and 0<s<2k−1n+2/2n, the following is true:(155)fLk,n2ℝd2≤ℭk,n,sγk,nE2ns/2k−1n+2hLk,n2ℝ2∫ℝ2y2sGhk,nfy,ν2dμk,ny,ν,where f∈PWk,nE and the constant ℭk,n,s is the same as given in Proposition 5.1.
Swapping the functionsf and ℱk,nf in Proposition 5.1 yields the below mentioned inequality.Corollary 5.3.
ForF⊂ℝ with 0<γk,nF<∞ and 0<t<2k−1n+2/2n, we have(156)∫Ffy2dγk,ny≤ℭk,n,tγk,nF2nt/2k−1n+2ξtℱk,nfLk,n2ℝ2,∀f∈Lk,n2ℝ,where ℭk,n,t is the same constant as mentioned in Proposition 5.1.
Adopting the strategy as in Theorem5.2 and implementing Corollary 5.3, the following inequality is obtained.Corollary 5.4.
ForF⊂ℝ with 0<γk,nF<∞ and 0<t<2k−1n+2/2n, for all f∈Lk,n2ℝ, we have(157)∫ℝ∫FGhk,nfy,ν2dμk,ny,ν≤ℭk,n,tγk,nF2nt/2k−1n+2hLk,n2ℝ2∫ℝξ2tℱk,nfξ2dγk,nξ,where ℭk,n,t is the same constant as mentioned in Proposition 5.1.
For each subsetF of ℝ, we consider the following generalization of Paley–Wiener spaces:(158)GPWk,nF≔f∈Lk,n2ℝ:∀ν∈ℝ,suppGhk,nf.,ν⊂F.
Applications of (45), Corollary 5.4, and the definition of generalized Paley–Wiener spaces GPWk,nF yield the following inequality.Corollary 5.5.
Given are two subsetsE,F⊂ℝ satisfying 0<γk,nE,γk,nF<∞. Also, let 0<s,t<2k−1n+2/2n; then,(i)
Iff∈GPWk,nF, we have(159)fLk,n2ℝd2≤ℭk,n,tγk,nF2nt/2k−1n+2∫ℝξ2tℱk,nfξ2dγk,nξ.(ii)
For anyf∈PWk,nE∩GPWk,nF, we have(160)fLk,n2ℝds+t≤ℭk,n,ts/2ℭk,n,st/2γk,nEγk,nF2ts/2k−1n+2/n∫ℝξ2tℱk,nfξ2dγk,nξs/2∫ℝ2y2sGhk,nfy,ν2dμk,ny,νt/2.
We now formulate yet another variant of Heisenberg-type uncertainty principle pertaining to the DGT defined in (41).Theorem 5.3.
LetGhk,nf be the deformed Gabor transform of any f∈Lk,n2ℝ and 0<p<2k−1n+2/2n and q>0. Then,(161)fLk,n2ℝd2≤Ck,n,p,qypGhk,nfLμk,n2ℝ22q/p+qξqℱk,nfLk,n2ℝ2p/p+q,where(162)Ck,n,p,q=2nMk,n/2k−1n+22np/2k−1n+2ℭk,n,phLk,n2ℝ2q/p+qpqq/p+q+qpq/p+q.Proof.
Letq>0 and r>0. Then, for 0<p<2k−1n+2/2n, we have(163)fLk,n2ℝd2=ℱk,nfLk,n2ℝd2=∫−rrℱk,nfξ2dγk,nξ+∫ℝ/−r,rℱk,nfξ2dγk,nξ.
Using Theorem5.2 and some simple calculation, we obtain(164)∫−rrℱk,nfξ2dγk,nξ≤2nMk,n/2k−1n+22np/2k−1n+2ℭk,n,phLk,n2ℝ2r2p∫ℝ2y2pGhk,nfy,ν2dμk,ny,ν.
Besides, it also follows that(165)∫ℝ/−r,rℱk,nfξ2dγk,nξ≤r−2q∫ℝξ2qℱk,nfξ2dγk,nξ.
Combining relations (163)–(165), we get(166)fLk,n2ℝd2≤2nMk,n/2k−1n+22np/2k−1n+2ℭk,n,phLk,n2ℝ2r2pypGhk,nfLμk,n2ℝ22+r−2qξqℱk,nfLk,n2ℝ2.
We choose(167)r=qhLk,n2ℝ2p2nMk,n/2k−1n+22np/2k−1n+2ℭk,n,p1/2p+2qξqℱk,nfLk,n2ℝypGhk,nfLμk,n2ℝ21/p+q,yielding 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).
Letη,p∈ℝ satisfy 0<η<2k−1n+2/n and p⩾1. For every measurable subset T⊂ℝ2 with 0<μk,nT<∞ and f∈Lk,n2ℝd, there exists Ck,nη,p>0 with(168)∫TGhk,nfy,νpdμk,ny,ν1/p≤Ck,nη,pμk,nT1/pp+1y,νηGhk,nfLμk,n2ℝ222k−1n+2/2k−1+ηn+2p+1fLk,n2ℝhLk,n2ℝ2k−1+ηn+2p+1−22k−1n+2/2k−1+ηn+2p+1.Proof.
Consider the normalized casefLk,n2ℝ=hLk,n2ℝ=1. For each r>1, we obtain(169)Ghk,nfLμk,npT≤Ghk,nf1B20,rLμk,npT+Ghk,nf1B2c0,rLμk,npT,where B20,r≔y,ν∈ℝ2:y,ν≤r denotes the ball on ℝ2 with radius r. However, for every η∈0,2k−1n+2/n, relation (43) and Hölder’s inequality implies that(170)Ghk,nf1B20,rLμk,npT=∫ℝ2Ghk,nfy,νp1B20,ry,ν1Ty,νdμk,ny,ν1/p≤Ghk,nfLμk,n∞ℝ2p/p+1∫ℝ2Ghk,nfy,νp/p+11B20,ry,ν1Ty,νdμk,ny,ν1/p≤μk,nT1/pp+1Ghk,nf1B20,rLμk,n1ℝ21/p+1≤μk,nT1/pp+1y,νηGhk,nfLμk,n2ℝ21/p+1y,ν−η1B20,rLμk,n2ℝ21/p+1.
On the flip side, we also have(171)y,ν−η1B20,rLμk,n2ℝ2≤nMk,nΓ2k−1n+2/2n22k−1−ηn+2Γ2k−1n+2/nr2k−1−ηn+2/n.
Therefore,(172)Ghk,nf1B20,rLμk,npT≤μk,nT1/pp+1nMk,nΓ2k−1n+2/2n22k−1−ηn+2Γ2k−1n+2/n1/2p+1r2k−1−ηn+2/np+1y,νηGhk,nfLμk,n2ℝ21/p+1.
Using the inequality due to Hölder together with (44), it follows that(173)Ghk,nf1B2c0,rLμk,npT≤Ghk,nfLμk,n∞ℝ2p−1/p+1∫ℝ2Ghk,nfy,ν2p/p+11B2c0,ry,ν1Ty,νdμk,ny,ν1/p≤μk,nT1/pp+1∫ℝ2Ghk,nfy,ν21B2c0,ry,νdμk,ny,ν1/p+1≤μk,nT1/pp+1y,νηGhk,nfLμk,n2ℝ22/p+1r−2η/p+1.
Therefore, for everyη∈0,2k−1n+2/n, we have(174)∫TGhk,nfy,νpdμk,ny,ν1/p≤μk,nT1/pp+1y,νηGhk,nfLμk,n2ℝ21/p+1nMk,nΓ2k−1n+2/2n22k−1−ηn+2Γ2k−1n+2/n1/2p+1r2k−1−ηn+2/np+1+y,νηGhk,nfLμk,n2ℝ21/p+1r−2η/p+1.
In particular, the inequality holds for(175)r0=2k−1−ηn+2Γ2k−1n+2/nnMk,nΓ2k−1n+2/2n2n/2k−1+ηn+22nη2k−1−ηn+2np+1/2k−1+ηn+2y,νηGhk,nfLμk,n2ℝ2n/2k−1+ηn+2,and hence,(176)∫TGhk,nfy,νpdμk,ny,ν1/p≤μk,nT1/pp+12k−1−ηn+2Γ2k−1n+2/nnMk,nΓ2k−1n+2/2n22nη/2k−1+ηn+2p+1y,νηGhk,nfLμk,n2ℝ222k−1n+2/2k−1+ηn+2p+12k−1−ηn+22nη2nη/2k−1+ηn+22k−1+ηn+22k−1−ηn+2.
Or equivalently,(177)∫TGhk,nfy,νpdμk,ny,ν1/p≤Ck,nη,pμk,nT1/pp+1y,νηGhk,nfLμk,n2ℝ222k−1n+2/2k−1+ηn+2p+1fLk,n2ℝhLk,n2ℝ22k−1n+2/2k−1+ηn+2p+1.Remark: forh∈Lk,n,e2ℝ, we define the modulation of h by ν as follows [20]:(178)ℳνh≔ℱk,nτνk,nℱk,nh2.Also, the generalized Gabor transformVhk,n is given by(179)∀y,ν∈ℝ2,Vhk,nfy,ν≔∫ℝfxτ−1nyk,nℳνhy¯dγkx.It is clear that(180)Vhk,n=Gℱk,nhk,n.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 Ghk,n are valuables for the integral transform Vhk,n and it is suffice to replace h by ℱk,nh 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.



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