# Clifford-Valued Shearlet Transforms on Cl(P,Q)-Algebras

Authors: Firdous A. Shah; Aajaz A. Teali; Mawardi Bahri
Journal: Journal of Mathematics
 (2022)

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## Abstract

The shearlet transform is a promising and powerful time-frequency tool for analyzing nonstationary signals. In this article, we introduce a novel integral transform coined as the Clifford-valued shearlet transform on Cl(p,q) algebras which is designed to represent Clifford-valued signals at different scales, locations, and orientations. We investigated the fundamental properties of the Clifford-valued shearlet transform including Parseval’s formula, isometry, inversion formula, and characterization of range using the machinery of Clifford Fourier transforms. Moreover, we derived the pointwise convergence and homogeneous approximation properties for the proposed transform. We culminated our investigation by deriving several uncertainty principles such as the Heisenberg–Pauli–Weyl uncertainty inequality, Pitt’s inequality, and logarithmic and local-type uncertainty inequalities for the Clifford-valued shearlet transform.


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## Body

## 1. Introduction

Wavelet transforms have been proved to be a successful tool for analyzing nontransient signals and have been applied in a number of fields including signal and image processing, differential and integral equations, sampling theory, quantum mechanics, medicine, and so on [1]. However, the efficiency of the wavelet transforms is considerably reduced when applied to higher dimensional signals as they are not able to capture the geometric features like edges and corners at different scales efficiently. The detection of such geometric features in nontransient signals is often highly desirable in numerous practical applications such as medical imaging, remote sensing, crystallography, and several other areas. To circumvent these constraints, a number of novel directional representation systems have been introduced and employed in recent years, such as the wedgelets, ridgelets, ripplets, curvelets, contourlets, surfacelets, brushlets, and shearlets. Among all these geometrical wavelet systems, the shearlet systems have been widely acknowledged and emerged as one of the most effective frameworks for representing multidimensional data because they are nonisotropic nature, and they offer optimally sparse representations [2], allow compactly supported analysing elements [3], are associated with fast decomposition and reconstruction algorithms, and provide a unified treatment of continuum and digital data [4, 5].Clifford algebras have dethroned both the Grossmann’s exterior algebra and Hamilton’s quaternion algebra in the sense that they incorporate both the geometrical and algebraic features of Euclidean space into a single structure [6]. As a result, the theory of Clifford algebras has attained an overwhelming response and gained a respectable status in higher-dimensional signal and image processing mainly due to the reason that such algebras encompass all dimensions at once unlike the multidimensional tensorial approach with tensor products of one-dimensional phenomena. This true multidimensional nature allows specific constructions of higher dimensional signal and image processing tools including the Clifford Fourier transforms [7, 8], Clifford Gabor transforms, Clifford wavelet transforms, and other integral transforms in general [9–13].Motivated and inspired by the contemporary developments in the theory of shearlet transforms abreast the profound applicability of the Clifford algebras, we introduce the notion of Clifford-valued shearlet transforms onClp,q algebras in the context of multidimensional signal analysis. Unlike the conventional shearlet transform, the proposed transform inherits both the geometric and algebraic properties of shearlet transforms and Clifford algebras. Although a meek analgoue of shearlet transform in the Clifford domain has been proposed in [14], it only deals with the Cl0,nalgebra, where n=3mod4. Therefore, the centre piece of this study is to construct the Clifford-valued shearlets and the corresponding shearlet transforms in the most general setting Clp,q by employing translations, sharing, scaling, and spinning elements. Besides, we study the basic properties of the Clifford-valued shearlet transforms including Parseval’s and inversion formulae and range theorem using the machinery of Clifford Fourier transforms. Moreover, we derive the pointwise convergence and homogeneous approximation properties for the proposed transform. Finally, we formulate some uncertainty inequalities including the classical Heisenberg–Pauli–Weyl inequality, Pitt’s inequality, and logarithmic inequality for the Clifford shearlet transforms.The structure of this article is as follows. Section2 deals with the preliminaries of Clifford algebras, whereas a comprehensive analysis of the general Clifford-valued shearlet transforms is carried out in Section 3. In Section 4, we study the homogeneous approximation properties for proposed transform. Several uncertainty principles for the proposed transform are also being studied in Section 5. Finally, a conclusion is summarized in Section 6.


## 2. Basics of Clifford Algebras

In this section, we present a brief overview of the Clifford algebras including the definitions of Clifford Fourier transforms, spin group, and some unitary operators.The Clifford algebraClp,q is a noncommutative, associative algebra generated by the orthonormal basis e1,e2,…,en of the n-dimensional Euclidean space ℝn governed by the multiplication rule:(1)eiej+ejei=2εiδij,i,j=1,2,…,n,where n=p+q, εi=+1 for i=1,2,…,p and εi=−1 for i=p+1,p+2,…,n, with δij denoting the usual Kronecker’s delta function. The noncommutative product and the additional axiom of associativity generates the 2n− dimensional Clifford geometric algebra Clp,q, which can be decomposed as(2)Clp,q=⊕k=0nClp,qk,where Clp,qk denotes the space of k-vectors given by(3)Clp,qk≔spanei1,ei2,…,eik;i1≤i2≤…≤ik.Any general element of the Clifford algebra is called a multivector and every multivectorM∈Clp,q can be represented in the following form:(4)M=∑AMAeA=M0+M1+⋯+Mn,MA∈ℝ,A⊂1,2,…,n,where eA=ei1ei2…eik and i1≤i2≤…≤ik. Moreover, ⋅k is called as the grade k-part of M, and ⋅0,⋅1,⋅2,…, respectively, denote the scalar part, vector part, bivector part, and so on. The Clifford conjugate of a multivector M∈Clp,q is given by(5)M¯=∑r=0n−1rr−1/2M¯r,where the scalar product of multivectors M and N¯ is defined as(6)ScMN¯=MN=M⋆N¯=∑AMANA.Moreover, for any pair of multivectorsM,N∈Clp,q, it can be easily verified that(7)MN≤2nMN.We now intend to recall the fundamental notion of Clifford Fourier transforms inLrℝp,q,Clp,q, 1≤r<∞ as(8)Lrℝp,q,Clp,q=f:ℝp,q⟶Clp,q;fr=∫ℝp,qfxrdnx1/r<∞.It is imperative to mention that any functionf∈Lrℝp,q,Clp,q can be expressed as a combination of the real-valued functions fA and the basis elements eA as(9)fx=∑AfAxeA.Due to the noncommutativity of Clifford-valued functions, several analogues of the Clifford Fourier transforms have been introduced in the literature. However, we shall be interested in following definition due to Bahri et al. [15].Definition 1.
LetI∈Clp,q be a square root of −1. The Clifford Fourier transform of any function f∈L1ℝp,q,Clp,q is defined by(10)ℱClfxξ=12πn/2∫ℝp,qfxe−Iυξ,xdnx,where nx,ξ∈ℝp,q, dnx=dx1dx2…dxn, υ:ℝp,q×ℝp,q⟶ℝ, and υξ,x=ξ1x1+ξ2x2+⋯+ξnxn.
The inversion and Plancherel formulae associated with the Clifford Fourier transform (10) are given by(11)fx=12πn/2∫ℝp,qℱClfxξeIυξ,xdnξ,f,gL2ℝp,q,Clp,q=ℱClf,ℱClgL2ℝp,q,Clp,q, In this case, the inner product of two multivector functions f and g is described through(12)f,gL2ℝp,q,Clp,q=∫ℝp,qfxgx¯dnx,and its scalar part is given by(13)f,gL2ℝp,q,Clp,q=∫ℝp,qfxgx2dnx=∫ℝp,qScfxgx¯dnx=Sc∫ℝp,qfxgx¯dnx.
For an efficient representation of Clifford-valued functions, we employ the spin elements obtained from the spin group as defined below.Definition 2.
The spin-group is a double covering of special orthogonal group ofℝn and is defined by(14)Spinn=r∈Clp,q+;r¯r=rr¯=1,where Clp,q+ is a subgroup of the invertible elements in the Clifford algebra Clp,q.
To facilitate the construction of Clifford-valued shearlets, we define the fundamental unitary operators acting on the spaceLrℝp,q. For a>0, s∈ℝn−1 and b∈ℝn, and the scaling, shearing, spin-rotation, and translation operators are denoted by DAa, DSs, ℛr, Tb, respectively, and are defined as(15)DAaΨx=detAa−1/2ΨAa−1x,DSsΨx=ΨSs−1x,ℛrΨx=rΨr¯xrr¯,TbΨx=Ψx−b,and the matrices involved in equation (15) are(16)Aa=a0n−1T0n−1sgnaa1/nIn−1,Ss=1sT0n−1In−1,where sT=s1,s2,…,sn−1, and  sgn⋅ and 0 denote the well-known Signum function and the null vector, respectively. Moreover, the composition of the scaling matrix Aa and the shearing matrix Ss is given by(17)SsAa=asgnaa1/ns1sgnaa1/ns2sgnaa1/ns3⋯sgnaa1/nsn−10sgnaa1/n00⋯⋯⋮⋮⋮⋮⋮⋮00000sgnaa1/n.


## 3. The Clifford-Valued Shearlet Transform onClp,q Algebras

In this section, we shall construct the Clifford-valued shearlets onClp,q algebras by using the combined action of the scaling, sharing, spin-rotation and translation operators. Besides, we study the fundamental properties of the Clifford-valued shearlet transform including Parseval’s formula, inversion formula, and obtain a complete characterization of the range. Prior to that, we shall demonstrate that the novel family of Clifford-valued shearlets is endowed with an affine group structure.Consider that the setG=ℝ+×Spinn×ℝn−1×ℝn endowed with the binary operation ⊙ is defined as(18)a,r,s,b⊙a′,r′,s′,b′=aa′,r+r′,s+a1−1/ns′,b+SsAab′,where a,a′∈ℝ+,s,s′∈ℝn−1,b,b′∈ℝn,r,r′∈Spinn. Clearly, 1,0,0n−1,0n, is the neutral element of G, whereas a−1,−r,−a1/n−1s,−Aa−1Ss−1b is the inverse of any arbitrary element a,r,s,b∈G. Moreover, it is easy to verify that(19)a,r,s,b⊙a′,r′,s′,b′⊙a″,r″,s″,b″=a,r,s,b⊙a′,r′,s′,b′⊙a″,r″,s″,b″.Hence, we conclude that G,⊙ constitutes a group and is formally called as the similitude group of dilations, translations, shearing, and spinning.Furthermore, we claim that the left Haar measure onG is given by dη=dadrdn−1sdnb/an+1. In fact, for any function f∈L2G,Clp,q, we have(20)∫Gfa,r,s,b⊙a′,r′,s′,b′dη=∫ℝ+×Spinn×ℝn×ℝnfaa′,r+r′,s+a1−1/ns′,b+SsAab′dadrdn−1sdnban+1.Making use of the substitutiona˜:=aa′,r˜:=rr′,s˜:=s+a1−1/ns′,b˜:=b+SsAab′, i.e., da=a′−1da˜,dr=dr˜,dn−1s=a−n−12/ndn−1s˜,dnb=a−2+1/ndnb˜, the above expression becomes(21)∫Gfa,r,s,b⊙a′,r′,s′,b′dη=∫ℝ+×ℝn×Spinn×ℝnfa˜,r˜,s˜,b˜a′−1da˜dr˜a−n−12/ndn−1s˜a−2+1/ndnb˜a˜/a′n+1=∫ℝ+×ℝn×Spinn×ℝnfa˜,r˜,s˜,b˜da˜dr˜dn−1s˜dnb˜a˜n+1,which validates the claim that dη=dadrdn−1sdnb/an+1 is indeed the left Haar measure on G.Next, we shall construct a novel class of shearlet systems onClp,q algebras by the combined action of the scaling DAa, sharing DSs, spin-rotation ℛr, and translation Tb operators on any analyzing function Ψ∈L2ℝp,q,Clp,q.For anya∈ℝ+,s∈ℝn−1,b∈ℝn, and r∈Spinn, consider the family of analyzing functions:(22)Ψa,s,brx=DAaDSsℛrTbΨx=a1/2n−1rΨAa−1Ss−1r¯x−brr¯,which is called as the family of Clifford-valued shearlets on the geometric algebra-Cℓp,q. The system of functions (22) satisfies the following properties:(i)
The system (22) is a dense subspace of L2ℝn,Clp,q(ii)
The following norm equality holds good:(23)Ψa,s,brL2ℝn,Clp,q=ΨL2ℝn,Clp,q.(iii)
The Clifford Fourier transform of the family of functionsΨa,s,brx reads(24)ℱClΨa,s,brξ=a1−1/2nℱClrΨ⋅r¯rSsAaξr¯e−Iυξ,b.Next, we shall present the notion of an admissible Clifford-valued shearlet on the space of Clifford-valued functionsL2ℝn,Clp,q.Definition 3.
(Admissibility). A nontrivial functionΨ∈L2ℝn,Clp,q is called an admissible Clifford-valued shearlet if(25)CΨ=∫ℝ+×ℝn−1×SpinnℱClrΨ⋅r¯rSsAaξr¯¯ℱClrΨ⋅r¯rSsAaξr¯dadn−1sdran2−n+1/n,which is an invertible multivector and finite, i.e., ξ∈ℝp,q.Remark 1.
It is worth noticing thatℱClrΨ⋅r¯0=0, for ξ=0; that is, Ψx=∑AΨAxeA. and(26)∫ℝp,qΨAxeAe−Iυ0,xdnx=0,which in turn implies that for every component ΨA of the Clifford-valued shearlet Ψ is zero; that is, (27)∫ℝp,qΨAxdnx=0.
Based on the novel family of Clifford-valued shearlets defined in equation (22), we have the following main definition of the continuous Clifford-valued shearlet transform.Definition 4.
The continuous Clifford-valued shearlet transform of any multivector signalf∈L2ℝn,Clp,q with respect to an analysing Clifford-valued shearlet Ψ∈L2ℝp,q,Clp,q is defined by(28)CSΨfa,r,s,b=f,Ψa,s,brL2ℝp,q,Clp,q=∫ℝp,qfxΨa,s,brx¯dnx.where Ψa,s,brx is given by equation (22).
The corresponding spectral representation of the Clifford-valued shearlet transform is(29)CSΨfa,r,s,b=a1−1/2n∫ℝnℱClfξeIυξ,bℱClrΨ⋅r¯rSsAaξr¯¯dnξ.
We now present an example for the lucid illustration of the proposed Clifford-valued shearlet transform (28).Example 1.
Consider the Clifford-valued Hermite wavelets [16]as(30)ψnx=−1n∂xnexp−x22,∂xn=∂n∂x1+∂n∂x2+⋯+∂n∂xn.
Therefore, the corresponding Clifford-valued shearlets ofψnx are obtained as(31)Ψa,s,brx=detAa−1/2rAa−1Ss−1x−bnexp−Aa−1Ss−1x−b22r¯,and the Clifford-valued shearlet transform (28) of any function f∈L2ℝn,Cln, with respect to the analyzing shearlets (31) can be computed as(32)CSΨfa,r,s,b=detAa−1/2∫ℝnfxrAa−1Ss−1x−bn×exp−Aa−1Ss−1x−b22r¯dnx.
For simplicity, we shall compute the two-dimensional Clifford-valued shearlet transform for the given functionf with respect to the shearlets:(33)Ψa,s,brx1,x2=detAa−1/2Aa−1Ss−1x1−b12,x2−b22×exp−Aa−1Ss−1x1−b1,x2−b222,where Aa=a00a1/2,Ss=1s01. After simplifying, we obtain(34)Ψa,s,brx1,x2=ax1−b12−s−a3/2x2−b22×exp−x1−b12+s−a−1/2x2−b222a2.
The two-dimensional analyzing shearletsψa,s,brx1,x2 given by equation (34) at different values of a,s,r, and b are plotted in Figure 1. The parameters a and s determine the scaling anisotropy and the decaying rate of shearlets providing more accurate location and orientation. In comparison with wavelets, shearlets not only inherits advantages of wavelets s=0 but also provide detailed information of position, normal and curvature of discontinuities.
The Clifford-valued shearlet transform offx1,x2=exp−x12+x22/2 is computed as(35)CSψfa,r,s,b=a∫ℝ2x1−b12−s−a3/2x2−b22×exp−x1−b12+s−a−1/2x2−b22+a2x12+x222a2dx1dx2.
For different values ofa,s,r, and b, the corresponding Clifford-valued shearlet transforms of fx1,x2 with respect the analysing shearlets (34) are depicted in Figure 2 after computing the integrals (35) in Mathematica software. From the simulation, we infer that the Clifford-valued shearlet transform enables a precise characterization of location, orientation, and curvature of discontinuities in two dimensional signals.
In the following theorem, we assemble some of the basic properties of the Clifford-valued shearlet transform (28).Figure 1
Two-dimensional analyzing shearletsψa,s,brx1,x2 given by equation (34) at different values of a,r,b, and s. (a) 2D-shearlets at a = 1, b = 1, and s = 0. (b) Contour plot of 2D-shearlets at a = 1, b = 1, and s = 0. (c) 2D-shearlets at a = 1/2, b = 1 and s = 1. (d) Contour plot of 2D-shearlets at a = 1/2, b = 1 and s = 1.
(a)(b)(c)(d)Figure 2
Two-dimensional Clifford-valued shearlet transforms offx1,x2=exp−x12+x22/2 with respect to analyzing function Ψa,s,brx1,x2 given by equation (35). (a) Clifford-valued ST of f at a = 1/2, b = 1, and s = 1/2. (b) Clifford-valued ST of f at a = b = 1, and s = 1. (c) Contour plot of Clifford-valued ST of f at a = 1/2, b = 1, and s = 1/2. (d) Contour plot of Clifford-valued ST of f at a = b = 1, and s = 1.
(a)(b)(c)(d)Theorem 1.
Ψa,s,brx1,x2for f,g∈L2ℝp,q,Clp,q, and admissible Clifford-valued shearlets Ψ and Φ. The continuous Clifford-valued shearlet transform (28) satisfies the following properties:(i)
Linearity:CSΨαf+βga,r,s,b=αCSΨfa,r,s,b+βCSΨga,r,s,b,α,β∈Clp,q(ii)
Anti−linearity:CSαΨ+βΦfa,r,s,b=CSΨfa,r,s,bα¯+CSΦfa,r,s,bβ¯(iii)
Translation covariance:CSΨTkfa,r,s,b=CSΨfa,r,s,b−k(iv)
Dilation covariance:CSΨD1/γfxa,r,s,b=CSDγΨfxa,r,s,b/γ,γ∈ℝ(v)
Parity:CSΨPfxa,r,s,b=−1nCSPΨfxa,r,s,−b,Pfx=f−x(vi)
Translation inΨ:CSTkΨfxa,r,s,b=CSΨfxa,r,s,b+kProof.
For the sake of brevity, we omit the proof.
In our next theorem, we show that the Clifford-valued shearlet transform sets up an isometry fromL2ℝp,q,Clp,q to L2ℝ+×ℝn−1×ℝn×Spinn,Clp,q.Theorem 2.
(Plancherel theorem). LetCSΨfa,r,s,b and CSΨga,r,s,b be the Clifford-valued shearlet transforms of the multivector signals f and g, respectively. Then, we have(36)∫ℝ+×ℝn−1×ℝn×SpinnScCSΨfa,r,s,bCSΨga,r,s,b¯dadn−1sdnbdran+1=2πn∫ℝp,qScfxCΨgx¯dnx=2πnfCΨ,gL2ℝp,q,Clp,q,where CΨ is given by equation (25).Proof.
.Invoking the spectral representation (29) of Clifford shearlet transforms, we obtain(37)∫ℝ+×ℝn−1×ℝn×SpinnScCSΨfa,r,s,bCSΨga,r,s,b¯dadn−1sdnbdran+1=∫ℝ+×ℝn−1×ℝn×Spinna2−1/nSc∫ℝnℱClfξeIυξ,bℱClrΨ⋅r¯rSsAaξr¯¯dnξ×∫ℝnℱClgξ′eIυξ′,bℱClrΨ⋅r¯rSsAaξ′r¯¯¯dnξ′dadn−1sdnbdran+1,=∫ℝ+×ℝn−1×ℝn×Spinna2−1/n∫ℝn∫ℝnScℱClfξeIυξ,bℱClrΨ⋅r¯rSsAaξr¯¯×ℱClrΨ⋅r¯rSsAaξ′r¯e−Iυξ′,bℱClgξ′¯dnξdnξ′dadn−1sdnbdran+1..
Then, equation (37) can be rewritten as(38)∫ℝ+×ℝn−1×ℝn×SpinnScCSΨfa,r,s,bCSΨga,r,s,b¯dadn−1sdnbdran+1=∫ℝ+×ℝn−1×ℝn×Spinn∫ℝn∫ℝnScℱClfξeIυξ,be−Iυξ′,b×ℱClrΨ⋅r¯rSsAaξr¯¯ℱClrΨ⋅r¯rSsAaξ′r¯ℱClgξ′¯dnξdnξ′dadrdn−1sdnban2−n+1/n=2πn∫ℝn∫ℝnScℱClfξ12πn∫ℝneIυξ−ξ′,bdnb×∫ℝ+×Spinn×ℝn−1ℱClrΨ⋅r¯rSsAaξr¯¯ℱClrΨ⋅r¯rSsAaξ′r¯dadrdn−1san2−n+1/n×ℱClgξ′¯dnξdnξ′=2πn∫ℝn∫ℝnScℱClfξδξ−ξ′×∫ℝ+×Spinn×ℝn−1ℱClrΨ⋅r¯rSsAaξr¯¯ℱClrΨ⋅r¯rSsAaξ′r¯dadrdn−1san2−n+1/n×ℱClgξ′¯dnξdnξ′=2πn∫ℝnScℱClfξ×∫ℝ+×Spinn×ℝn−1ℱClrΨ⋅r¯rSsAaξr¯¯ℱClrΨ⋅r¯rSsAaξr¯dadrdn−1san2−n+1/n×ℱClgξ¯dnξ=2πn∫ℝnScℱClfξCΨℱClgξ¯dnξ=2πn∫ℝp,qScfxCΨgx¯dnx.
This completes the proof of Theorem2.Corollary 1.
Forf=g, we have the following identity:(39)∫ℝ+×ℝn−1×ℝn×SpinnCSΨfa,r,s,b2dadn−1sdnbdran+1=2πn∫ℝp,qScfxCΨfx¯dnx.
By takingΨ∈L2ℝp,q,Clp,q with CΨ=1, the Clifford-valued shearlet transform CSΨfa,r,s,b becomes an isometry from L2ℝp,q,Clp,q to L2ℝ+×ℝn−1×ℝn×Spinn,Clp,q.
The next theorem guarantees the reconstruction of the input Clifford-valued signal from the corresponding Clifford-valued shearlet transform.Theorem 3 (Inversion formula).
Any Clifford-valued signalf∈L2ℝp,q,Clp,q can be reconstructed from the Clifford-valued shearlet transform CSΨfa,r,s,b via the formula:(40)fx=12πn∫ℝ+×ℝn−1×ℝn×SpinnCSΨfa,r,s,bΨa,s,brxCΨ−1dadn−1sdnbdran+1.Proof.
Implication of Plancherel theorem of Clifford-valued shearlet transform (36) for every g∈L2ℝp,q,Clp,q yields that(41)2πnfCΨ,gL2ℝp,q,Clp,q=CSΨf,CSΨgL2G,Clp,q=∫ℝ+×ℝn−1×ℝn×SpinnScCSΨfa,r,s,bCSΨga,r,s,b¯dadn−1sdnbdran+1=∫ℝ+×ℝn−1×ℝn×SpinnScCSΨfa,r,s,b∫ℝp,qgxΨa,s,brx¯¯dnxdadn−1sdnbdran+1=∫ℝ+×ℝn−1×ℝn×Spinn∫ℝp,qScCSΨfa,r,s,bΨa,s,brxgx¯dnxdadn−1sdnbdran+1=∫ℝp,q∫ℝ+×ℝn−1×ℝn×SpinnScCSΨfa,r,s,bΨa,s,brxdadn−1sdnbdran+1gx¯dnx=∫ℝ+×ℝn−1×ℝn×SpinnCSΨfa,r,s,bΨa,s,brxdadn−1sdnbdran+1,gL2ℝp,q,Clp,q,where we used the Fubini–Tonelli theorem in getting the second last step. Since g∈L2ℝp,q,Clp,q is arbitrary, we have(42)2πnfxCΨ=∫ℝ+×ℝn−1×ℝn×SpinnCSΨfa,r,s,bΨa,s,brxdadn−1sdnbdran+1,or equivalently(43)fx=12πn∫ℝ+×ℝn−1×ℝn×SpinnCSΨfa,r,s,bΨa,s,brxCΨ−1dadn−1sdnbdran+1.
This completes the proof of Theorem3.
The next theorem presents a characterization of the range of the Clifford-valued shearlet transformCSℋΨ. The result follows as a consequence of the reconstruction formula (40) and the well known Fubini theorem.Theorem 4.
(Characterization of range ofCSΨ). If h=CSΨf∈L2G,Clp,q, let Ψ be an admissible Clifford-valued shearlet. Then, h is a Clifford-valued shearlet transform of a function f∈L2G,Clp,q if and only if it satisfies the reproducing property:(44)ha′,r′,s′,b′=12πn∫Gha,r,s,bΨa,s,brCΨ−1,Ψa′,s′,b′r′L2ℝp,q,Clp,qdadn−1sdnbdran+1.Proof.
Leth belongs to the range of the Clifford-valued shearlet transform CSΨ. Then, there exist a Clifford-valued function f∈L2ℝp,q,Clp,q such that CSΨf=h. In order to show that h satisfies equation (44), we proceed as(45)ha′,r′,s′,b′=CSΨfa′,r′,s′,b′=∫ℝp,qfxΨa′,s′,b′r′¯dnx=∫ℝp,q12πn∫GCSΨfa,r,s,bΨa,s,brxCΨ−1dadn−1sdnbdran+1Ψa′,s′,b′r′¯dnx=12πn∫GCSΨfa,r,s,b∫ℝp,qΨa,s,brxCΨ−1Ψa′,s′,b′r′¯dnxdadn−1sdnbdran+1=12πn∫GCSΨfa,r,s,bΨa,s,brCΨ−1,Ψa′,s′,b′r′L2ℝp,q,Clp,qdadn−1sdnbdran+1=12πn∫Gha,r,s,bΨa,s,brCΨ−1,Ψa′,s′,b′r′L2ℝp,q,Clp,qdadn−1sdnbdran+1.
Conversely, suppose that an arbitrary functionh∈L2G,Clp,q satisfies equation (44). Then, we show that there exists f∈L2ℝp,q,Clp,q, such that CSΨf=h. Assume that(46)fx=12πn∫Gha,r,s,bΨa,s,brxCΨ−1dadn−1sdnbdran+1.
Then, it can be easily verified that(47)fL2ℝp,q,Clp,q2=12π2nCΨ−12Ψa,s,brL2ℝp,q,Clp,q2ha,r,s,bL2G,Clp,q2,which implies that f∈L2ℝp,q,Clp,q. Moreover, as a consequence of the well-known Fubini theorem and inversion Theorem (40), we have(48)CSΨfa′,r′,s′,b′=∫ℝp,qfxΨa′,s′,b′r′x¯dnx=∫ℝp,q12πn∫GCSΨfa,r,s,bΨa,s,brCΨ−1dadn−1sdnbdran+1Ψa′,s′,b′r′x¯dnx=12πn∫GCSΨfa,r,s,b∫ℝp,qΨa,s,brCΨ−1Ψa′,s′,b′r′x¯dnxdadn−1sdnbdran+1=12πn∫Gha,r,s,bΨa,s,brCΨ−1,Ψa′,s′,b′r′L2ℝp,q,Clp,qdadn−1sdnbdran+1=ha′,r′,s′,b′.
This evidently completes the proof of theorem.Corollary 2.
For an admissible Clifford shearletΨ∈L2ℝp,q,Clp,q, the range of the Clifford shearlet transform 28) is a reproducing kernel in L2ℝp,q,Clp,q with kernel that can be given by(49)KΨa,r,s,b,a′,r′,s′,b′=12πnΨa,s,brCΨ−1,Ψa′,s′,b′r′L2ℝp,q,Clp,q.


## 4. HAP Property for the Clifford-Valued Shearlet Transforms

Homogeneous approximation property (HAP) means that the approximation rate in a reconstruction of signal is essentially invariant under time-scale shifts. The HAP is being extensively used for studying frame density [17]. In this section, we investigate the homogeneous approximation property for the proposed Clifford-valued shearlet transforms. Initially, we shall present some results related to the pointwise convergence of the reconstruction formula (40).Theorem 5.
LetCSΨfa,r,s,b be the Clifford-valued shearlet transform of any f∈L2ℝp,q,Clp,q such that(50)fM,Nx=12πn∫MN∫ℝn×ℝn−1×SpinnCSΨfa,r,s,bΨa,s,brxCΨ−1drdn−1sdnbdaan+1,N>M>0,where Ψ∈L2ℝp,q,Clp,q is an admissible Clifford-valued shearlet with CΨ≠0, real valued. Then, we have(51)ℱClfM,Nξ=ℱClfξ∫MN∫ℝn−1×SpinnℱClrΨ⋅r¯rSsAaξ′r¯2CΨ−1drdn−1sdaan2−n+1/n.Proof.
ForM,N∈ℝ+, we define(52)fM,Nx=12πn∫MN∫ℝn×ℝn−1×SpinnCSΨfa,r,s,bΨa,s,brxCΨ−1drdn−1sdnbdaan+1.
Then, the application of Schwartz’s inequality implies that(53)∫MN∫ℝn×ℝn−1×SpinnCSΨfa,r,s,bΨa,s,brxCΨ−1drdn−1sdnbdaan+1≤∫MN∫ℝn×ℝn−1×SpinnCSΨfa,r,s,b2drdn−1sdnb1/2×∫ℝn×ℝn−1×SpinnΨa,s,brx2drdn−1sdnb1/2CΨ−1daan+1=∫MN∫ℝn×ℝn−1×SpinnCSΨfa,r,s,b2drdn−1sdnb1/2×Ψa,s,brxL2Rn×Rn−1×SpinnCΨ−1daan+1≤∫MN∫ℝn×ℝn−1×SpinnCSΨfa,r,s,b2drdn−1sdnbdaan+11/2×ΨL2Rn×Rn−1×SpinnCΨ−1∫MNdaan+11/2≤2πnfCΨ,fL2ℝp,q,Clp,q1/2ΨL2Rn×Rn−1×SpinnCΨ−11nM−n−N−n1/2<∞.
This shows thatfM,N is well defined on ℝ2.
Next, we show thatfM,N is uniformly continuous on ℝn. For any x,x′∈ℝn, we have(54)fM,Nx−fM,Nx′=12πn∫MN∫ℝn×ℝn−1×SpinnCSΨfa,r,s,bΨa,s,brx−Ψa,s,brx′CΨ−1drdn−1sdnbdaan+1≤12πn∫MN∫ℝn×ℝn−1×SpinnCSΨfa,r,s,b2drdn−1sdnbdaan+11/2×12πn∫MN∫ℝn×ℝn−1×SpinnΨa,s,brx−Ψa,s,brx′2drdn−1sdnbdaan+11/2CΨ−1≤12πn/2fCΨ,fL2ℝp,q,Clp,q1/2×∫MN∫ℝn×ℝn−1×SpinnΨa,s,brx−Ψa,s,brx′2drdn−1sdnbdaan+11/2CΨ−1.
From equation (54), we observe that fM,Nx−fM,Nx′⟶0 as x−x′⟶0. Thus, we conclude that fM,N is uniformly continuous on ℝp,q.
Moreover, for anyg∈L1∩L2ℝp,q,Clp,q, we have(55)fM,N,gL2ℝn,Clp,q=∫ℝnScfM,Nxgx¯dnx=Sc∫ℝn12πn∫MN∫ℝn×ℝn−1×SpinnCSΨfa,r,s,bΨa,s,brxCΨ−1drdn−1sdnbdaan+1gx¯dnx=12πn∫MN∫ℝn×ℝn−1×SpinnScCSΨfa,r,s,b∫ℝnΨa,s,brxgx¯dnxCΨ−1drdn−1sdnbdaan+1=12πn∫MN∫ℝn×ℝn−1×SpinnScCSΨfa,r,s,bCSΨga,r,s,b¯dnxCΨ−1drdn−1sdnbdaan+1=12πnSc∫MN∫ℝn×ℝn−1×Spinna1−1/2n∫ℝnℱClfξeIυξ,bℱClrΨ⋅r¯rSsAaξr¯¯dnξ×a1−1/2n∫ℝnℱClgξ′eIυξ,bℱClrΨ⋅r¯rSsAaξ′r¯¯dnξ′¯CΨ−1drdn−1sdnbdaan+1=12πnSc∫MN∫ℝn×ℝn−1×Spinn∫ℝnℱClfξeIυξ,bℱClrΨ⋅r¯rSsAaξr¯¯dnξ×∫ℝnℱClrΨ⋅r¯rSsAaξ′r¯e−Iυξ′,bℱClgξ′¯dnξ′CΨ−1drdn−1sdnbdaan2−n+1/n=12πnSc∫MN∫ℝn×ℝn−1×Spinn∫ℝn∫ℝnℱClfξeIυξ,be−Iυξ′,bℱClrΨ⋅r¯rSsAaξr¯¯dnξ×ℱClrΨ⋅r¯rSsAaξ′r¯ℱClgξ′¯dnξ′CΨ−1drdn−1sdnbdaan2−n+1/n=Sc∫MN∫ℝn×ℝn−1×Spinn∫ℝnℱClfξ12πn∫ℝneIυξ−ξ′,bdnbℱClrΨ⋅r¯rSsAaξr¯¯dnξ×ℱClrΨ⋅r¯rSsAaξr¯ℱClgξ′¯dnξ′CΨ−1drdn−1sdnbdaan2−n+1/n=Sc∫MN∫ℝn×ℝn−1×Spinn∫ℝnℱClfξδξ−ξ′ℱClrΨ⋅r¯rSsAaξr¯¯×ℱClrΨ⋅r¯rSsAaξ′r¯ℱClgξ′¯dnξ′CΨ−1drdn−1sdaan2−n+1/n=Sc∫MN∫ℝn×ℝn−1×Spinn∫ℝnℱClfξℱClrΨ⋅r¯rSsAaξr¯¯×ℱClgξℱClrΨ⋅r¯rSsAaξr¯¯¯dnξCΨ−1drdn−1sdaan2−n+1/n=∫MN∫ℝn−1×Spinn∫ℝnScℱClfξℱClrΨ⋅r¯rSsAaξr¯2ℱClgξ¯dnξCΨ−1drdn−1sdaan2−n+1/n=∫ℝnScℱClfξ∫MN∫ℝn−1×SpinnℱClrΨ⋅r¯rSsAaξr¯2CΨ−1drdn−1sdaan2−n+1/nℱClgξ¯dnξ.
Invoking scalar part for the Clifford Fourier transform, we can deduce that(56)ℱfM,Nξ=ℱClfξ∫MN∫ℝn−1×SpinnℱClrΨ⋅r¯rSsAaξr¯2CΨ−1drdn−1sdaan2−n+1/n.
This completes the proof of Theorem5.Theorem 6.
LetΨ∈L2ℝp,q,Clp,q be an admissible Clifford-valued shearlet. Then, for any f∈L1∩L2ℝp,q,Clp,q), we have(57)limM⟶0,N⟶∞f−fM,N∞=0,limM⟶0,N⟶∞f−fM,N2=0.Proof.
Using Parseval’s formula for the Clifford Fourier transforms together with an application of Theorem5, we have(58)f−fM,NL∞ℝn,Clp,q≤f−fM,NL1ℝn,Clp,q=ℱClfξ−ℱClfM,NξL1ℝn,Clp,q=‖ℱClfξ−ℱClfξ∫MN∫ℝn−1×Spinn×ℱClrΨ⋅r¯rSsAaξr¯2drdn−1sdaan2−n+1/nCΨ−1‖L1ℝn,Clp,q=‖ℱClfξ1−∫MN∫ℝn−1×Spinn×ℱClrΨ⋅r¯rSsAaξr¯2drdn−1sdaan2−n+1/nCΨ−1‖L1ℝn,Clp,q=∫ℝnℱClfξ‖1−∫MN∫ℝn−1×Spinn×ℱClrΨ⋅r¯rSsAaξr¯2drdn−1sdaan2−n+1/nCΨ−1|dnξ.
SinceΨ is given to be admissible, it follows that(59)∫MN∫ℝn−1×SpinnℱClrΨ⋅r¯rSsAaξr¯2drdn−1sdaan2−n+1/n≤∫ℝ+∫ℝn−1×SpinnℱClrΨ⋅r¯rSsAaξr¯2drdn−1sdaan2−n+1/n=CΨ<∞.
Therefore, we have(60)limM⟶0,N⟶∞1−∫MN∫ℝn−1×SpinnℱClrΨ⋅r¯rSsAaξr¯|2drdn−1sdaan2−n+1/nCΨ−1=0.
Using dominated convergence theorem in equation (58), we conclude that(61)limM⟶0,N⟶∞f−fM,NL∞ℝn,Clp,q=0.
Proceeding in a manner similar to the above case, we can show that(62)limM⟶0,N⟶∞f−fM,NL2ℝn,Clp,q=0.
This completes the proof of Theorem6.
In the sequel, we study the homogeneous approximation property for the proposed Clifford-valued shearlet transforms. Prior to that, we introduce some notations as given below:
For everya′,r′,s′,b′∈L2ℝ+×Spinn×ℝn−1×ℝn,Clp,q and M>N,P>0, we denote(63)QM,N;P=−N,−M∪N,M×Spinn×−P,Pn−1×−P,Pn,a′,s′,b′,r′QM,N;P=a′,s′,b′,r′a,s,b,r=a′a,s′+a′1−1/ns+s′,b′+Ss′Aa′b,r′r,where a∈−N,−M∪N,M,r∈Spinn,s∈−P,Pn−1 and b∈−P,Pn.Theorem 7.
LetΨ∈L2ℝp,q,Clp,qbe an admissible Clifford-valued shearlet with CΨ≠0, real valued. Then, for any f∈L2ℝp,q,Clp,q and ε>0, there exist some constants N>M>0,P>0, such that for any a′,r′,s′,b′∈L2ℝ+×Spinn×ℝn−1×ℝn,Clp,q, with any 0<M′≤M,N≤N′ and P′≥P, we have(64)fa′,s′,b′r′−∫a,s,b,r∈Q′fa′,s′,b′r′,Ψa,s,brCΨ−1Ψa,s,brdadn−1sdnbdran+1L2ℝp,q,Clp,q2<ε,where a′,s′,b′,r′QM′,N′;P′=Q′.Proof.
For an arbitraryg∈L2ℝp,q,Clp,q, we have(65)fa′,s′,b′r′−∫a,s,b,r∈Q′fa′,s′,b′r′,Ψa,s,brCΨ−1Ψa,s,brdadn−1sdnbdran+1L2ℝp,q,Clp,q2=sup‖g‖=1fa′,s′,b′r′−∫a,s,b,r∈Q′fa′,s′,b′r′,Ψa,s,brCΨ−1Ψa,s,brdadn−1sdnbdran+1,g2=sup‖g‖=1fa′,s′,b′r′,g∫a,s,b,r∈Q′fa′,s′,b′r′,Ψa,s,brCΨ−1Ψa,s,brdadn−1sdnbdran+1,g2=sup‖g‖=1fa′,s′,b′r′,g−∫a,s,b,r∈Q′fa′,s′,b′r′,Ψa,s,brCΨ−1Ψa,s,brdadn−1sdnbdran+12=sup‖g‖=1∫a,s,b,r∉Q′fa′,s′,b′r′,Ψa,s,brCΨ−1Ψa,s,brdadn−1sdnbdran+12≤sup‖g‖=1∫a,s,b,r∉Q′fa′,s′,b′r′,Ψa,s,brCΨ−12dadn−1sdnbdran+1×∫a,s,b,r∉Q′Ψa,s,br,g2dadn−1sdnbdran+1=∫a,s,b,r∉Q′fa′,s′,b′r′,Ψa,s,br2CΨ−12dadn−1sdnbdran+1×sup‖g‖=1∫a,s,b,r∉Q′g,Ψa,s,br2dadn−1sdnbdran+1=∫a,s,b,r∉Q′fa′,s′,b′r′,Ψa,s,br2CΨ−12dadn−1sdnbdran+1×sup‖g‖=1∫a,s,b,r∉Q′CSΨga,s,b,r2dadn−1sdnbdran+1=∫a,s,b,r∉Q′fa′,s′,b′r′,Ψa,s,br2CΨ−12dadn−1sdnbdran+1×CΨ=∫a,s,b,r∉QM′,N′;P′f,Ψa,s,br2dadn−1sdnbdran+1CΨ−12×CΨ.
By choosingN and P large enough and M arbitrary small, we can make R. H. S as small as we need. This completes the proof of Theorem 7.


## 5. Uncertainty Principles for the Clifford-Valued Shearlet Transforms

In this section, we shall establish several uncertainty inequalities including Heisenberg–Pauli–Weyl uncertainty inequality, Pitt’s inequality, and logarithmic and local uncertainty inequality for the Clifford-valued shearlet transform as defined by equation (28). Prior to establishing the uncertainty principle for the Clifford-valued shearlet transform, we have the following lemma which shall be employed for deriving certain uncertainty inequalities and whose proof follows directly from the Parseval’s and inversion formulae of the Clifford Fourier transforms.Lemma 1.
LetΨ∈L2ℝp,q,Clp,q be an admissible Clifford-valued shearlet. Then, for any f∈L2ℝp,q,Clp,q), we have(66)ℱClCSΨfa,r,s,bξ=2πn/2a1−1/2nℱClfξℱClrΨ⋅r¯rSsAaξr¯¯.Theorem 8.
(Heisenberg–Weyl inequality). LetCSΨfa,r,s,b be the Clifford-valued shearlet transform of any Clifford-valued function f∈L2ℝp,q,Clp,q). Then, the following inequality follows(67)bCSΨfa,r,s,bL2G,Clp,qξℱClfξCΨL2ℝp,q,Clp,q≥12fxCΨ,fxL2G,Clp,q.Proof.
For any Clifford-valued functionf∈L2ℝp,q,Clp,q, the Heisenberg–Paul–Weyl inequality for the Clifford Fourier transforms [8, 18] is given by(68)∫ℝnb2fb2dnb1/2∫ℝnξ2ℱClfξdnξ1/2≥122πn/2∫ℝnfb2dnb.
ConsideringCSΨfa,r,s,b as a function of b and replacing f by CSΨfa,r,s,b in (68), we get(69)∫ℝnb2CSΨfa,r,s,b2dnb1/2∫ℝnξ2ℱClCSΨfa,r,s,bξ2dnξ1/2≥122πn/2∫ℝnCSΨfa,r,s,b2dnb.
We now integrate the above inequality with respect to measuredrdn−1sda/an+1, and using Schwartz inequality, to obtain(70)∫ℝ+×ℝn−1×Spinn∫ℝnb2CSΨfa,r,s,b2dnbdrdn−1sdaan+11/2×∫ℝ+×ℝn−1×Spinn∫ℝnξ2ℱcCSΨfa,r,s,bξ2dnξdrdn−1sdaan+11/2≥122πn/2∫ℝ+×ℝn−1×Spinn∫ℝnCSΨfa,r,s,b2drdn−1sdaan+1dnb.
Using Lemma1 together with Fubini theorem, we obtain(71)∫ℝn×ℝ+×ℝn−1×Spinnb2CSΨfa,r,s,b2dnbdrdn−1sdaan+11/2×∫ℝ+×ℝn−1×Spinn∫ℝnξ22πn/2a1−1/2nℱClfξℱClrΨ⋅r¯rSsAaξr¯¯2dnξdrdn−1sdaan+11/2≥122πn/2∫ℝn×ℝ+×ℝn−1×SpinnCSΨfa,r,s,b2drdn−1sdaan+1dnb.
Equivalently, we have(72)∫Gb2CSΨfa,r,s,b2dη1/2×∫ℝnξ2ℱClfξ2∫ℝ+×ℝn−1×SpinnℱClrΨ⋅r¯rSsAaξr¯¯2dnξdrdn−1sdaan2−n+1/ndnξ1/2≥122πn∫GCSΨfa,r,s,b2dη.
Using the definition ofCΨ in L. H. S and Corollary 1 in R. H. S, we obtain the desired result as follows(73)bCSΨfa,r,s,bL2G,Clp,qξℱClfξCΨL2ℝp,q,Clp,q≥12fxCΨ,fxL2G,Clp,q.
This completes the proof of Theorem8.Remark 2.
For real-valuedCΨ, Theorem 5 boils down to(74)bCSΨfa,r,s,bL2G,Clp,qξℱClfξL2ℝp,q,Clp,q≥CΨ2fxL2G,Clp,q2.
The classical Pitt’s inequality expresses a fundamental relationship between a sufficiently smooth functionf and the corresponding Clifford Fourier transform [19]. We derive the Pitt’s type inequality for the proposed Clifford-valued shearlet transform (28). The Schwartz space on Cℓp,q algebras is given by(75)Sℝp,q,Clp,q=f∈C∞ℝp,q,Clp,q:supt∈ℝp,qtα∂tβft<∞,where C∞ℝp,q,Clp,q is the class of smooth functions, and α,β denote multiindices, and ∂t denotes the usual partial differential operator.Theorem 9.
(Pitt’s inequality forCSΨ). For any f∈Sℝp,q,Clp,q, the Pitt’s inequality for the Clifford-valued shearlet transform (28) is given by(76)∫ℝnξ−λℱClfξ2dnξ≤Cλ2π2∫GbλCSΨfa,r,s,b2CΨ−1dη,where CΨ is the admissibility condition of Clifford-valued shearlet, and Cλ is given by(77)Cλ=πλΓ′n−λ/4Γn+λ/42,0≤λ<n,where Γ⋅ denotes the well-known Euler’s gamma function.Proof.
ConsideringCSΨfa,r,s,b as a function of the translation variable b, the Pitt’s inequality in the Clifford Fourier domain implies 13:(78)∫ℝnξ−λℱCℓCSΨfa,r,s,bξ2dnξ≤Cλ2πn∫ℝnbλCSΨfa,r,s,b2dnb,which upon integration with respect to the measure drdn−1sda/an+1 yields(79)∫ℝ+×ℝn−1×Spinn∫ℝnξ−λℱClCSψfa,r,s,bξ2dnξdrdn−1sdaan+1≤Cλ2πn∫ℝ+×ℝn−1×Spinn∫ℝnbλCSΨfa,r,s,b2dnbdrdn−1sdaan+1.
Invoking Lemma1, we can express the inequality (79) in the following manner:(80)∫ℝ+×ℝn−1×Spinn∫ℝnξ−λ2πn/2a1−1/2nℱClfξℱClrΨ⋅r¯rSsAaξr¯¯2dnξdrdn−1sdaan+1≤Cλ2πn∫ℝ+×ℝn−1×Spinn∫ℝnbλCSΨfa,r,s,b2dnbdrdn−1sdaan+1.
Equivalently, we have(81)∫ℝnξ−λℱClfξ2∫ℝ+×ℝn−1×SpinnℱClrΨ⋅r¯rSsAaξr¯2drdn−1sdaan2−n+1/ndnξ≤Cλ2π2n∫ℝ+×ℝn−1×Spinn∫ℝnbλCSΨfa,r,s,b2dnbdrdn−1sdaan+1.
SinceΨ is an admissible Clifford shearlet, inequality (81) boils down to(82)∫ℝnξ−λℱClfξ2CΨdnξ≤Cλ2π2n∫GbλCSΨfa,r,s,b2dη.which is the desired Pitt’s inequality for the Clifford-valued shearlet transform.Remark 3.
Forλ=0, equality which holds in equation (76) is equivalent to equation (39).
Next, we shall formulate the logarithmic uncertainty principle for the Clifford-valued shearlet transformCSΨfa,r,s,b given by equation (28).Theorem 10.
(Logarithmic uncertainty principle). For anyf∈Sℝp,q,Clp,q, the Clifford-valued shearlet transform CSΨfa,r,s,b satisfies the following logarithmic estimate of the uncertainty inequality:(83)12πn∫GCSΨfa,r,s,b2lnbdη+2πn∫ℝnℱClfξ2CΨlnξdnξ≥Γ′n/4Γn/4−lnπfCΨ,fL2G,Clp,q,provided the left hand side of this inequality is defined.Proof.
For the Clifford-valued functionf∈Sℝp,q,Clp,q, the logarithmic uncertainty inequality in the Clifford Fourier domain yields [18](84)∫ℝnfb2lnbdnb+2πn∫ℝnℱClfξ2lnξdnξ≥Γ′n/4Γn/4−lnπ∫ℝnfb2dnb.
Upon replacingfb by CSΨfa,r,s,b in the above inequality, we obtain(85)∫ℝnCSΨfa,r,s,b2lnbdnb+2πn∫ℝnℱClCSΨfa,r,s,bξ2lnξdnξ≥Γ′n/4Γn/4−lnπ∫ℝnCSΨfa,r,s,b2dnb.
Integrating equation (85) with respect to measure drdn−1sda/an+1 and then invoking the Fubini theorem, we obtain(86)∫GCSΨfa,r,s,b2lnbdnbdrdn−1sdaan+1+2πn∫GℱClCSΨfa,r,s,bξ2lnξdnξdrdn−1sdaan+1≥Γ′n/4Γn/4−lnπ∫GCSΨfa,r,s,b2dnbdrdn−1sdaan+1.
Using Lemma1, the inequality (86) can be further simplified as(87)∫GCSΨfa,r,s,b2lnbdnbdrdn−1sdaan+1+2πn∫G2πn/2a1−1/2nℱClfξℱClrΨ⋅r¯rSsAaξr¯¯2lnξdnξdrdn−1sdaan+1×lnξdnξdrdn−1sdaan+1≥Γ′n/4Γn/4−lnπ∫GCSΨfa,r,s,b2dnbdrdn−1sdaan+1.
Alternatively, the above inequality can be rewritten as(88)∫GCSΨfa,r,s,b2lnbdη+2π2n∫ℝnℱClfξ2×∫ℝ+×ℝn−1×SpinnℱClrΨ⋅r¯rSsAaξr¯2drdn−1sdaan2−n+1/nlnξdnξ≥Γ′n/4Γn/4−lnπ∫GCSΨfa,r,s,b2dη.
Noting thatΨ is admissible and using Corollary 1, we obtain the desired result as(89)12πn∫GCSΨfa,r,s,b2lnbdη+2πn∫ℝnℱClfξ2CΨlnξdnξ≥Γ′n/4Γn/4−lnπfCΨ,fL2G,Clp,q.
This completes the proof of Theorem10.
In the following, we establish a local-type uncertainty principle for the Clifford-valued sharelet transformCSΨf defined by equation (28). More precisely, we shall demonstrate that the portion of CSΨ lying outside some given set ℳ of finite Lebesgue measure cannot be arbitrarily small.Theorem 11.
(Concentration ofCSΨ in small sets). Let Ψ∈L2ℝp,q,Clp,q be an admissible Clifford-valued shearlet satisfying 0<a1/2n−1Ψ2μℳ/CΨ<1. Then, for any measurable subset ℳ of G=ℝ+×ℝn−1×ℝn×Spinn and f∈L2ℝp,q,Clp,q, we have(90)CSΨfa,r,s,bL2Ec,Clp,q≥CΨ1−a1/2n−1μℳΨL2ℝp,q,Clp,q2CΨ1/2fL2ℝp,q,Clp,q,where μℳ denotes the measure of ℳ.Proof.
Using the definition of Clifford-valued shearlet transforms, we have(91)CSΨfa,r,s,bL2ℝp,q,Clp,q=a1/2n−1∫ℝp,qfxr¯ΨAa−1Ss−1r¯x−br¯rdnxL2ℝp,q,Clp,q≤a1/2n−1∫ℝp,qfxr¯ΨAa−1Ss−1r¯x−br¯rdnx.
By virtue of Holders inequality, we have(92)CSΨfa,r,s,bL2ℝp,q,Clp,q≤a1/2n−1fL2ℝp,q,Clp,qΨL2ℝp,q,Clp,q.
On the other hand, we can write(93)CSΨfa,r,s,bL2G,Clp,q2=∫∭GCSΨfa,r,s,bL2ℝp,q,Clp,q2dη=∫∭ℳCSΨfa,r,s,bL2ℝp,q,Clp,q2dη+∫∭ℳcCSΨfa,r,s,bL2ℝp,q,Clp,q2dη≤a1/2n−1μℳfL2ℝp,q,Clp,q2ΨL2ℝp,q,Clp,q2+CSΨfa,r,s,bL2Ec,Clp,q2.
Application of Corollary1 for the real-valued CΨ implies that(94)CΨfL2ℝp,q,Clp,q2≤a1/2n−1μℳfL2ℝp,q,Clp,q2ΨL2ℝp,q,Clp,q2+CSΨfa,r,s,bL2Ec,Clp,q2,or(95)CSΨfa,r,s,bL2Ec,Clp,q≥CΨ−a1/2n−1μℳΨL2ℝp,q,Clp,q21/2fL2ℝp,q,Clp,q=CΨ1−a1/2n−1μℳΨL2ℝp,q,Clp,q2CΨ1/2fL2ℝp,q,Clp,q.
This completes the proof of Theorem11.


## 6. Conclusion

In the present study, we formulated the notion of continuous Clifford-valued shearlet transform on the generalized geometric algebra Clp, q. The proposed transform has the advantage of efficiently handling Clifford-valued signals at various scales, positions and orientations while upholding the affine structure. Besides, studying the fundamental aspects of the Clifford-valued shearlet transform, the homogeneous approximation property is also investigated in detail. Nevertheless, some prominent uncertainty inequalities, such as the Hesienberg–Puali–Weyl logarithmic and local uncertainty principles are obtained at the end.



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*Source: 7848503-2022-09-02.xml*