Generalized Uncertainty Principles associated with the Quaternionic Offset Linear Canonical Transform
aa r X i v : . [ m a t h . C A ] S e p Generalized Uncertainty Principles associatedwith the Quaternionic Offset Linear CanonicalTransform
Youssef El Haoui , * , Said Fahlaoui , Eckhard Hitzer Department of Mathematics and Computer Sciences, Faculty of Sciences, University Moulay Ismail,Meknes 11201, Morocco Dr. rer. nat. of Theoretical Physics, International Christian University,Osawa 3-10-2, Mitaka-shi 181-8585 Tokyo, JapanE-MAIL: [email protected], [email protected], [email protected]
Abstract
The quaternionic offset linear canonical transform (QOLCT) can be defined as a generalization ofthe quaternionic linear canonical transform (QLCT). In this paper, we define the QOLCT, we derive therelationship between the QOLCT and the quaternion Fourier transform (QFT). Based on this fact, weprove the Plancherel formula, and some properties related to the QOLCT. Then, we generalize somedifferent uncertainty principles (UPs), including Heisenberg-Weyls UP, Hardys UP, Beurlings UP, andlogarithmic UP to the QOLCT domain in a broader sense.
Key words:
Quaternion Fourier transform; Quaternionic linear canonical transform; Quaternionic Off-set Linear Canonical Transform, Uncertainty principle.
The QFT plays an important role in the representation of signals. It transforms a real (or quaternionic)2D signal into a quaternion valued frequency domain signal. In [7], the authors provide the Heisenberg’sinequality and the Hardy’s UP for the two-sided QFT. The authors in [10] generalize the Beurling’s UP tothe QFT domain. It is well known that the LCT provides a more general framework for a number of famouslinear integral transforms in signal processing and optics, such as Fourier transform FT, the fractional FT,the Fresnel transform, the Lorentz transform.The LCT was extended to the Clifford analysis by Kit Ian Kou et al [19] in the 2013s, to study the general-ized prolate spheroidal wave functions and their connection with energy concentration problems.In [20], the authors introduced the quaternion linear canonical transform (QLCT), which is a generalizationof the LCT in the framework of quaternion algebra.Several properties, such as the Parseval’s formula, and UP associated with the QLCT are established.In view of the fact that the OLCT is a generalization of the LCT, and has wide applications in signal pro-cessing and optics, one is interested to extend the OLCT to a quaternionc algebra framework.To the best of our knowledge, the generalization of the OLCT to a quaternionic algebra, and the study ofthe properties and UPs associated with this generalization, have not been carried out yet. Therefore, theresults in this paper are new in the literature.The main objective of the present study is to develop further technical methods in the theory of partialdifferential equations [9].In the present work, we study the QOLCT that transforms a real (or quaternionic) 2D signal into aquaternion-valued frequency domain signal. Some important properties of the two-sided QOLCT areestablished. A well known UPs for the two-sided QOLCT are generalized. * Corresponding author.
The quaternion algebra
In the present section we collect some basic facts about quaternions, which will be needed throughoutthe paper. For all what follows, let H be the Hamiltonian skew field of quaternions: H = { q = q + iq + jq + kq ; q , q , q , q ∈ R } which is an associative noncommutative four-dimensional algebra.where the elements i, j, k satisfy the Hamilton’s multiplication rules: ij = − ji = k ; jk = − kj = i ; ki = − ik = j ; i = j = k = − . In this way the quaternionic algebra can beseen as an extension of the complex field C .Quaternions are isomorphic to the Clifford algebra Cl ( , ) of R ( , ) : H ≅ Cl ( , ) (2.1)The scalar part of a quaternion q ∈ H is q denoted by Sc ( q ) , the non scalar part(or pure quaternion)of q is iq + jq + kq denoted by V ec ( q ) .The quaternion conjugate of q ∈ H , given by q = q − iq − jq − kq .is an anti-involution, namely, qp = p q, p + q = p + q, p = p. The norm or modulus of q ∈ H is defined by ∣ q ∣ Q = √ qq = √ q + q + q + q . Then, we have ∣ pq ∣ Q = ∣ p ∣ Q ∣ q ∣ Q . In particular, when q = q is a real number, the module ∣ q ∣ Q reduces to the ordinary Euclidean module ∣ q ∣ = √ q .It is easy to verify that ≠ q ∈ H implies : q − = q ∣ q ∣ Q . Any quaternion q can be written as q = ∣ q ∣ Q e µθ where e µθ is understood in accordance with Euler’sformula e µθ = cos ( θ ) + µ sin ( θ ) , where θ = artan ∣ V ec ( q )∣ Q Sc ( q ) , 0 ≤ θ ≤ π and µ := V ec ( q )∣ V ec ( q )∣ Q verifying µ = − .In this paper, we will study the quaternion-valued signal f ∶ R → H , f which can be expressed as f = f + if + jf + kf , f m ∶ R → R f or m = , , , . Let us introduce the canonical inner product for quaternion valuedfunctions f, g ∶ R → H , as follows: < f, g >= ∫ R f ( t ) g ( t ) dt, dt = dt dt . Hence, the natural norm is given by ∣ f ∣ ,Q = √< f, f > = ( ∫ R ∣ f ( t )∣ Q dt ) , and the quaternion module L ( R , H ) , is given by L ( R , H ) = { f ∶ R → H , ∣ f ∣ ,Q < ∞} . We denote by S ( R , H ) , the quaternion Schwartz space of C ∞ - functions f , from R to H , such thatfor all m, n ∈ N sup t ∈ R ,α + α ≤ n (( + ∣ t ∣) m ∣ ∂ α + α ∂t α ∂t α f ( t )∣ Q ) < ∞ , where ( α , α ) ∈ N . Besides the quaternion units i, j, k , we will use the following real vector notation: t = ( t , t ) ∈ R , ∣ t ∣ = t + t , f ( t ) = f ( t , t ) , dt = dt dt , and so on. The QFT which has been defined by Ell [8], is a generalization of the classical Fourier transform (CFT)using a quaternionic algebra framework. Several known and useful properties, and theorems of this ex-tended transform are generalizations of the corresponding properties, and theorems of the CFT with somemodifications (e.g., [5], [6], [13], [8]). The QFT belongs to the family of Clifford Fourier transformationsbecause of (2.1). There are three different types of QFT, the left-sided QFT , the right-sided QFT, andtwo-sided QFT [21].Let us define the two-sided QFT and provide some properties used in the sequel.
Definition 2.1. (Two-sided QFT with respect to two pure unit quaternions λ ; µ [17])Let λ, µ ∈ H , λ = µ = − , be any two pure unit quaternions.For f in L ( R , H ) , the two-sided QFT with respect to λ ; µ is F λ,µ { f }( u ) = ∫ R e − λu t f ( t ) e − µu t dt, where t, u ∈ R . (2.2)We define a new module of F { f } λ,µ as follows : ∥ F λ,µ { f }∥ Q ∶= ¿ÁÁÀ m = ∑ m = ∣ F λ,µ { f m }∣ Q . (2.3)Furthermore, we define a new L -norm of F { f } as follows : ∥ F λ,µ { f }∥ ,Q ∶= √ ∫ R ∥ F λ,µ { f } ( y )∥ Q dy. (2.4)It is interesting to observe that ∥ F λ,µ { f }∥ Q is not equivalent to ∣ F λ,µ { f }∣ Q unless f is real valued. Lemma 2.2. (Dilation property), see example 2 on page 50 [5]Let k , k be a positive scalar constants, we have F λ,µ { f ( t , t )} ( u k , u k ) = k k F λ,µ { f ( k t , k t )} ( u , u ) .
3y following the proof of theorem 3.2 in [6], and replacing i by λ , j by µ we obtain the next lemma. Lemma 2.3. (QFT Plancherel)Let f ∈ L ( R , H ) , then ∫ R ∥ F λ,µ { f } ( u )∥ Q du = π ∫ R ∣ f ( t )∣ Q dt. Lemma 2.4. If f ∈ L ( R , H ) , ∂ m + n ∂t m ∂t n f exist and are in L ( R , H ) for m, n ∈ N , then F λ,µ { ∂ m + n ∂t m ∂t n f } ( u ) = ( λu ) m F λ,µ { f } ( u ) ( µu ) n . Proof. See [[5], Thm. 2.10].
Lemma 2.5.
Inverse QFT [15]If f ∈ L ( R , H ) , and F λ,µ { f } ∈ L ( R , H ) , then the two-sided QFT is an invertible transform and itsinverse is given by f ( t ) = ( π ) ∫ R e λu t F λ,µ { f ( t )}( u ) e µu t du. Kit Ian Kou et al [19] introduce the quaternionic linear canonical transform (QLCT). They consider apair of unit determinant two-by-two matrices A = [ a b c d ] , A = [ a b c d ] ∈ R × , with unit determinant, that is a d − b c = , a d − b c = , Eckhard Hitzer [14] generalize the definitions of [19] to be:The two-sided QLCT of signals f ∈ L ( R , H ) , is defined as L λ,µA ,A { f }( u ) = ∫ R K λA ( t , u ) f ( t ) K µA ( t , u ) dt. with λ, µ ∈ H , two pure unit quaternions, λ = µ = − , including the cases λ = ± µ,K λA ( t , u ) = √ λ πb e λ ( a t − t u + d u )/ b , K µA ( t , u ) = √ µ πb e µ ( a t − t u + d u )/ b , In [19], for λ = i and µ = j , the right-sided QLCT and its properties, including an UP are studied in somedetail.We now generalize the definitions of [17], [15] as follows: Definition 3.1.
Let A l = [∣ a l b l c l d l ∣ τ l η l ] ,parameters a l , b l , c l , d l , τ l , η l ∈ R such as a l d l − b l c l = , for l = , . The two-sided quaternionic offset linear canonical transform (QOLCT) of a signal f ∈ L ( R , H ) , isgiven by O λ,µA ,A { f ( t )}( u ) = ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ ∫ R K λA ( t , u ) f ( t ) K µA ( t , u ) dt, b , b ≠ , √ d e λ ( c d ( u − τ ) + u τ ) f ( d ( u − τ ) , t ) K µA ( t , u ) , b = , b ≠ , √ d K λA ( t , u ) f ( t , d ( u − τ )) e µ ( c d ( u − τ ) + u τ ) , b ≠ , b = , √ d d f (( d ( u − τ ) , d ( u − τ )) e λ ( c d ( u − τ ) + u τ ) e µ ( c d ( u − τ ) + u τ ) , b = b = , (3.1)4 λA ( t , u ) = √ λ πb e λ ( a t − t ( u − τ )− u ( d τ − b η )+ d ( u + τ )) b , for b ≠ ,and K µA ( t , u ) = √ µ πb e µ ( a t − t ( u − τ )− u ( d τ − b η )+ d ( u + τ )) b , for b ≠ , with √ λ = e − λ π , √ µ = e − µ π . The left-sided and right-sided QOLCTs can be defined correspondingly by placing the two kernelfactors both on the left or on the right, respectively.We note that when τ = τ = η = η =
0, the two-sided QOLCT reduces to the QLCT.Also, when A = A = [∣ − ∣ ] , the conventional two-sided QFT is recovered. Namely, O λ,µA ,A { f ( t )}( u ) = √ λ π ( ∫ R e − λt u f ( t ) e − µt u dt ) √ µ π = π e − λ π F λ,µ { f } ( u , u ) e − µ π , where F λ,µ { f } is the QFT of f given by (2.2).For simplicity’s sake, in this paper we restrict our attention to the two-sided QLCTs of 2D quaternion-valued signals. Note that when b b = b = b = b b ≠ in this paper, without loss of generality, we set b l > ( l = , ) , The following lemma gives the relationships of two-sided QOLCTs and two-sided QFTs of 2D quaternion-valued signals.
Lemma 3.2.
The QOLCT of a signal f ∈ L ( R , H ) can be reduced to the QFT O λ,µA ,A { f ( t )} ( u , u ) = F λ,µ { h ( t )} ( u b , u b ) , with h ( t ) = √ πλb e λ [ − b u ( d τ − b η ) + d b ( u + τ ) + b t τ + a b t ] f ( t ) e µ [ − b u ( d τ − b η ) + d b ( u + τ ) + b t τ + a b t ] √ πµb . (3.2)Proof. From the definition of the QOLCT, we have O λ,µA ,A { f ( t )} ( u , u ) = ∫ R K λA ( t , u ) f ( t ) K µA ( t , u ) dt = ∫ R √ πλb e λ [ a b t − b t ( u − τ ) − b u ( d τ − b η ) + d b ( u + τ )] f ( t ) √ πµb × e µ [ a b t − b t ( u − τ ) − b u ( d τ − b η ) + d b ( u + τ )] dt = ∫ R e − λ b t u [ √ πλb e λ [ − b u ( d τ − b η ) + d b ( u + τ ) + b t τ + a b t ] f ( t ) e µ [ − b u ( d τ − b η ) + d b ( u + τ ) + b t τ + a b t ] × √ πµb ] ] e − µ b t u dt = F λ,µ { h ( t )} ( u b , u b ) ◻ Due to the lemma 3.2 and proposition 3.1, theorem 3.1 in [6], the following properties are easily shown.
Theorem 3.3.
Let f ∈ L ( R , H ) . Then its QOLCT satisfies:The map f → O λ,µA ,A { f } is real linear. That is, for α, β ∈ R , we have O λ,µA ,A { αf + βg } = α O λ,µA ,A { f } + β O λ,µA ,A { g } . lim ∣ u ∣ → ∞ ∥ O λ,µA ,A { f } ( u )∥ Q = . O λ,µA ,A { f } is uniformly continuous on R . Following the proof of theorems 11 and 12 in [2], and by straightforward computation we derive shiftand modulation properties for the QOLCT
Theorem 3.4.
Let f ∈ L ( R , H ) , with t, u ∈ R , constants ξ = ( ξ , ξ ) , k = ( k , k ) ∈ R . We have:t-Shift property O λ,µA ,A { f ( t − k )}( u ) = λ [( k u − a k ) b c − k a ( d τ − b η )] b O λ,µA ,A { f ( t )}( u − k a , u − k a ) e µ [( k u − a k ) b c − k a ( d τ − b η )] b . Modulation property O λ,µA ,A { e λt ξ f ( t ) e µt ξ }( u ) = e − λ [ d ( b ξ − b u ) + ξ ( d τ − b η )] O λ,µA ,A { f ( t }( u − b ξ , u − b ξ ) e − µ [ d ( b ξ − b u ) + ξ ( d τ − b η )] . Theorem 3.5. If f and O λ,µA ,A { f } are in L ( R , H ) , then the inverse transform of the QOLCT can bederived from that of the QFT. Proof. Indeed, Let g ( t ) = e λ b t τ + λ a b t f ( t ) e µ b t τ + µ a b t . (3.3)We have O λ,µA ,A { f ( t )} ( u , u ) = √ πλb e − λ b u ( d τ − b η ) + λ d b ( u + τ ) F λ,µ { g ( t )} ( u b , u b ) × e − µ b u ( d τ − b η ) + µ d b ( u + τ ) √ πµb = e − λ ( a t − t ( u − τ )) b K λA ( t , u ) F λ,µ { g ( t )} ( u b , u b ) K µA ( t , u ) e − µ ( a t − t ( u − τ )) b . As K A m ( t m , u m ) K A m ( t m , u m ) = πb m , m = , . We easily obtain F λ,µ { g ( t )} ( u b , u b ) = ( π ) b b K λA ( t , u ) e λ ( a t − t ( u − τ )) b O λ,µA ,A { f ( t )} ( u , u ) K µA ( t , u ) × e µ ( a t − t ( u − τ )) b . From lemma 2.5, it follows that g ( t ) = ( π ) ∫ R e λt u F λ,µ { g ( t )} ( u ) e µt u du = b b ∫ R K λA ( t , b u ) e λ ( a t + t τ ) b O λ,µA ,A { f ( t )} ( b u , b u ) K µA ( t , b u ) × e µ ( a t + t τ ) b du. Or, equivalently e λ b t τ + λ a b t f ( t ) e µ b t τ + µ a b t = b b ∫ R K λA ( t , b u ) e λ ( a t + t τ ) a b O λ,µA ,A { f ( t )} ( b u , b u ) × K µ A ( t , b u ) e µ ( a t + t τ ) a b du. It means that f ( t ) = b b ∫ R K λA ( t , b u )O λ,µA ,A { f ( t )} ( b u , b u ) K µA ( t , b u ) du = ∫ R K λA ( t , u )O λ,µA ,A { f ( t )} ( u , u ) K µA ( t , u ) du. which is the inverse transform of the QOLCT. This proves the theorem. ◻ Theorem 3.6. (Plancherel’s theorem of the QOLCT)Every 2D quaternion-valued signal f ∈ L ( R , H ) and its QOLCT are related to the Plancherel identity inthe following way: ∥O λ,µA ,A { f }∥ Q, = ∣ f ∣ Q, . Proof. Let h ( t ) be rewritten in the form of (3.2).By the definition of the norm ∥ . ∥ Q, and lemma 2.2 and lemma 3.2, we have ∥ O λ,µA ,A { f }∥ Q, = ∫ R ∥ O λ,µA ,A { f } ( u )∥ Q du = ∫ R ∥ F λ,µ { h ( t )} ( u b , u b )∥ Q du = ∫ R ∥ b b F λ,µ { h ( b t , b t )} ( u , u )∥ Q du = b b ∫ R ∥ F λ,µ { h ( b t , b t )} ( u , u )∥ Q du. From lemma 2.3 we get ∫ R ∥ F λ,µ { h ( b t , b t )} ( u , u )∥ Q du = π ∫ R ∣ h ( b t , b t )∣ Q dt. Let s l = b l t l , for l = , , we have ∫ R ∣ h ( b t , b t )∣ Q dt = b b ∫ R ∣ h ( s , s )∣ Q ds = π b b ∫ R ∣ f ( s , s )∣ Q ds. The last statement follows from ∣ h ( t )∣ Q = π √ b b ∣ f ( t ) ∣ Q ,6herefore, we get ∥ O λ,µA ,A { f }∥ Q, = ∫ R ∣ f ( s )∣ Q ds = ∣ f ∣ Q, . This ends the proof. ◻ Lemma 3.7. If f ∈ L ( R , H ) , ∂ l + n ∂t l ∂t n f exist and are in L ( R , H ) for l, n ∈ N , then1. ∫ R u ∥ O λ,µA ,A { f } ( u )∥ Q du = b ∫ R ∣ λ ( a b t + τ b ) f ( t ) + ∂∂t f ( t )∣ Q dt, ∫ R u ∥ O λ,µA ,A { f } ( u )∥ Q du = b ∫ R ∣( a b t + τ b ) f ( t ) µ + ∂∂t f ( t )∣ Q dt. Proof. Let h ( t ) be rewritten in the form of (3.2). For the first statement, using lemma 2.4 shows that F λ,µ { ∂∂t f } ( u b , u b ) = λ u b F λ,µ { f } ( u b , u b ) , Then, using lemma 3.2, (2.3), lemma 2.3, and the above equality we get ∫ R u ∥ O λ,µA ,A { f } ( u )∥ Q du = ∫ R u ∥ F λ,µ { h ( t )} ( u b , u b )∥ Q du = ∫ R ∑ m = m = ∣ u F λ,µ { h m ( t )} ( u b , u b )∣ Q du = ∫ R ∑ m = m = ∣ λ b F λ,µ { ∂∂t h m } ( u b , u b )∣ Q du = b ∫ R ∥ F λ,µ { ∂∂t h } ( u b , u b )∥ Q du = b b ∫ R ∥ F λ,µ { ∂∂t h } ( u b , u b )∥ Q du b du b = π b b ∫ R ∣ ∂∂t h ( t )∣ Q dt. where the last equation is the consequence of using lemma 2.3.Moreover, using ∣ ∂∂t h ( t )∣ Q =∣ ∂∂t ( √ πλb e λ [ − b u ( d τ − b η ) + d b ( u + τ ) + b t τ + a b t ] f ( t ) e µ [ − b u ( d τ − b η ) + d b ( u + τ ) + b t τ + a b t ] √ πµb )∣ Q = π √ b b ∣ e λ [ − b u ( d τ − b η ) + d b ( u + τ ) + b t τ + a b t ] [ λ ( a b t + τ b ) f ( t ) + ∂∂t f ( t )]∣ Q = π √ b b ∣ λ ( a b t + τ b ) f ( t ) + ∂∂t f ( t ) ∣ Q . We further get ∫ R u ∥ O λ,µA ,A { f } ( u )∥ Q du = b ∫ R ∣ λ ( a b t + τ b ) f ( t ) + ∂∂t f ( t )∣ Q dt. To prove the statement 2, we argue in the same spirit as in the previous proof.Applying lemma 3.2, (2.3), lemma 2.4 and lemma 2.3, we have ∫ R u ∥ O λ,µA ,A { f } ( u )∥ Q du = ∫ R u ∥ F λ,µ { h ( t )} ( u b , u b )∥ Q du = ∫ R ∑ m = m = ∣ F λ,µ { h m ( t )} ( u b , u b ) µu ∣ Q du = ∫ R ∑ m = m = ∣ b F λ,µ { ∂∂t h m } ( u b , u b ) µ ∣ Q du = b ∫ R ∥ F λ,µ { ∂∂t h } ( u b , u b )∥ Q du = b b ∫ R ∥ F λ,µ { ∂∂t h } ( u b , u b )∥ Q du b du b = π b b ∫ R ∣ ∂∂t h ( t )∣ Q dt. Since ∣ ∂∂t h ( t )∣ Q = π √ b b ∣ f ( t ) µ ( a b t + τ b ) + ∂∂t f ( t ) ∣ Q , It follows that ∫ R u ∥ O λ,µA ,A { f } ( u )∥ Q du = b ∫ R ∣( a b t + τ b ) f ( t ) µ + ∂∂t f ( t )∣ Q dt. ◻ xample 3.8. (The QOLCT of a Gaussian quaternionic function)Consider a Gaussian quaternionic function f ( t ) = βe − ( α t + α t ) , where β = β β , and β = β + λβ , β = β + µβ , and β , β , β , β ∈ R , and α , α are realpositive constants.The QOLCT of f is given by O λ,µA ,A { f ( t )} ( u , u ) = β [ ∫ R K λA ( t , u ) e − α t e − α t K µA ( t , u ) dt ] β , = β ∫ R K λA ( t , u ) e − α t dt ∫ R K µA ( t , u ) e − α t dt β . We have ∫ R K λA ( t , u ) e − α t dt = √ πλb [ ∫ R e λ ( a b t − t ( u − τ ) b ) e − α t dt ] e λ ( − u ( d τ − b η ) + d ( u + τ )) b , Since ∫ R e λ ( a b t − t ( u − τ ) b ) e − α t dt = ∫ R e − ( α − a b λ )[ t + λ ( u − τ ) b ( α − a b λ ) ] dt e − ( ( u − τ ) b ) ( α − a b λ ) = √ πα − a b λ e − ( u − τ ) b ( α b − a λ ) α = √ b π b α − a λ e − ( u − τ ) b ( α b + a ) ( α b + a λ ) = √ b π b α − a λ e − α ( u − τ ) ( α b + a ) e − a ( u − τ ) b ( α b + a ) λ where the second equality follows from ∫ R e − z ( t + z ′ ) dt = √ πz , for z, z ′ ∈ C ,Re(z) > , (Gaussian integral withcomplex offset).Then ∫ R K λA ( t , u ) e − α t dt = √ b α λ + a λ e − α ( u − τ ) ( α b + a ) e λ ( − u ( d τ − b η ) + d ( u + τ ) − a ( u − τ ) α b + a ) b . We deduce that O λ,µA A { f ( t )} ( u , u ) = e − [ α ( u − τ ) ( α b + a ) + α ( u − τ ) ( α b + a ) ] β √ b α λ + a λ e λ ( − u ( d τ − b η ) + d ( u + τ ) − a ( u − τ ) α b + a ) b × √ b α µ + a µ e µ ( − u ( d τ − b η ) + d ( u + τ ) − a ( u − τ ) α b + a ) b β . ◻ Some properties of the QOLCT are summarized in Table 1.
In harmonic analysis, the UP states that a non-trivial function and its FT cannot both be sharplylocalized. The UP plays an important role in signal processing, and quantum mechanics. In quantummechanics , UP asserts that one cannot make certain of the position and momentum of the particule atthe same time, i.e., increasing the knowledge of the position decreases the knowledge of the momentum,and vice versa. There are many different forms of UPs in the time-frequency plane, such as Heisenberg-Weyl’s UP, Hardy’s UP, Beurling’s UP, and logarithmic UP, and so on in terms of different notations oflocalization. As far as we know, in 2013, Kit-Ian Kou et al [20] extended the Heisenberg-type UP to theQLCT. Recently Mawardi et al [1] established the logarithmic UP associated with the QLCT. Consideringthat the QOLCT is a generalized version of the QLCT quaternionic Fourier, and so of the QFT, it is naturaland interesting to study the simultaneous localization of a function and its QOLCT by further extendingthe aforementioned UPs to the QOLCT domain. Therefore, in this section, we prove and generalize theHeisenberg-Weyl’s UP, Hardy’s UP, Beurling’s UP, and logarithmic UP to 2D quaternion-valued signalsusing the two-sided QOLCT.
Proposition 4.1. ([7], Thm. 4.1) et f ( t ) = ∣ f ( t ) ∣ Q e u ( t ) θ ( t ) .If f, ∂∂t k f, t k f ∈ L ( R , H ) for k = , , then ∣ t k f ( t ) ∣ ,Q ∥ ξ k F λ,µ { f ( t )} ( πξ )∥ ,Q ≥ π ∣ f ( t ) ∣ ,Q + COV t k , with COV t k ∶= π ∫ R ∣ f ( t ) ∣ Q ∣ t k ( ∂∂t k e u ( t ) θ ( t ) ) ∣ Q dt. The equation holds if and only if f ( t ) = De − a k t k e u ( t ) θ ( t ) and ∂∂t k e u ( t ) θ ( t ) = δ k t k , where a , a > , D ∈ R + and δ , δ are pure quaternions. Theorem 4.2.
Suppose that f, ∂∂t k f, t k f ∈ L ( R , H ) for k = , , then ∣ t k f ( t )∣ ,Q ∥ ξ k πb k O λ,µA ,A { f ( t )} ( ξ )∥ ,Q ≥ π ∣ f ( t )∣ ,Q + COV t k ξ , Where
COV t k ξ ∶= π ( ∫ R (∣ f ( t )∣ Q ) ∣ t k ( ∂∂t k e u ( t ) θ ( t ) )∣ Q dt ) , and e u ( t ) θ ( t ) = ∣ f ( t )∣ Q √ λ e − λ b ξ ( d τ − b η ) + λ d b ( ξ + τ ) + λ b t τ + λ a b t f ( t ) e µ b t τ + µ a b t − µ b ξ ( d τ − b η ) + µ d b ( ξ + τ ) √ µ . The equation holds if and only if f ( t ) = De − a k t k e u ( t ) θ ( t ) and ∂∂t k e u ( t ) θ ( t ) = δ k t k where a , a > D ∈ R + and δ , δ are pure quaternions. Proof. Let h ( t ) be rewritten as (3.2).Since ∂∂t k f, t k f ∈ L ( R , H ) , and ∣ h ( t )∣ Q = π √ b b ∣ f ( t )∣ Q . we get ∂∂t k h, t k h ∈ L ( R , H ) , and ∣ t k h ( t )∣ ,Q = ∫ R t k ∣ h ( t )∣ Q dt = π b b ∣ t k f ( t )∣ ,Q , ∣ h ( t )∣ ,Q = π b b ∣ f ( t )∣ ,Q By lemma 3.2, we have ∥ ξ k F λ,µ { h ( t )} ( πξ )∥ ,Q = ∥ ξ k O λ,µA ,A { f ( t )} ( πb ξ , πb ξ )∥ ,Q = π b b ∥ ξ k πb k O λ,µA ,A { f ( t )} ( ξ )∥ ,Q . Hence, it follows from proposition 4.1. (∣ t k h ( t )∣ ,Q )( ξ k ∥ F λ,µ { h ( t )} ( πξ )∥ ,Q )) ≥ π ∥ h ( t )∥ ,c + COV t k , (4.1)With COV t k = π ∫ R ∣ h ( t )∣ Q ∣ t k ( ∂∂t k e u ( t ) θ ( t ) )∣ Q dt, and e u ( t ) θ ( t ) = ∣ h ( t )∣ Q h ( t ) = π √ b b ∣ f ( t )∣ Q √ πλb e − λ b ξ ( d τ − b η ) + λ d b ( ξ + τ ) + λ b t τ + λ a b t f ( t )× e µ b t τ + µ a b t − µ b ξ ( d τ − b η ) + µ d b ( ξ + τ ) √ πµb = ∣ f ( t )∣ Q √ λ e − λ b ξ ( d τ − b η ) + λ d b ( ξ + τ ) + λ b t τ + λ a b t f ( t )× e µ b t τ + µ a b t − µ b ξ ( d τ − b η ) + µ d b ( ξ + τ ) √ µ . (4.1) implies that ∣ t k f ( t )∣ ,Q ∥ ξ k πb k O λ,µA ,A { f ( t )} ( ξ )∥ ,Q ≥ π ∣ f ( t )∣ ,Q + π b b COV t k . After straightforward calculation one obtains π b b COV t k = π ( ∫ R (∣ f ( t )∣ Q ) ∣ t k ( ∂∂t k e u ( t ) θ ( t ) )∣ Q dt ) . By proposition 4.1, the equation holds in (4.1) if and only if ∂∂t k e u ( t ) θ ( t ) = δ k t k , and ∣ h ( t )∣ Q = Ce − a k t k , that is ∣ f ( t )∣ Q = π √ b b Ce − a k t k , where a , a > , C, D ∈ R + and δ , δ are pure quaternions. This proves the theorem. ◻ Hardy’s theorem [11] is a qualitative UP, it states that it is impossible for a function and its Fouriertransform to decrease rapidly simultaneously. The following proposition is the Hardy’s UP for the Two-sided QFT.
Proposition 4.3. ([7], Thm. 5.3)Let α and β are positive constants .Suppose f ∈ L ( R , H ) with f ( t ) ∣ Q ≤ Ce − α ∣ t ∣ , t ∈ R . ∣ F λ,µ { f } ( u ) ∣ Q ≤ C ′ e − β ∣ u ∣ , u ∈ R . for some positive constants C, C ′ . Then, three cases can occur :(i) f αβ > , then f = .(ii) f αβ = , then f ( t ) = Ae − α ∣ t ∣ , whit A is a quaternion constant.(iii) f αβ < , then there are infinitely many such functions f . On the basis of proposition 4.3, we give the Hardy’s UP in the QOLCT domains.
Theorem 4.4.
Let α and β are positive constants. Suppose f ∈ L ( R , H ) with ∣ f ( t )∣ Q ≤ Ce − α ∣ t ∣ , t ∈ R . (4.2) ∣ O λ,µA ,A f ( b u , b u )∣ Q ≤ C ′ e − β ∣ u ∣ , u ∈ R . (4.3) for some positive constants C, C ′ . Then, three cases can occur :(i) f αβ > , then f = .(ii) f αβ = , then f ( t ) = Ae − α ∣ t ∣ e − λ a b t − λ b t τ e − µ a b t − µ b t τ , where A is a quaternion constant.(iii) f αβ < , then there are infinitely many f . Proof. Let g ( t ) be rewritten in the form of (3.3), we have O λ,µA ,A { f ( t )} ( u , u ) = √ πλ e − λ b u ( d τ − b η ) + λ d b ( u + τ ) F λ,µ { g ( t )} ( u b , u b )× e − µ b u ( d τ − b η ) + µ d b ( u + τ ) √ πµb . (4.4)Since ∣ g ( t )∣ Q = ∣ f ( t )∣ Q , We get g ∈ L ( R , H ) and ∣ g ( t )∣ Q ≤ Ce − α ∣ t ∣ . On the other hand, by (4.4) and (4.3) we obtain ∣ F λ,µ { g ( t )} ( u , u )∣ Q = π √ b b ∣ O λ,µA ,A { f ( t )} ( b u , b u )∣ Q ≤ π √ b b C ′ e − β ∣ u ∣ . Therefore, it follows from proposition 4.3 that,If αβ = then g ( t ) = Ae − α ∣ t ∣ , for some constant A. Hence f ( t ) = Ae − α ∣ t ∣ e − λ b t τ − λ a b t e − µ b t τ − µ a b t . If αβ > then g = , so f = .If αβ < , then there are infinitely many such functions f , that verify (4.2) and ( (4.3).This completes the proof. ◻ It follows from theorem 4.4 that it is impossible for f and its two-sided QOLCT to both decrease veryrapidly. 10 .3 Beurling’s uncertainty principle Beurling’s UP [4], [16] is a variant of Hardy’s UP. It implies the weak form of Hardy’s UP immediatly.The following proposition is the Beurling’s UP for the Two-sided QFT.
Proposition 4.5. [10]Let f ∈ L ( R , H ) and d ≥ satisfy ∫ R ∫ R ∣ f ∣ Q ∥F{ f }( y )∥ Q ( + ∣ x ∣ + ∣ y ∣) d e π ∣ x ∣∣ y ∣ dxdy < ∞ , Then f ( x ) = P ( x ) e − a ∣ x ∣ , a.e.Where a > and P is a polynomial of degree < d − . In particular, f is identically 0 when d ≤ . On the basis of proposition 4.5, we give the Beurlings’UP in the QOLCT domains.
Theorem 4.6.
Let f ∈ L ( R , H ) and d ≥ satisfy ∫ R ∫ R ∣ f ( t )∣ Q ∥O λ,µA ,A { f }( b u ,b u )∥ Q ( + ∣ t ∣ + ∣ u ∣) d e ∣ t ∣∣ y ∣ dtdu < ∞ , Then f ( t ) = e − a ∣ t ∣ √ πλb e λ b u ( d τ − b η ) − λ d b ( u + τ ) − λ b t τ − λ a b t P ( t ) e µ b u ( d τ − b η ) − µ d b ( u + τ ) − µ b t τ − µ a b t ×√ πµb , a.e.Where a > and P is a quaternion polynomial of degree < d − . In particular, f = a.e. when d ≤ . Proof. Let h ( t ) be rewritten in the form of (3.2), we have h ∈ L ( R , H ) . It follows from lemma 3.2, and ∣ h ( t )∣ Q = π √ b b ∣ f ( t ) ∣ Q That, ∫ R ∫ R ∣ h ( t )∣ Q ∥F λ,µ { h }( u )∥ Q ( + ∣ t ∣ + ∣ u ∣) d e ∣ t ∣∣ u ∣ dtdu = ∫ R ∫ R ∣ h ( t )∣ Q ∥O λ,µA ,A { f }( b u ,b u )∥ Q ( + ∣ t ∣ + ∣ u ∣) d e ∣ t ∣∣ u ∣ dtdu = π √ b b ∫ R ∫ R ∣ f ( t )∣ Q ∥O λ,µA ,A { f }( b u ,b u )∥ Q ( + ∣ t ∣ + ∣ u ∣) d e ∣ t ∣∣ u ∣ dtdu < ∞ . Then by proposition 4.5, we get h ( t ) = P ( t ) e − a ∣ t ∣ a.e. where a > and P is a quaternion polynomial ofdegree < d − . i.e. f ( t ) = e − a ∣ t ∣ √ πλb e λ b u ( d τ − b η ) − λ d b ( u + τ ) − λ b t τ − λ a b t P ( t ) e µ b u ( d τ − b η ) − µ d b ( u + τ ) − µ b t τ − µ a b t ×√ πλb . In particular, f = a.e. when d ≤ . ◻ The logarithmic UP [3] is a more general form of Heisenberg type UP, its localization is measured interms of entropy. It is derived by using Pitt’s inequality.
Lemma 4.7. (Pitt’s inequality for the two-sided QFT [6])For f ∈ S ( R , H ) , and ≤ α < , ∫ R ∣ u ∣ − α ∥ F i,j { f ( t )} ( u , u )∥ Q du ≤ C α ∫ R ∣ t ∣ α ∣ f ( t )∣ Q dt. With C α ∶= π α [ Γ ( − α )/ Γ ( + α )] , and Γ ( . ) is the Gamma function. Theorem 4.8.
Under the assumptions of lemma 4.7, one has ∫ R ∣( z b , z b )∣ − α ∥O i,jA ,A { f } ( z , z )∥ Q dz ≤ C α π ∫ R ∣ t ∣ α ∣ f ( t )∣ Q dt. (4.5)Proof. Let h ( t ) be rewritten in the form of (3.2), with λ = i, and µ = j .It’s clear that h ∈ S ( R , H ) , and ∣ h ( t )∣ Q = π √ b b ∣ f ( t )∣ Q . Let O i,jA ,A { f } ( u ) be rewritten as (3.1), we have by lemma 3.211 i,jA ,A { f ( t )} ( u , u ) = F i,j { h ( t )} ( u b , u b ) . By lemma 4.7, we obtain ∫ R ∣ u ∣ − α ∥ O i,jA ,A { f } ( b u , b u )∥ Q du = ∫ R ∣ u ∣ − α ∥ F i,j { h ( t )} ( u , u )∥ Q du ≤ C α ∫ R ∣ t ∣ α ∣ h ( t )∣ Q dt = C α π b b ∫ R ∣ t ∣ α ∣ f ( t )∣ Q dt. Let z = b u and z = b u , we have b b ∫ R ∣( z b , z b )∣ − α ∥ O i,jA ,A { f } ( z )∥ Q dz ≤ C α π b b ∫ R ∣ t ∣ α ∣ f ( t )∣ Q dt, i.e., ∫ R ∣( z b , z b )∣ − α ∥ O i,jA ,A { f } ( z )∥ Q dz ≤ C α π ∫ R ∣ t ∣ α ∣ f ( t )∣ Q dt. ◻ Theorem 4.9. (Logarithmic UP for the QOLCT)Let f ∈ S ( R , H ) , then ∫ R ln ( ∣ z b , z b ∣) ∥ O i,jA ,A { f } ( z )∥ Q dz + ∫ R ln (∣ t ∣) ∣ f ( t )∣ Q dt ≥ A ∫ R ∣ f ( t )∣ Q dt, (4.6) with A = ln ( ) + Γ ′ ( )/ Γ ( ) . Proof. Let f ∈ S ( R , H ) , ≤ α < , D α = C α π = α [ Γ ( − α )/ Γ ( + α )] , and Φ ( α ) ∶= ∫ R ∣( z b , z b )∣ − α ∥O i,jA ,A { f } ( z )∥ Q dz − D α ∫ R ∣ t ∣ α ∣ f ( t )∣ Q dt. By differentiating Φ ( α ) , we have Φ ′ ( α ) = − ∫ R ln ( ∣ z b , z b ∣)∣( z b , z b )∣ − α ∥ O i,jA ,A { f } ( z )∥ Q dz − D ′ α ∫ R ∣ t ∣ α ∣ f ( t )∣ Q dt − D α ∫ R ln (∣ t ∣)∣ t ∣ α ∣ f ( t )∣ Q dt, whith D ′ α = − ln ( ) − α [ Γ ( − α )/ Γ ( + α )] + − α [ − Γ ( − α ) Γ ′ ( − α ) Γ ( + α ) − Γ ( − α ) Γ ( + α ) Γ ′ ( + α )] / Γ ( + α ) . We have D = and D ′ = − ln ( ) − Γ ′ ( )/ Γ ( ) . Because of (4.5), we see that Φ ( α ) ≤ for ≤ α < , also by theorem 3.6 we have Φ ( ) = . Then Φ ′ ( + ) = lim α → + Φ ( α ) − Φ ( ) α ≤ . Therefore ( ln ( ) + Γ ′ ( )/ Γ ( ) ) ∫ R ∣ f ( t )∣ Q dt ≤ ∫ R ln (∣ z b , z b ∣)∥ O i,jA ,A { f } ( z )∥ Q dz + ∫ R ln (∣ t ∣)∣ f ( t )∣ Q dt. From which the theorem follows. ◻ Remark 4.10.
Applying Jensen’s inequality to (4.6), we can show that the logarithmic UP implies Heisenberg-Weyl’s inequality (theorem 4.2) .
In this paper, we first presented a new generalization of the QLCT and so of the QFT, namely theQOLCT. Second, We established some properties of the QOLCT including the Plancherel’s formula. Then,we derive three UPs in the QOLCT domain: Heisenberg-Weyl’s UP, Hardy’s UP and its variant-Beurling’sUP. These three UPs assert that it is impossible for a non-zero function and its QOLCT to both decreasevery rapidly. Finally, we generalize Pitt’s inequality to the QOLCT domain, and then obtain a logarithmicUP associated with QOLCT. In the future work, we will consider these UPs for the offset linear canonicaltransform in Clifford analysis. 12 eferences [1] M. Bahri, R. Ashino, Logarithmic uncertainty principle for quaternion linear canonical transform. In:Proceeding of 2016 International Conference on Wavelet Analysis and Pattern Recognition (ICWAPR,2016), pp. 140-145 (2016).[2] M. Bahri, R. Ashino, A Simplified Proof of Uncertainty Principle for Quaternion Linear Canonical Trans-form, Abstract and Applied Analysis, Volume 2016, Article ID 5874930, 11 pages,http://dx.doi.org/10.1155/2016/5874930.[3] W. Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123(6), pp. 1897-1905 (1995).[4] A. Beurling, The collect works of Arne Beurling, Birkhauser. Boston (1989), 1-2.[5] T. B ¨ulow, Hypercomplex spectral signal representations for the processing and analysis of images.Ph.D. Thesis, Institut f ¨ur Informatik und Praktische Mathematik, University of Kiel (1999), Germany.[6] L.P. Chen, K.I. Kou, M.S. Liu, Pitt’s inequality and the uncertainty principle associated with the quater-nion Fourier transform, J. Math. Anal. Appl. , vol. 423, no. 1, pp. 681-700, 2015.[7] Y. El Haoui and S. Fahlaoui, The Uncertainty principle for the two-sided quaternion Fourier transform,Mediterr. J. Math. (2017) doi:10.1007/s00009-017-1024-5.[8] T.A. Ell, Quaternion-Fourier transfotms for analysis of two-dimensional linear time-invariant partialdifferential systems. In: Proceeding of the 32nd Conference on Decision and Control, San Antonio,Texas, pp. 1830-1841 (1993).[9] C. L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9(2) (September 1983).[10] S. Fahlaoui and Y. El Haoui, Beurling’s theorem for the two-sided quaternion Fourier transform,https://arxiv.org/abs/1711.04142, Conference paper, ICCA 11, Gent, Belgium, 7-11 Aug. 2017.[11] G.H. Hardy, A theorem concerning Fourier transform, J. London Math. Soc. 8 (1933), pp. 227-231.[12] W. Heisenberg, ¨Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik.Zeitschrift f ¨ur Physik 43, pp. 172-198 (1927).[13] E. Hitzer, Quaternion Fourier Transform on Quaternion Fields and Generalizations, Adv. Appl. Cliff.Algs., (2007) vol. 17(3): pp. 497-517. https://doi.org/10.1007/s00006-007-0037-8.[14] E. Hitzer, New Developments in Clifford Fourier Transforms, in N. E. Mastorakis, et al. (eds.), Adv.in Appl. and Pure Math., Proc. of the 2014 Int. Conf. on Pure Math., Appl. Math., Comp. Methods(PMAMCM 2014), Santorini, Greece, July 2014, Math. & Comp. in Sci. & Eng., Vol. 29, 7 pages.[15] E. Hitzer, Two-Sided Clifford Fourier Transform with Two Square Roots of -1 in Cl(p; q) Adv. Appl.Cliffrd Algebras, 24 (2014), pp. 313-332, DOI:10.1007/s00006-014-0441-9.[16] L. H ¨ormander, A uniqueness theorem of Beurling for Fourier transform pairs, Ark. F ¨or Math., 2(1991),pp. 237-240.[17] E. Hitzer, S. J. Sangwine, The Orthogonal 2D Planes Split of Quaternions and Steerable QuaternionFourier Transformations, in E. Hitzer, S.J. Sangwine (eds.), ”Quaternion and Clifford Fourier transformsand wavelets”, Trends in Mathematics 27, Birkhauser, Basel, 2013, pp. 15-39.
DOI ∶ . / − − − − , Preprint: http://arxiv.org/abs/1306.2157.[18] H. Huo, Uncertainty Principles for the Offset Linear Canonical Transform, Circuits Syst Signal Pro-cess (2018), https://doi.org/10.1007/s00034-018-0863-z.1319] K. I. Kou, J. Morais, Y. Zhang, Generalized prolate spheroidal wave functions for offset linear canon-ical transform in clifford analysis, Mathematical Methods in the Applied Sciences 36 (9) (2013), pp.1028-1041. doi:10.1002/mma.2657.[20] K. I. Kou, J. Y. Ou, J. Morais, On uncertainty principle for quaternionic linear canonical transform,Abstract and Applied Analysis, 2013, Article ID 725952, 14 pages.[21] S.C. Pei, J.J. Ding, J.H. Chang, Efficient implementation of quaternion Fourier transform, convolution,and correlation by 2-D complex FFT. IEEE Trans. Signal Process. 49(11), pp. 2783-2797 (2001).[22] Y. Yang, P. Dang, and T. Qian, Tighter uncertainty principles based on quaternion Fourier transform,Adv. Appl. Clifford Algebras 26 (2016), 479-497.14et f and g ∈ L ( R , H ) , the constants α and β ∈ R , u ∈ R ,A l = [∣ a l b l c l d l ∣ τ l η l ] , parameters a l , b l , c l , d l , τ l , η l ∈ R such that a l d l − b l c l = , for l = , . Property Function QOLCTReal linearityPlancherel’s identity αf + βg ∣ f ∣ Q, = α O λ,µA ,A { f } ( u ) + β O λ,µA ,A { g } ( u )∥ O λ,µA ,A { f }∥ Q,2