Wigner-Ville distribution associated with the quaternion offset linear canonical transforms
aa r X i v : . [ m a t h . C A ] N ov WIGNER-VILLE DISTRIBUTION ASSOCIATED WITHTHE QUATERNION OFFSET LINEAR CANONICAL TRANSFORMS
MOHAMMED EL KASSIMI , YOUSSEF EL HAOUI AND SAÏD FAHLAOUI
Abstract.
The Wigner-Ville distribution (WVD) and quaternion offset linear canon-ical transform (QOLCT) are a useful tools in signal analysis and image processing.The purpose of this paper is to define the Wigner-Ville distribution associated withquaternionic offset linear canonical transform (WVD-QOLCT). Actually, this trans-form combines both the results and flexibility of the two transform WVD and QOLCT.We derive some important properties of this transform such as inversion and Plancherelformulas, we establish a version of Heisenberg inequality, Lieb’s theorem and we givethe Poisson summation formula for the WVD-QOLCT. keywords:
Wigner-Ville distribution, Offset linear canonical transform, linear canon-ical transform, quaternionic transform,Heisenberg uncertainty.1.
Introduction
The Fourier transformation used for a simple description of the input-output relation-ships of the filters linear, occupies a privileged place in the theory and signal processing.However, this transformation can not give a temporal signal, it only gives a global fre-quency information: its natural field of application is analysis stationary signals. So,as soon as we consider modulated signals or non-process stationary the Fourier trans-form becomes insufficient to study this type of signal. One solution to this problem isto associating to directly search a tool adapted to the study of non-stationary signal,without direct reference to the methods resulting from the stationary case. In this case,a particular axis of interest has been manifested for many years to a proposed trans-formation in Quantum Mechanics by E. P. Wigner [27] in 1932. This transformationallows to define what we will call the distribution of Wigner-Ville (WVD) in referenceand tribute to J. City which first introduced this same notion in Signal Theory. In re-cent years, this distribution has served as a useful analysis tool in many fields as diverseas optics, biomedical engineering, signal processing and image processing. Due to thelarge applications of the linear canonical transform (LCT)[28] in several area includingradar analysis, signal processing and optics [22, 23, 25]. The LCT has received atten-tion since 1970 is introduced integral transform with four parameters (a,b,c,d) [8][21].A lot of authors were interested to study LCT. This transform is also known under theaffine Fourier transform [1], and the generalized Fresnel Fourier transform [17]. More-over the Fourier transform [5] and the Fresnel transform [12] are all special cases ofthe LCT. In [23], the LCT is generalized by introducing two extra parameters, onecorresponding to time shift and an other to frequency modulation. This generalized ofLCT is called offset LCT (OLCT)[24, 29], and it is known under six parameters linear transform. These two parameters make the OLCT more general and flexible than LCT,in consequence the OLCT can apply to most electrical and optical signal systems. Thetwo-sided quaternionic Fourier transform (QFT) was introduced in [9]. The QFT hasmany application in large domains, in [9] the QFT used in analysis of 2D linear timeinvariant dynamic systems, In [4] the authors used the QFT to design a digital colorimage water marking scheme, in [26] the QFT is used for filtering color images.The main objective of this work is the combination between the WVD, QFT and theOLCT, in order to get the Quaternion Offset Wigner-Ville distribution associated tolinear canonical transforms (WVD-QOLCT). The paper is organized as follows, in sec-tion 2, we recall the main results about the quaternion algebra and harmonic analysisrelated to QFT, QLCT and QOLCT. In section 3, we introduce the WVD-QOLCT, and establish its important properties. The section 4 is devoted to give the ana-logue of Heisenberg’s inequality, Poisson summation formula, and Lieb’s theorem forthe WVD-QOLCT. In section 5, we conclude this paper.2.
Preliminaries
The quaternion algebra.
In the present section we collect some basic facts about quaternions, which will beneeded throughout the paper. For all what follows, let H be the Hamiltonian skew fieldof 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 be seen as an extension of the complex field C .Quaternions are isomorphic to the Clifford algebra Cl (0 , of R (0 , : H ∼ = Cl (0 , . (2.1)The scalar part of a quaternion q ∈ H is q denoted by Sc ( q ), the non scalar part(orpure 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 + 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 ordinaryEuclidean module | q | = √ q .It is easy to verify that 0 = 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 accordancewith Euler’s formula 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 = − λ be a pure unit quaternion, λ = − , clearly, we have for all x ∈ R , | e λx | Q = 1 . (2.2)In this paper, we will study the quaternion-valued signal f : R → H , f which can beexpressed as f = f + if + jf + kf , with f m : R → R f or m = 0 , , , . Let us introduce the canonical inner product forquaternion valued functions f, g : R → H , as follows: < f, g > = Z R f ( t ) g ( t ) dt, dt = dt dt . (2.3)Hence, the natural norm is given by | f | ,Q = q < f, f > = ( Z R | f ( t ) | Q dt ) , and the quaternion module L ( R , H ), is given by L ( R , H ) = { f : R → H , | f | ,Q < ∞} . Furthermore, for 2 < p < ∞ , we introduce the quaternion modules L p ( R , H ) , as L p ( R , H ) = { f : R → H , | f | pp,Q = Z R | f ( x ) | pQ dx < ∞} . From (2.3), we obtain the quaternion Schwartz’s inequality ∀ f, g ∈ L (cid:16) R , H (cid:17) : (cid:12)(cid:12)(cid:12)(cid:12)Z R f ( x ) g ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) Q ≤ Z R | f ( x ) | Q dx Z R | g ( x ) | Q dx. 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 . M. EL KASSIMI, Y. EL HAOUI, AND S. FAHLAOUI
The general two-sided quaternion Fourier transform.
In this subsection, we begin by defining the two-sided QFT, and reminder someproperties for this transform,Let us define the two-sided QFT and provide some properties used in the sequel.
Definition 2.1 ([14]) . Let λ, µ ∈ H , be any two pure unit quaternions, i.e., λ = µ = − . For f in L ( R , H ) , the two-sided QFT with respect to λ ; µ is F λ,µ { f } ( u ) = Z R e − λu t f ( t ) e − µu t dt, where t, u ∈ R . (2.4)We define a new module of F { f } λ,µ as follows : (cid:13)(cid:13)(cid:13) F λ,µ { f } (cid:13)(cid:13)(cid:13) Q := vuut m =3 X m =0 |F λ,µ { f m }| Q . (2.5)Furthermore, we define a new L -norm of F { f } as follows : (cid:13)(cid:13)(cid:13) F λ,µ { f } (cid:13)(cid:13)(cid:13) ,Q := sZ R kF λ,µ { f } ( y ) k Q dy. (2.6)It is interesting to observe that (cid:13)(cid:13)(cid:13) F λ,µ { f } (cid:13)(cid:13)(cid:13) Q is not equivalent to (cid:12)(cid:12)(cid:12) F λ,µ { f } (cid:12)(cid:12)(cid:12) Q unless f isreal valued. Lemma 2.2 (Dilation property) . ( see page 50 in [6]) Let k , k be a positive scalar constants, we have F λ,µ { f ( t , t ) } (cid:18) u k , u k (cid:19) = k k F λ,µ { f ( k t , k t ) } ( u , u ) . (2.7)By following the proof of Theorem (3 .
2) in [7], and replacing i by λ , j by µ we obtainthe next lemma. Lemma 2.3. (QFT Plancherel)Let f ∈ L ( R , H ) , then Z R (cid:13)(cid:13)(cid:13) F λ,µ { f } ( u ) (cid:13)(cid:13)(cid:13) Q du = 4 π Z R | f ( t ) | Q dt. (2.8) 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 . (2.9)Proof. See ([6], Thm. 2.10). Lemma 2.5. [Inverse QFT] (see [16] ) If f ∈ L ( R , H ) , and F λ,µ { f } ∈ L ( R , H ) , then the two-sided QFT is an invertibletransform and its inverse is given by f ( t ) = 1(2 π ) Z R e λu t F λ,µ { f ( t ) } ( u ) e µu t du. (2.10)3. The offset quaternionic linear canonical transform
Morais et al [18] introduce the quaternionic linear canonical transform (QLCT). Theyconsider two real matrixes A = " a b c d , A = " a b c d ∈ R × . with a d − b c = 1 , a d − b c = 1 , Eckhard Hitzer [15] generalize the definitions of [18] to be: the two-sided QLCT ofsignals f ∈ L ( R , H ), is defined by L λ,µA ,A { f } ( u ) = R R K λA ( t , u ) f ( t ) K µA ( t , u ) dt, b , b = 0; √ d e λ c d u f ( d u , t ) K µA ( t , u ) , b = 0 , b = 0; √ d K λA ( t , u ) f ( t , d u ) e µ c d u , b = 0 , b = 0; √ d d e λ c d u f ( d u , d u ) e µ c d u , b = b = 0 . (3.1)with λ, µ ∈ H , denote two pure unit quaternions, λ = µ = −
1, including the cases λ = ± µ,K λA ( t , u ) = 1 √ λ πb e λ ( a t − t u + d u ) / b , K µA ( t , u ) = 1 √ µ πb e µ ( a t − t u + d u ) / b , In [18], the properties of the right-sided QLCT and its uncertainty principles are studiedin detail. El Haoui et al [11] introduced and studied the QOLCT, and established itsproperties and uncertainty principles. Let’s give the definitions of Quaternionic offsetlinear canonical transform as follows:
Definition 3.1.
Let A l = "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a l b l c l d l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ l η l ,the parameters a l , b l , c l , d l , τ l , η l ∈ R such that a l d l − b l c l = 1 , for l = 1 , the two-sided quaternionic offset linear canonical transform (QOLCT) of a signal f ∈ L ( R , H ) , is given by M. EL KASSIMI, Y. EL HAOUI, AND S. FAHLAOUI O λ,µA ,A { f ( t ) } ( u ) = R R K λA ( t , u ) f ( t ) K µA ( t , u ) dt, b , b = 0; √ d e λ ( c d ( u − τ ) + u τ ) f ( d ( u − τ ) , t ) K µA ( t , u ) , b = 0 , b = 0; √ d K λA ( t , u ) f ( t , d ( u − τ )) e µ ( c d ( u − τ ) + u τ , b = 0 , b = 0; √ d d e λ ( c d ( u − τ ) + u τ ) f ( d ( u − τ ) , d ( u − τ )) e µ ( c d ( u − τ ) + u τ ) ,b = b = 0 . Where K λA ( t , u ) = 1 √ λ πb e λ ( a t − t ( u − τ ) − u ( d τ − b η )+ d ( u + τ )) b , f or b = 0 , (3.2) and K µA ( t , u ) = 1 √ µ πb e µ ( a t − t ( u − τ ) − u ( d τ − b η )+ d ( u + τ )) b , f or b = 0(3.3) with √ λ = e − λ π , √ µ = e − µ π . The left-sided and right-sided QOLCTs can be defined by placing the two kernel factorsboth on the left or on the right, respectively.
We remark that, when τ = τ = η = η =0, the two-sided QOLCT reduces to theQLCT.Also, when A = A = "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , the conventional two-sided QFT is recovered.Namely, O λ,µA ,A { f ( t ) } ( u ) = 1 √ λ π ( Z R e − λt u f ( t ) e − µt u dt ) 1 √ µ π = 12 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 (2.4).The following lemma gives the relationships of two-sided QOLCTs and two-sidedQFTs 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 ) } (cid:18) u b , u b (cid:19) , (3.4) with h ( t ) = 1 √ 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 . By using lemma 2.5 and (3.4), we get the inversion formula for the QOLCT,
Theorem 3.3. If f and O λ,µA ,A { f } are in L ( R , H ) , then the inverse transform ofthe QOLCT can be derived from that of the QFT, and we have f ( t ) = Z R K λA ( t , u ) O λ,µA ,A { f ( t ) } ( u , u ) K µA ( t , u ) du. Theorem 3.4. (Plancherel’s theorem of the QOLCT)Every 2D quaternion-valued signal f ∈ L ( R , H ) and its QOLCT are related to thePlancherel identity in the following way: (cid:13)(cid:13)(cid:13) O λ,µA ,A { f } (cid:13)(cid:13)(cid:13) ,Q = | f | ,Q . (3.5)4. Wigner-Ville distribution associated with quaternionic offsetlinear canonical transform
The Fourier transform is a powerful tool to study the stationary signals, but it hasbecome not sufficient for characterize the non-stationary signals. However, in practice,most natural signals are non stationary. In order to study a non stationary signal theWigner-Ville distribution has become a suite tool for the analysis of the non stationarysignals.In this section, we are going to give the definition of Wigner-Ville distribution associatedwith the quaternionic offset linear canonical transform WVD-QOLCT, then, we willinvestigate its important properties, and establish the Heisenberg uncertainty principle,Poisson summation formula and Lieb’s theorem related for the WVD-QOLCT.
Definition 4.1.
Let A l = "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a l b l c l d l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ l η l , with a l , b l , c l , d l , τ l , η l ∈ R such that a l d l − b l c l = 1 , for l = 1 , .The Wigner-Ville distribution associated with the two-sided quaternionic offset linearcanonical transform (WVD-QOLCT) of a signal f ∈ L ( R , H ) , is given by M. EL KASSIMI, Y. EL HAOUI, AND S. FAHLAOUI W A ,A f,g ( t, u ) = R R K λA ( s , u ) f ( t + s ) g ( t − s ) K µA ( s , u ) ds, b , b = 0 , √ d e λ ( c d ( u − τ ) + u τ ) f ( t + d ( u − τ )2 , t + s ) × g ( t − d ( u − τ )2 , t − s ) K µA ( s , u ) , b = 0 , b = 0; √ d K λA ( s , u ) f ( t + s , t + d ( u − τ )2 ) × g ( t − s , t − d ( u − τ )2 ) e µ ( c d ( u − τ ) + u τ , b = 0 , b = 0; √ d d e λ ( c d ( u − τ ) + u τ ) f ( t + d ( u − τ ) , t + d ( u − τ )) × g ( t − d ( u − τ ) , t − d ( u − τ )) e µ ( c d ( u − τ ) + u τ ) , b = b = 0 . where K λA ( s , u ) , and K µA ( s , u ) , are given respectively by (3.2) , and (3.3) . Remark 4.2.
It’s clear that if we take h f,g ( t, s ) = f ( t + s ) g ( t − s ) for all t, s ∈ R , we have, W A ,A f,g ( t, u ) = O λ,µA ,A { h f,g ( t, s ) } ( u ) . (4.1) We note that when we take τ l = η l = 0 , l = 1 , the WVD-QOLCT reduces to theWVD-QLCT [2] . And by using (3.4), we obtain the relation between WVD-QOLCT and QFT:
Lemma 4.3. W A ,A f,g ( t, u ) = F λ,µ { k f,g ( t, s ) } (cid:18) u b , u b (cid:19) , (4.2)where k f,g ( t, s ) = √ πλb e λ [ − b u ( d τ − b η )+ d b ( u + τ )+ b s τ + a b s ] h f,g ( t, s ) × e µ [ − b u ( d τ − b η )+ d b ( u + τ )+ b s τ + a b s ] 1 √ πµb . Now, we give the inversion formula for the WVD-QOCLT
Theorem 4.4. If f, g and W A ,A f,g are in L ( R , H ) , then, the inverse transform of QWVD-OCLT isgiven by f ( v ) = 1 | g | ,Q Z R Z R K λA (cid:18) v + ε , u (cid:19) W A ,A f,g ( v + ε , u ) K µA (cid:18) v + ε , u (cid:19) g ( ε ) dudε (4.3) Proof.
By the equation (4.1) W A ,A f,g ( t, u ) = O λ,µA ,A { f ( t + . g ( t − . } ( u ) . then, by theorem3.3, we obtain f ( t + s g ( t − s Z R K λA ( t , u ) W A ,A f,g ( t, u ) K µA ( t , u ) du. By taking v = t + s and ε = t − s we get t = v + ε and f ( v ) g ( ε ) = Z R K λA (cid:18) v + ε , u (cid:19) W A ,A f,g ( v + ε , u ) K µA (cid:18) v + ε , u (cid:19) du. (4.4)Multiplying both sides of (4.4) from the right by g and integrating with respect to dε we get f ( v ) Z R | g ( ε ) | dε = Z R Z R K λA (cid:18) v + ε , u (cid:19) W A ,A f,g ( v + ε , u ) K µA (cid:18) v + ε , u (cid:19) g ( ε ) dudε. (4.5)Consequently, f ( v ) = 1 | g | ,Q Z R Z R K λA (cid:18) v + ε , u (cid:19) W A ,A f,g ( v + ε , u ) K µA (cid:18) v + ε , u (cid:19) g ( ε ) dudε. (4.6) (cid:3) The following theorem gives the Plancherel’s identity fo for the WVD-QOLCT,
Theorem 4.5 (Plancherel’s theorem for WVD-QOLCT) . Let f, g ∈ L ( R , H ) , then we have, (cid:13)(cid:13)(cid:13) W A ,A f,g (cid:13)(cid:13)(cid:13) ,Q = | f | ,Q | g | ,Q . (4.7) Proof.
We have by the equality (4.1) W A ,A f,g ( t, u ) = O λ,µA ,A { h f,g ( t, . ) } ( u ) , and the Plancherel formula for the QOLCT (3.5) (cid:13)(cid:13)(cid:13) O λ,µA ,A { f } (cid:13)(cid:13)(cid:13) ,Q = | f | ,Q . So (cid:13)(cid:13)(cid:13) W A ,A f,g (cid:13)(cid:13)(cid:13) ,Q = (cid:13)(cid:13)(cid:13) O λ,µA ,A { h f,g } (cid:13)(cid:13)(cid:13) ,Q = | h f,g | ,Q = Z R Z R (cid:12)(cid:12)(cid:12)(cid:12) f ( t + s g ( t − s (cid:12)(cid:12)(cid:12)(cid:12) Q dtds ! = (cid:18)Z R Z R | f ( u ) g ( v ) | Q dudv (cid:19) = Z R | f ( u ) | du Z R | g ( v ) | dv = | f | ,Q | g | ,Q . (cid:3) Definition 4.6.
A couple α = ( α , α ) of non negative integers is called a multi-index.One denotes | α | = α + α and α ! = α ! α ! and, for x ∈ R x α = x α x α Derivatives are conveniently expressed by multi-indices ∂ α = ∂ | α | ∂x α ∂x α Next, we obtain the Schwartz space as ([19]) S ( R , H ) = { f ∈ C ∞ ( R , H ) : sup x ∈ R (1 + | x | k ) | ∂ α f ( x ) | < ∞} , where C ∞ ( R , H ) is the set of smooth function from R to H .The following theorem is the Heisenberg’s theorem for QOLCT (see [11]), Theorem 4.7 (Heisenberg QOLCT) . Suppose that f, ∂∂s k f, s k f ∈ L ( R , H ) for k = 1 , , then | s k f ( s ) | ,Q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ξ k πb k O λ,µA ,A { f ( s ) } ( ξ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ,Q ≥ π | f ( s ) | ,Q . (4.8)The next theorem states the Heisenberg’s uncertainty principle for the WVD-QOLCT. Theorem 4.8.
Let f, g ∈ S ( R , H ) . We have the following inequality (cid:18)Z R Z R | s k f ( t + s g ( t − s | Q dsdt (cid:19) Z R Z R k ξ k πb k W A ,A f,g ( t, ξ ) k Q dξdt ! ≥ π | f | ,Q | g | ,Q . (4.9) Proof.
Let h f,g , be rewritten as in remark 4.2.As f, g ∈ S ( R , H ), we obtain that h f,g ( ., s ) ∈ L ( R , H ) . Therefore by applying (4.9), we get | s k h f,g ( ., s ) | ,Q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ξ k πb k O λ,µA ,A { h f,g ( ., s ) } ( ξ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ,Q ≥ π | h f,g ( ., s ) | ,Q . According to (4.1), we obtain | s k h f,g ( t, s ) | ,Q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ξ k πb k W A ,A f,g ( t, ξ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ,Q ≥ π | h f,g ( t, s ) | ,Q . Then, we have, | s k f ( t + s g ( t − s | ,Q k ξ k πb k W A ,A f,g ( t, ξ ) k ,Q ≥ π | h f,g ( t, s ) | ,Q . (4.10)By taking the square root on both sides of (4.10)and integrating both sides with respectto dt, we get Z R (cid:18)Z R | s k f ( t + s g ( t − s | Q ds (cid:19) Z R k ξ k πb k W A ,A f,g ( t, ξ ) k Q dξ ! dt ≥ π Z R Z R | h f,g ( t, s ) | Q dsdt. (4.11)Now, by applying the Schwartz’s inequality to the left hand side of (4.11), and using(4.7), we obtain (cid:18)Z R Z R | s k f ( t + s g ( t − s | Q dsdt (cid:19) Z R Z R k ξ k πb k W A ,A f,g ( t, ξ ) k Q dξdt ! ≥ π | f | ,Q | g | ,Q . Therefore, the proof is complete. (cid:3)
Poisson summation formula.
It is well known that, the Poisson summation formula play an important role inmathematics, due to its various applications in signal processing. In this section wegeneralize the above mentioned formula into WVD-QOLCT domaine.
Proposition 4.9. (see [7] ) Let f ∈ L ( R , H ) , then X ( k ,k ) ∈ Z f ( s + k , s + k ) = X ( k ,k ) ∈ Z e πik s b f ( k , k ) e πjk s (4.12) where b f is the QFT of f defined by b f ( ξ ) = R R e − πis ξ f ( s ) e − πjt ξ ds. Now, we give a version of Poisson summation formula for the WVD-QOLCT,
Theorem 4.10.
Let f, g ∈ L ( R , H ) , then X ( k ,k ) ∈ Z e ib ( s + k ) τ + i a b ( s + k ) f ( t + s g ( t − s e jb ( s + k )+ j a b ( s + k ) = q πib [ X ( k ,k ) ∈ Z e πik s e πik ( d τ − b η ) − i d b (4 π b k + τ ) W A ,A f,g ( t, (2 πb k , πb k )) × e πjk s e πjk ( d τ − b η ) − j d b (4 π b k + τ ) ] q πjb Proof.
Let ω f,g ( t, s ) = e i b s τ + i a b s f ( t + s ) g ( t − s ) e j b s τ + j a b s . As f, g ∈ L ( R , H ) , we have by Hölder’s inequality ω f,g ∈ L ( R , H ), then by proposi-tion 4.9 we have X ( k ,k ) ∈ Z ω f,g ( t, s + k , s + k ) = X ( k ,k ) ∈ Z e πik s F i,j { ω f,g ( t, ( s , s )) } (2 πk , πk ) e πjk s . Applying (4.2) leads to X ( k ,k ) ∈ Z e ib ( s + k ) τ + i a b ( s + k ) f ( t + s g ( t − s e jb ( s + k ) τ + j a b ( s + k ) = q πib X ( k ,k ) ∈ Z e πik s e πik ( d τ − b η ) − i d b (4 π b k + τ ) W A ,A f,g ( t, (2 πb k , πb k )) × e πjk s e πjk ( d τ − b η ) − j d b (4 π b k + τ ) q πjb . (cid:3) Lieb’s theorem.
In this part of this paper, we are going to give a version of Lieb’s theorem for theWVD-QOLCT.In the following theorem[3], we state Lieb’s theorem related to the QLCT.
Theorem 4.11. If ≤ p ≤ and let q be such that p + q = 1 , then, for all f ∈ L p ( R , H ) , it holds that |L i,jA ,A { f }| q,Q ≤ | b b | − + q π | f | p,Q (4.13) Proof.
For the proof see [3]. (cid:3)
Theorem 4.12 (Lieb’s theorem associated with the WVD-QOLCT) . Let ≤ p < ∞ and f, g ∈ L ( R , H ) . Then Z R Z R | W A ,A f,g ( t, u ) | pQ dudt ≤ C | b b | − p +1 (2 π ) p | f | p ,Q | g | p ,Q , where C is a positive constant. Before proving this theorem, we need the following lemma,
Lemma 4.13.
Let A l = "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a l b l c l d l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ l η l , and B l = a l b l c l d l ! , with a l d l − b l c l = 1 for l = 1 , .For f ∈ L ( R , H ) , we have the relation: O λ,µA ,A { f } ( u ) = e λ (2 t τ − u ( d τ − b η )) e µd τ b L B ,B { f } ( u ) e µd τ b e µ (2 t τ − u ( d τ − b η )) . (4.14) Proof.
To prove this lemma we just use the definitions of the QOLCT and QLCT toobtain the result. (cid:3)
Now we give a demonstration of the theorem 4.12
Proof. We have by the equation (4.14), (cid:18)Z R |W A ,A f,g ( t, u ) | pQ du (cid:19) p = (cid:18)Z R |O λ,µA ,A { f ( t + s g ( t − s } ( u ) | pQ du (cid:19) p = (cid:18)Z R |L λ,µB ,B { f ( t + s g ( t − s } ( u ) | pQ du (cid:19) p ≤ | b b | − + p π (cid:18)Z R | f ( t + s g ( t − s | qQ ds (cid:19) q . In the last equality we used (4.13).Furthermore, Z R |W A ,A f,g ( t, u ) | pQ du ≤ | b b | − p +1 (2 π ) p (cid:18)Z R | f ( t + s g ( t − s | qQ ds (cid:19) pq integrating both sides of the last equality with respect to dt yields Z R (cid:18)Z R | W A ,A f,g ( t, u ) | pQ du (cid:19) dt ≤ | b b | − p +1 (2 π ) p Z R (cid:18)Z R | f ( t + s g ( t − s | qQ ds (cid:19) pq dt. Using relation (3.3) in the proof of theorem 1 in [20], we have Z R (cid:18)Z R | f ( t + s g ( t − s | qQ ds (cid:19) pq dt ≤ C [ | f | ,Q | g | ,Q ] p , where C is a positive constant.Consequently, we obtain Z R Z R | W A ,A f,g ( t, u ) | pQ dudt ≤ C | b b | − p +1 (2 π ) p | f | p ,Q | g | p ,Q . (cid:3) Conclusion
Firstly, we introduced an extension of the Winger-Ville distribution to the quaternionalgebra by means of the quaternionic offset linar canonical Fourier transform (QOLCT),namely the WVD-QOLCT transform. Secondly, the Plancherel theorem and the inver-sion formula have been demonstrated. Thirdly, Heisenberg’s uncertainty principle andPoisson summation formula associated with WVD-QOLCT were established by usingthe theorems obtained for the QFT and QOLCT. Finally the Lieb’s theorem related tothe WVD-QOLCT transform was formulated by applying the Lieb’s theorem for theQLCT.
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Department of Mathematics and Computer Sciences, Faculty of Sciences,Equipe d’Analyse Harmonique et Probabilités, University Moulay Ismaïl,BP 11201 Zitoune, Meknes, Morocco
Mohammed El kassimi
E-mail address : [email protected] Youssef El haoui
E-mail address : [email protected] Saïd Fahlaoui
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