The two-sided Gabor quaternion Fourier transform and some uncertainty principles
aa r X i v : . [ m a t h . C A ] N ov THE TWO-SIDED GABOR QUATERNIONIC FOURIER TRANSFORM ANDSOME UNCERTAINTY PRINCIPLES
S.FAHLAOUI AND M.EL KASSIMI
Abstract.
In this paper, we define a new transform called the Gabor quaternionic Fourier transform(GQFT), which generalizes the classical windowed Fourier transform to quaternion valued-signals,we give several important properties such as the Plancherel formula and inversion formula. Finally,we establish the Heisenberg uncertainty principles for the GQFT. keywords:
Quaternion algebra, Quaternionic Fourier transform, Gabor Fourier transform, Heisen-berg uncertainty principle.
As it is known, the quaternion Fourier transform (QFT) is a very useful mathematical tool. It hasbeen discussed extensively in the literature and has proved to be powerful and useful in some theories.In [1, 6, 12] the authors used the (QFT) to extend the colour image analysis. Researchers in [5] appliedthe QFT to image processing and neural computing techniques. The QFT is a generalisation of thereal and complex Fourier transform (FT), but it is ineffective in representing and computing localinformation about quaternionic signals. A lot of papers have been devoted to the extension of thetheory of the windowed FT to the quaternionic case. Recently Bülow and Sommer [6, 7] extend theWFT to the quaternion algebra. They introduced a special case of the GQFT known as quaternionicGabor filters. They applied these filters to obtain a local two-dimensional quaternionic phase. In[2] Bahri et al. studied the right sided windowed quaternion Fourier transform. In [14] the authorsstudied two-sided windowed (QFT) for the case when the window has a real valued. Moreover, they alsopointed out that the extension of the windowed Fourier transforms to the quaternionic case by meansof a two-sided QFT is rather complicated in view of the non-commutativity. So for that, this paperattempts to study the two-sided quaternionic Gabor Fourier transform (GQFT) with the window hasa quaternionic valued and some important properties are derived. We start by reminding some resultsof two-sided quaternionic Fourier transform (QFT), we give some examples, to show the differencebetween the GQFT and WFT, and we establish important properties of the GQFT like inversionformula, Plancherel formula, using a version of Heisenberg uncertainty principle for two-sided QFT toprove a generalized uncertainty principle for GQFT. H :The quaternion algebra H is defined over R with three imaginary units i, j and k obey the Hamilton’smultiplication rules, ij = − ji = k, jk = − kj = i, ki = − ik = j (1.1) i = j = k = ijk = − H is non-commutative, one cannot directly extend various results on complex numbersto a quaternion. For simplicity, we express a quaternion q as the sum of scalar q , and a pure 3Dquaternion q. Every quaternion can be written explicitly as: q = q + iq + jq + kq ∈ H , q , q , q , q ∈ R , The conjugate of quaternion q is obtained by changing the sign of the pure part, i.e. q = q − iq − jq − kq
41 S.FAHLAOUI AND M.EL KASSIMI
The quaternion conjugation is a linear anti-involution p = p, p + q = p + q, pq = qp, ∀ p, q ∈ H The modulus | q | of a quaternion q is defined as | q | = p qq = q q + q + q + q , | pq | = | p || q | . It is straight forward to see that | pq | = | p || q | , | q | = | q | , p, q ∈ H In particular, when q = q is a real number, the module | q | reduces to the ordinary Euclidean modulus,i.e. | q | = √ q q . A function f : R → H can be expressed as f ( x, y ) := f ( x, y ) + if ( x, y ) + jf ( x, y ) + kf ( x, y ) , where ( x, y ) ∈ R × R .We introduce an inner product of functions f, g defined on R with values in H as follows < f, g > L ( R , H ) = Z R f ( x ) g ( x ) dx If f = g we obtain the associated norm by k f k = < f, f > = Z R | f ( x ) | dx The space L ( R , H ) is then defined as L ( R , H ) = { f | f : R → H , k f k < ∞} And we define the norm of L ( R , H ) by k f k L ( R , H ) = k f k The quaternion Fourier transform (QFT) is an extension of Fourier transform proposed by Ell [13].Due to the non-commutative properties of quaternion, there are three different types of QFT, the leftsided QFT, the right sided QFT and the two-sided QFT [17]. In this paper we only treat the two-sidedQFT. We now review the definition and some properties of the two-sided QFT[16].
Definition 2.1 (Quaternion Fourier transform) . The two-sided quaternion Fourier transform (QFT)ofa quaternion function f ∈ L ( R , H ) is the function F q ( f ) : R → H defined by:for ω = ( ω , ω ) ∈ R × R F q ( f )( w ) = Z R e − πix .ω f ( x ) e − πjx .ω dx (2.1) where dx = dx dx This transform can be inverted by means of:
Theorem 2.2. If f, F q ( f ) ∈ L ( R , H ) , then, f ( x ) = F − q F q ( f )( x ) = Z R e πix .ω F q ( f )( ω ) e πjx .ω dω (2.2) Theorem 2.3 (Plancherel theorem for QFT ) . If f ∈ L ( R , H ) then k f k = kF q ( f ) k (2.3) Proof.
See [16] (cid:3)
Definition 2.4.
A quaternion window function is a non null function ϕ ∈ L ( R , H ) HE TWO-SIDED GABOR QUATERNIONIC FOURIER TRANSFORM AND SOME UNCERTAINTY PRINCIPLES 3
Based on the above formula 2.1 for the QFT, we establish the following definition of the two-sidedGabor quaternionic Fourier transform (GQFT).
Definition 2.5.
We define the GQFT of f ∈ L ( R , H ) with respect to non-zero quaternion windowfunction ϕ ∈ L ( R , H ) as, G ϕ f ( ω, b ) = Z R e − πix ω f ( x ) ϕ ( x − b ) e − πjx .ω dx (2.4)Note that the order of the exponentials in 2.4 is fixed because of the non-commutativity of theproduct of quaternion.The energy density is defined as the modulus square of GQFT 2.5 given by | G ϕ f ( ω, b ) | = | Z R e − iπx ω f ( x ) ϕ ( x − b ) e − jπx ω dx | (2.5)The equation 2.5 is often called a spectogram which measures the energy of a quaternion-valuedfunction f in the position-frequency neighbourhood of ( ω, b ). For illustrative purposes, we shall discuss examples of the GQFT. We begin with a straightforwardexample.
Example 1
Consider the two-dimensional window function defined by ϕ ( x ) = , for − ≤ x ≤ − ≤ x ≤ , otherwise (2.6)Obtain the GQFT of the function defined as follows f ( x ) = e − x − x , ≤ x ≤ + ∞ and 0 ≤ x ≤ + ∞ ;0 , otherwise (2.7)By applying the definition of the GQFT we have G ϕ f ( ω, b ) = Z b m Z b m e − i πx ω e − x − x e − j πx ω dx dx , with m = max (0 , − b ); m = max (0 , − b ) , = Z b m e − x (1+ i πω ) dx Z b m e − x (1+ j πω ) dx , = [ e − x (1+ i πω ) ( − − i πω ) ] b m [ e − x (1+ j πω ) ( − − j πω ) ] b m , = 1( − − i πω )( − − j πω ) ( e − (1+ b )(1+ i πω ) − e − m (1+ i πω ) )( e − (1+ b )(1+ j πω ) − e − m (1+ j πω ) ) . Example 2
Given the window function of the two-dimensional Haar function defined by: ϕ ( x ) = , for 0 ≤ x ≤ and 0 ≤ x ≤ ; − , for ≤ x ≤ ≤ x ≤ , otherwise; (2.8) S.FAHLAOUI AND M.EL KASSIMI find the GQFT of the Gaussian function f ( x ) = e − ( x + x ) .From definition 2.5 we obtain G ϕ { f } ( ω, b ) = Z R e − i πx ω f ( x ) ϕ ( x − b ) e − j πx ω dx, = Z + b b e − i πx ω e − x dx Z + b b e − x e − j πx ω dx , − Z b + b e − i πx ω e − x dx Z b + b e − x e − j πx ω dx , by completing squares, we have G ϕ { f } ( ω, b ) = Z + b b e − ( x + iπω ) e − ( ω π ) dx Z + b b e − ( x + jπω ) e − ( ω π ) dx − Z b + b e − ( x + iπω ) e − ( ω π ) dx Z b + b e − ( x + jπω ) e − ( ω π ) dx making the substitutions y = x + iπω and y = x + jπω in the above expression we immediatelyobtain : G ϕ { f } ( ω, b ) = e − ( ω + ω ) π ( Z + b + iπω b + iπω e − y dy Z + b + jπω b + jπω e − y dy − Z b + iπω + b + iπω e − y dy Z b + jπω + b + jπω e − y dy , )= e − ( ω + ω ) π ( Z b + iπω ( − e − y ) dy + Z + b + iπω e − y dy ) × ( Z b + jπω ( − e − y ) dy + Z + b + jπω e − y dy ) − e − ( ω + ω ) π ( Z + b + iπω ( − e − y ) dy + Z b + iπω e − y dy ) × ( Z + b + jπω ( − e − y ) dy + Z b + jπω e − y dy ) , (2.9)Equation 2.9 can be written in the form G ϕ { f } ( ω, b ) = e − ( ω + ω ) π { [ − qf (1 + b + iπω ) + qf ( 12 + b + iπω )] × [ − qf (1 + b + jπω ) + qf ( 12 + b + jπω )] − [ − qf ( 12 + b + iπω ) + qf (1 + b + iπω )] × [ − qf ( 12 + b + jπω ) + qf (1 + b + jπω )] } Where, qf ( x ) = R x e − t dt. In this section, we are going to to give some properties for the Gabor quaternionic Fourier transform.
Theorem 3.1.
Let f ∈ L ( R , H ) ; and ϕ ∈ L ( R , H ) be a non zero quaternionic window function.Then, we have ( G ϕ { T y f } ( ω, b ) = e − iπy ω ( G ϕ f )( ω, b − y ) e − jπx .ω where T y f ( x ) = f ( x − y ) ; and y = ( y , y ) ∈ R Proof.
We have G ϕ { T y f } ( w, b ) = Z R e − πix ω f ( x ) ϕ ( x − b ) e − πjx .ω dx we take t = x − y , then G ϕ { T y f } ( w, b ) = Z R e − iπ ( t + y ) ω f ( x ) ϕ ( t + y − b ) e − jπ ( t + y ) ω dt = e − iπy ω Z R e − iπt ω f ( x ) ϕ ( t + y − b ) e − jπt ω dt e − iπy ω = e − iπy ω G ϕ { f } ( ω, b − y ) e − jπx .ω . HE TWO-SIDED GABOR QUATERNIONIC FOURIER TRANSFORM AND SOME UNCERTAINTY PRINCIPLES 5 (cid:3)
Theorem 3.2.
Let ϕ ∈ L ( R , H ) a quaternion window function. Then we have G e ϕ ( e f )( ω, b ) = G ϕ { f } ( − ω, − b ) Where e ϕ ( x ) = ϕ ( − x ) ; ∀ ϕ ∈ L ( R , H ) Proof.
A direct calculation allows us to obtain for every f ∈ L ( R , H ) G e ϕ ( e f )( ω, b ) = Z R e − iπx ω f ( − x ) ϕ ( − ( x − b )) e − jπx ω dx = Z R e − iπ ( − x )( − ω ) f ( − x ) ϕ ( − x − ( − b )) e − jπ ( − x )( − ω ) dx = G ϕ { f } ( − ω, − b ) (cid:3) For establishing an inversion formula and Plancherel identity for GQFT we use the fact that, theGQFT can be expressed in terms of two-sided quaternionic Fourier transform. G ϕ { f } ( ω, b ) = F q { f ( . ) ϕ ( . − b ) } ( ω ) Theorem 3.3 (Inversion formula) . Let ϕ be a quaternion window function. Then for every function f ∈ L ( R , H ) can be reconstructed by : f ( x ) = 1 k ϕ k Z R Z R e iπx ω G ϕ f ( w, b ) e jπx ω ϕ ( x − b ) dωdb Proof.
We have G ϕ { f } ( ω, b ) = Z R e − iπx ω f ( x ) ϕ ( x − b ) e − jπx ω dx then G ϕ { f } ( ω, b ) = F q ( f ( x ) ϕ ( x − b )) (3.1)Taking the inverse of two-sided QFT of both sides of 3.1 we obtain f ( x ) ϕ ( x − b ) = F − q G ϕ f ( ω, b )( x )= Z R e iπx ω G ϕ { f } ( ω, b ) e jπx ω dω, (3.2)Multiplying both sides of 3.2 from the right and integrating with respect to db we get f ( x ) Z R | ϕ ( x − b ) | db = Z R Z R e iπx ω G ϕ f ( w, b ) e jπx ω ϕ ( x − b ) dωdb then, f ( x ) = 1 k ϕ k Z R Z R e iπx ω G ϕ f ( w, b ) e jπx ω ϕ ( x − b ) dωdb Set C ϕ = k ϕ k R and assume that 0 < C ϕ < ∞ . Then the inversion formula can also written as f ( x ) = 1 C ϕ Z R Z R e iπx ω G ϕ f ( w, b ) e jπx ω ϕ ( x − b ) dωdb (cid:3) Theorem 3.4 (Plancherel theorem ) . Let ϕ be quaternion window function and f ∈ L ( R , H ) , then we have kG ϕ { f }k = k f k k ϕ k (3.3) S.FAHLAOUI AND M.EL KASSIMI
Proof.
We have kG ϕ { f }k = kF q ( f ( x ) ϕ ( x − b )) k = k f ( x ) ϕ ( x − b ) k (3.4)= Z R Z R | f ( x ) | | ϕ ( x − b ) | dxdb = Z R | f ( x ) | dx Z R | ϕ ( t ) | dt (3.5)= k f k k ϕ k where in line 3.4 we use the the Plancherel theorem’s of QFT 2.3. (cid:3) In this section we demonstrate some versions of uncertainty principles and inequalities for the twosided quaternion windowed Fourier transform.
Before proving the Heisenberg uncertainty principle for GQFT, first, we are giving a version ofHeisenberg uncertainty for the QFT, that we will use it to demonstrate our result.
Theorem 4.1.
Let f ∈ L ( R , H ) be a quaternion-valued signal such that : x k f, ∂∂x k f ∈ L ( R , H ) for k = 1 , , then, (cid:18)Z R x k | f ( x ) | dx (cid:19) (cid:18)Z R ω k |F q ( f )( ω ) | dω (cid:19) ≥ π k f k , (4.1)To prove this theorem, we need the following result, Lemma 4.2.
Let f ∈ L ∩ L ( R , H ) . If ∂∂x k f exist and belong to L ( R , H ) for k = 1 , . Then (2 π ) Z R ω k |F ( f ( x ))( ω ) | dω = Z R | ∂∂x k f ( x ) | dx. (4.2) Proof.
See [11]. (cid:3)
We are going to prove the first theorem 4.1.
Proof.
For k ∈ ,
2. First, by applying lemma 4.2 and Plancherel’s theorem 3.3, we obtain R R x k | f ( x ) | dx R R ω k |F q ( f )( ω ) | d ω R R | f ( x ) | dx R R |F q ( f )( ω ) | dω == π ) R R x k | f ( x ) | dx R R | ∂∂x k f ( x ) | dω R R | f ( x ) | dx R R |F q ( f )( ω ) | dω = π ) R R x k | f ( x ) | dx R R | ∂∂x k f ( x ) | dω ( R R | f ( x ) | dx ) ≥ π ( R R ( ∂∂x k f ( x ) x k f ( x ) + x k f ( x ) ∂∂x k f ( x )) dx ) k f ( x ) k = 116 π ( R R x k ∂∂x k ( f ( x ) f ( x )) dx ) k f ( x ) k Second, using integration par parts, we further get,= 116 π ([ R R x k | f ( x ) | dx l ] x k =+ ∞ x k = −∞ − R R k f ( x ) k dx ) k f ( x ) k = 116 π then, (cid:18)Z R x k | f ( x ) | dx (cid:19) (cid:18)Z R ω k |F q ( f )( ω ) | dω (cid:19) ≥ π k f k (cid:3) Applying the Plancherel theorem for the QFT 2.3 to the right-hand side of 4.1, we get the followingcorollary,
Corollary 4.3.
Under the above assumptions, we have (cid:18)Z R x k |F − q {F q ( f ) } ( x ) | dx (cid:19) (cid:18)Z R ω k |F q ( f )( ω ) | dω (cid:19) ≥ π kF q ( f ) k (4.3)Now, we are going to establish a generalization of the Heisenberg type uncertainty principle for theGQFT. Theorem 4.4 (Heisenberg for GQFT) . Let ϕ ∈ L ( R , H ) be a quaternion window function and let G ϕ { f } ∈ L ( R , H ) be the GQFT of f such that ω k G ϕ { f } ∈ L ( R , H ) , k = 1 , . Then for every f ∈ L ( R , H ) we have the following inequality (cid:18)Z R x k | f ( x ) | dx (cid:19) (cid:18)Z R Z R ω k |G ϕ { f } ( ω, b ) | dωdb (cid:19) ≥ π k f k k ϕ k (4.4)In order to prove this theorem, we need to introduce the following lemmas. The first lemma calledthe Cauchy-Schwartz inequality, Lemma 4.5.
Let f, g ∈ L ( R , H ) be two quaternion on valued functions. Then the Cauchy-Schwartzinequality takes the form | Z R f ( x ) g ( x ) dx | ≤ Z R | f ( x ) | dx Z R | g ( x ) | dx Lemma 4.6.
Under the assumptions of theorem 4.4, we have k ϕ k Z R x k | f ( x ) | dx = Z R Z R x k |F − q {G ϕ { f } ( ω, b ) } ( x ) | dxdb (4.5) for k = 1 , .Proof. Applying elementary properties of quaternion, we get k ϕ k Z R x k | f ( x ) | dx = Z R x k | f ( x ) | dx Z R | ϕ ( x − b ) | db = Z R Z R x k | f ( x ) | | ϕ ( x − b ) | dxdb = Z R Z R x k | f ( x ) ϕ ( x − b ) | dxdb = Z R Z R x k |F − ( G ϕ { f } ( ω, b ))( x ) | dxdb (cid:3) Now, we are going to prove the theorem 4.4.
S.FAHLAOUI AND M.EL KASSIMI
Proof. (of theorem 4.4) Replacing the QFT of f by the GQFT of the left hand side of 4.3 in corollary4.3, we obtain (cid:18)Z R x k |F − q {G ϕ { f } ( ω, b ) } ( x ) | dx (cid:19) (cid:18)Z R ω k |G ϕ { f } ( ω, b ) | dω (cid:19) ≥ π (cid:18)Z R | G ϕ f ( ω, b ) | dω (cid:19) (4.6)we have, F − ( G ϕ { f } ( ω, b ))( x ) = f ( x ) ϕ ( x − b )Taking the square root on both sides of 4.6 and integrating both sides with respect to db we get Z R (cid:18)Z R x k |F − q {G ϕ { f } ( ω, b ) } ( x ) | dx (cid:19) (cid:18)Z R ω k |G ϕ { f } ( ω, b ) | dω (cid:19) db ≥ π Z R Z R |G ϕ { f } ( ω, b ) | dωdb (4.7)Applying the Cauchy-Schwartz inequality 4.5 to the left-hand side of 4.7 we obtain (cid:18)Z R Z R x k |F − q {G ϕ { f } ( ω, b ) } ( x ) | dxdb (cid:19) (cid:18)Z R Z R ω k G ϕ { f } ( ω, b ) | dωdb (cid:19) ≥ π Z R Z R |G ϕ { f } ( ω, b ) | dωdb (4.8)Using lemma 4.5 into the second term on the left-hand side of 4.8, and use the Plancherel’s formula3.4 into the right-hand side of 4.8, we obtain that (cid:18) k ϕ k Z R x k | f ( x ) | dx (cid:19) (cid:18)Z R Z R ω k |G ϕ { f } ( ω, b ) | dωdb (cid:19) ≥ π k f k k ϕ k (4.9)Now, simplifying both sides of 4.9 by k ϕ k , we get our result. (cid:3) Definition 5.1.
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 ([ ? ]) 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 .we have the logarithmic uncertainty principle for the QFT [ ? ] as follows Theorem 5.2 (QFT logarithmic uncertainty principle ) . For f ∈ S ( R , H ) , we have Z R ln | x || f ( x ) | dx + Z R ln | ω ||F Q { f } ( ω ) | dω ≥ Γ ′ ( t )Γ( t ) − lnπ ! Z R | f ( x ) | dx, (5.1) Where Γ ′ ( t ) = (cid:0) ddt (cid:1) and Γ( t ) is Gamma function. Remark 5.3.
If we apply Plancherl’s theorem for QFT [] to the right hand side of 5.1, we get Z R ln | x || f ( x ) | dx + Z R ln | ω ||F Q { f } ( ω ) | dω ≥ Γ ′ ( t )Γ( t ) − lnπ ! Z R |F Q { f } ( ω ) | dωdx, (5.2) HE TWO-SIDED GABOR QUATERNIONIC FOURIER TRANSFORM AND SOME UNCERTAINTY PRINCIPLES 9
Lemma 5.4.
Let ϕ ∈ S ( R , H ) a windowed quaternionic function and f ∈ S ( R , H ) . We have Z R Z R ln | x ||F − Q { G ϕ f ( ω, b ) } ( x ) | dxdb = k ϕ k L ( R , H ) Z R ln | x || f ( x ) | dx (5.3) Proof.
By a simple calculation we get, Z R Z R ln | x ||F − Q { G ϕ f ( ω, b ) } ( x ) | dxdb = Z R Z R ln | x || f ( x ) ϕ ( x − b ) | dxdb = Z R Z R ln | x || f ( x ) | | ϕ ( x − b ) | dxdb = Z R ln | x || f ( x ) | ( Z R | ϕ ( x − b ) | db ) dx (5.4)= k ϕ k L ( R , H ) Z R ln | x || f ( x ) | dx. To obtain the result of lemma 5.4 we use a substitution in 5.4. (cid:3)
Corollary 5.5.
For f ∈ S ( R , H ) , and ϕ ∈ S ( R , H ) , we have Z R ln | x ||F − Q F Q ( f )( x ) | dx + Z R ln | ω ||F Q { f } ( ω ) | dω ≥ Γ ′ ( t )Γ( t ) − lnπ !Z R |F Q ( ω ) | dω, (5.5) Theorem 5.6.
Let f ∈ S ( R , H ) and ϕ ∈ S ( R , H ) a quaternionc windowed function, we have thefollowing algorithmic inequality, k ϕ k L ( R , H ) Z R ln | x || f ( x ) | dx + Z R Z R ln | ω || G ϕ f ( ω, b ) | dωdb ≥ k ϕ k L ( R , H ) Γ ′ ( t )Γ( t ) − lnπ ! Z R Z R | f ( x ) | dx, (5.6) Proof.
For classical two-sided quaternionic Fourier transform, by theorem 5.2, Z R ln | x || f ( x ) | dx + Z R ln | ω ||F Q { f } ( ω ) | dω ≥ Γ ′ ( t )Γ( t ) − lnπ ! Z R | f ( x ) | dx, (5.7)we replace f by G ϕ f on both sides of 5.7, we get Z R ln | ω || G ϕ f ( ω, b ) | dω + Z R ln | x ||F Q { G ϕ f } ( x ) | dx ≥ Γ ′ ( t )Γ( t ) − lnπ !Z R | G ϕ f ( ω, b ) | dx, (5.8)Integrating both sides of this equation with respect to db , we obtain Z R Z R ln | ω || G ϕ f ( ω, b ) | dωdb + Z R Z R ln | x ||F Q { G ϕ f } ( x ) | dxdb ≥ Γ ′ ( t )Γ( t ) − lnπ !Z R Z R | G ϕ f ( ω, b ) | dxdb, (5.9)Applying lemma 5.4 into the second term on the left hand side of 5.9, yields Z R Z R ln | ω || G ϕ f ( ω, b ) | dωdb + k ϕ k L ( R , H ) Z R ln | x || f ( x ) | dx ≥ Γ ′ ( t )Γ( t ) − lnπ ! k ϕ k L ( R , H ) Z R Z R | G ϕ f ( ω, b ) | dxdb, (5.10)we applying the Placncherel formula, we obtain our desired result, k ϕ k L ( R , H ) Z R ln | x || f ( x ) | dx + Z R Z R ln | ω || G ϕ f ( ω, b ) | dωdb ≥ k ϕ k L ( R , H ) Γ ′ ( t )Γ( t ) − lnπ !Z R Z R | f ( x ) | dx, (5.11) (cid:3) References [1] D. Assefa, L. Mansinha, KF. Tiampo, et al.,
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HE TWO-SIDED GABOR QUATERNIONIC FOURIER TRANSFORM AND SOME UNCERTAINTY PRINCIPLES11
Department of Mathematics and Computer Sciences, Faculty of Sciences-Meknès,Equipe d’Analyse Harmonique et Probabilités, University Moulay Ismaïl,BP 11201 Zitoune, Meknes, Morocco
Saïd Fahlaoui
E-mail address : [email protected] Mohammed El kassimi
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