# Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle

Authors: Mawardi Bahri; Ryuichi Ashino
Journal: Journal of Mathematics
 (2019)

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## Abstract

A definition of the two-dimensional quaternion linear canonical transform (QLCT) is proposed. The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the definition of the classical linear canonical transform (LCT). Several useful properties of the QLCT are obtained from the properties of the QLCT kernel. Based on the convolutions and correlations of the LCT and QFT, convolution and correlation theorems associated with the QLCT are studied. An uncertainty principle for the QLCT is established. It is shown that the localization of a quaternion-valued function and the localization of the QLCT are inversely proportional and that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.


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## Body

## 1. Introduction

The quaternion Fourier transform (QFT) is a nontrivial generalization of the classical Fourier transform (FT) using the quaternion algebra. The QFT has been shown to relate to the other quaternion signal analysis tools, such as quaternion wavelet transform [1–3], fractional quaternion Fourier transform [4, 5], quaternionic windowed Fourier transform [6–9], and quaternion Wigner transform [10]. Because of the noncommutative property of quaternion multiplication, we obtain at least three different kinds of two-dimensional QFTs as follows (see [11–15]):(1)ℱqIfv=∫ℜ2e−μ1v⋅zfzdz,v⋅z=v1z1+v2z2,ℱqIIfv=∫ℜ2fze−μ1v⋅zdz,v⋅z=v1z1+v2z2,ℱqIIIfv=∫ℜ2e−μ1v1z1fze−μ2v2z2dz,dz=dz1dz2,where μ1 and μ2 are any two unit pure quaternions (μ12=μ22=−1) that are orthogonal to each other. These three QFTs are so-called left-sided, right-sided, and double-sided QFTs or type I, II, and III QFTs, respectively. As is well known, the linear canonical transform (LCT) is a general form of the FT, and the quaternion linear canonical transform (QLCT) is a generalization of the QFT in the LCT domain.In the recent years, the LCT and QLCT have received much attention. Kou et al. [16, 17], for instance, constructed the windowed LCT, with a local window function. It can reveal the local LCT-frequency contents and enjoys high concentrations and eliminates the cross term. It is used to study the generalized prolate spheroidal wave functions and the connection with energy concentration problems. In [18–20], Kou et al. also proposed the (right-sided) quaternion linear canonical transform (QLCT) which is a generalization of the quaternion Fourier transform (QFT) in the LCT domain. This generalization is obtained by replacing the Fourier kernel with the right-sided QFT (type III QFT) kernel in the LCT definition. Some important properties such as Parseval’s theorem, reconstruction formula, uncertainty principles, and asymptotic behaviour are discussed. Like the uncertainty principle for the QFT [3], they also showed that only a two-dimensional Gaussian signal minimizes the uncertainty. However, the convolution theorem is an important result of the QLCT which does not hold using this construction because of the noncommutative property of the right-sided quaternion Fourier kernel.Our attention in this article is to introduce a definition of the QLCT (compared to [21–23]). This definition is obtained by substituting the Fourier kernel with the type II QFT kernel (see [24]) in the LCT definition, which is essentially different from the kernel of the type III QFT. We investigate some important properties such as linearity, shift, and modulation and Plancherel’s and Parseval’s theorems. We study the convolution theorems associated with the QLCT, which can be useful in digital signal and image processing. Based on the convolution definitions of the LCT [25–27] and QFT, we also propose a new correlation definition for the QLCT and obtain its correlation theorems. We finally establish an uncertainty principle for the QLCT and show that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.We display here the organization of the paper: Section2 describes the preliminaries about the quaternion Fourier transform and its basic properties, which will be used in the next section. The construction of the QLCT is presented in Section 3; its important properties are also discussed in this section. The definition of the convolution in the QLCT domain is introduced and the theorem on the QLCT of the convolution of two quaternion-valued functions is constructed in Section 4. Section 5 provides the definition of the correlation in the QLCT domain and discusses the theorem on the QLCT of a correlation of two quaternion functions. Section 6 establishes an uncertainty principle for the QLCT, which shows that the spread of a quaternion-valued function and its QLCT are inversely proportional. It is shown that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.Throughout this paper,ℜ is used to denote a set of real numbers.


## 2. Preliminaries

### 2.1. Quaternion

The quaternion, which is a type of hypercomplex number, was originally invented by William Hamilton in 1843 [28]. It is a generalization of a complex number to a 4D algebra and is denoted by ℍ. Every element of ℍ can be written in the hypercomplex form as follows:(2)ℍ=q=q0+iq1+jq2+kq3:q0,q1,q2,q3∈ℜ.Here, the three different imaginary parts satisfy the following multiplication rules:(3)ij=−ji=k,jk=−kj=i,ki=−ik=j,i2=j2=k2=ijk=−1.For a quaternionq=q0+iq1+jq2+kq3∈ℍ, q0 is called the scalar part of q denoted by Scq and q=iq1+jq2+kq3 is called the pure part of q denoted by Vecq=iq1+jq2+kq3. A quaternionic conjugation is given by(4)q¯=q0−iq1−jq2−kq3.Any quaternionq can be represented in the polar form as (see [29])(5)q=qeμθ,q=qq¯=q02+q12+q22+q32,where θ=arctanScq/Vecq, 0≤θ≤π, is the eigen angle or phase of q and μ is an arbitrarily fixed unit quaternion such that μ2=−1. When q=1, q is a unit quaternion. Note that Euler’s formula holds for quaternions; that is, eμθ=cosθ+μsinθ. We also have eμθ=1. Hereinafter, besides the quaternion units i, j, and k and the vector part q of a quaternion q∈ℍ, we shall use the following real vector notation:(6)z=z1,z2∈ℜ2,fz=fz1,z2,z2=z12+z22,z⋅y=z1y1+z2y2,zy=z1y1,z2y2,and so on when there is no confusion. We may define the left quaternion inner product for two functions f and g by(7)f,gL2ℜ2;ℍ=∫ℜ2fzgz¯dz.We see that, forf=g, we obtain the norm induced by the above inner product as(8)fL2ℜ2;ℍ=∫ℜ2fz2dz1/2.We find it convenient to introduce the space as follows:(9)L2ℜ2;ℍ=f∣f:ℜ2→ℍ,fL2ℜ2;ℍ<∞.


### 2.2. Convolution Associated with QFT

The quaternion Fourier transform (QFT) was originally proposed by Ell [11] and applied in quaternion color image processing [12, 30]. Some results related to the fundamental properties of the QFT can be found in [31–33]. We introduce a definition of the QFT as follows.Definition 1.
The QFT off∈L2ℜ2;ℍ is the function ℱqf∈L2ℜ2,ℍ given by(10)ℱqfv=∫ℜ2fze−μv⋅zdz,where µ is an arbitrarily fixed pure quaternion such that μ2=−1. The inverse transform of the QFT is given by(11)fz=ℱq−1ℱqfz=12π2∫ℜ2ℱqfveμv⋅zdv,provided that the integral exists.Definition 2.
The convolution off∈L2ℜ2;ℍ and g∈L2ℜ2;ℍ, denoted by f⋆g, is defined by(12)f⋆gz=∫ℜ2fygz−ydy.
We have the following important result on the QFT of a convolution of two quaternion functions.Theorem 1 [34].
Letf, g∈L2ℜ2;ℍ have the following representations:(13)fz=f0z+if1z+jf2z+kf3z,gz=g0z+ig1z+jg2z+kg3z.
Then, we have(14)ℱqf⋆gv=ℱqgvℱqf0v+iℱqgvℱqf1v+jℱqgvℱqf2v+kℱqgvℱqf3v.It is easily seen that the convolution theorem in the FT domain is a special case of Theorem 1.


## 2.1. Quaternion

The quaternion, which is a type of hypercomplex number, was originally invented by William Hamilton in 1843 [28]. It is a generalization of a complex number to a 4D algebra and is denoted by ℍ. Every element of ℍ can be written in the hypercomplex form as follows:(2)ℍ=q=q0+iq1+jq2+kq3:q0,q1,q2,q3∈ℜ.Here, the three different imaginary parts satisfy the following multiplication rules:(3)ij=−ji=k,jk=−kj=i,ki=−ik=j,i2=j2=k2=ijk=−1.For a quaternionq=q0+iq1+jq2+kq3∈ℍ, q0 is called the scalar part of q denoted by Scq and q=iq1+jq2+kq3 is called the pure part of q denoted by Vecq=iq1+jq2+kq3. A quaternionic conjugation is given by(4)q¯=q0−iq1−jq2−kq3.Any quaternionq can be represented in the polar form as (see [29])(5)q=qeμθ,q=qq¯=q02+q12+q22+q32,where θ=arctanScq/Vecq, 0≤θ≤π, is the eigen angle or phase of q and μ is an arbitrarily fixed unit quaternion such that μ2=−1. When q=1, q is a unit quaternion. Note that Euler’s formula holds for quaternions; that is, eμθ=cosθ+μsinθ. We also have eμθ=1. Hereinafter, besides the quaternion units i, j, and k and the vector part q of a quaternion q∈ℍ, we shall use the following real vector notation:(6)z=z1,z2∈ℜ2,fz=fz1,z2,z2=z12+z22,z⋅y=z1y1+z2y2,zy=z1y1,z2y2,and so on when there is no confusion. We may define the left quaternion inner product for two functions f and g by(7)f,gL2ℜ2;ℍ=∫ℜ2fzgz¯dz.We see that, forf=g, we obtain the norm induced by the above inner product as(8)fL2ℜ2;ℍ=∫ℜ2fz2dz1/2.We find it convenient to introduce the space as follows:(9)L2ℜ2;ℍ=f∣f:ℜ2→ℍ,fL2ℜ2;ℍ<∞.


## 2.2. Convolution Associated with QFT

The quaternion Fourier transform (QFT) was originally proposed by Ell [11] and applied in quaternion color image processing [12, 30]. Some results related to the fundamental properties of the QFT can be found in [31–33]. We introduce a definition of the QFT as follows.Definition 1.
The QFT off∈L2ℜ2;ℍ is the function ℱqf∈L2ℜ2,ℍ given by(10)ℱqfv=∫ℜ2fze−μv⋅zdz,where µ is an arbitrarily fixed pure quaternion such that μ2=−1. The inverse transform of the QFT is given by(11)fz=ℱq−1ℱqfz=12π2∫ℜ2ℱqfveμv⋅zdv,provided that the integral exists.Definition 2.
The convolution off∈L2ℜ2;ℍ and g∈L2ℜ2;ℍ, denoted by f⋆g, is defined by(12)f⋆gz=∫ℜ2fygz−ydy.
We have the following important result on the QFT of a convolution of two quaternion functions.Theorem 1 [34].
Letf, g∈L2ℜ2;ℍ have the following representations:(13)fz=f0z+if1z+jf2z+kf3z,gz=g0z+ig1z+jg2z+kg3z.
Then, we have(14)ℱqf⋆gv=ℱqgvℱqf0v+iℱqgvℱqf1v+jℱqgvℱqf2v+kℱqgvℱqf3v.It is easily seen that the convolution theorem in the FT domain is a special case of Theorem 1.


## 3. Quaternion Linear Canonical Transform

The linear canonical transform (LCT) is a linear integral transformation with three free parameters which has widely been used in various fields such as spectral analysis, image processing, and optical system analysis [35, 36]. Several famous transforms such as the Fourier transform, the fractional Fourier transform, and the other transformations are special cases. This section will consider generalizing the LCT using the quaternion algebra. This extension is then called the quaternion linear canonical transform (QLCT).


### 3.1. Definition of QLCT

Based on the definition of the type II QFT, we obtain a definition of the QLCT by replacing the kernel of the FT with the kernel of the type II QFT in the classical LCT definition. The special linear group of degree 2 overℜ, that is, the group of all real 2×2 matrices with determinant one, is denoted by SL2,ℜ. Let(15)As=asbscsds∈SL2,ℜ,s=1,2.Whenb1b2≠0, we define the kernels KAs, s=1,2, of the QLCT by(16)KAszs,vs=12πμbse1/2μas/bszs2−2/bszsvs+ds/bsvs2,s=1,2.Definition 3.
The QLCT off∈L1ℜ2;ℍ is defined by(17)LA1,A2ℍfv=∫ℜ2fzKA1z1,v1KA2z2,v2dz,b1b2≠0,d1d2fd1v1,d2v2eμc1d1/2v12eμc2d2/2v22,b1b2=0,where eμcsds/2vs2, s=1,2, are called chirp signals in signal processing. Here after, we will deal with the case when b1b2≠0 because LA1,A2ℍfv is trivial for b1b2=0. When A1=A2=01−10, the QLCT reduces to the QFT, that is,(18)LA1,A2ℍfv=∫ℜ2fz12πμe−μv1z112πμe−μv2z2dz=∫ℜ2fze−μv⋅zdz12πμ=ℱqfv12πμ.
The polar form of the QLCT (17) can be found in [37].
The following describes the general relationship between the QLCT and the type II QFT of a signalf.Lemma 1.
The QLCT of a signalf with As=asbscsds∈SL2,ℜ, s=1,2, can be seen as the QFT of a signal f in the form(19)LA1,A2ℍfv=ℱqfzeμa1/2b1z12eμa2/2b2z22v1b1,v2b212πμb112πμb2eμd1/2b1v12eμd2/2b2v22.Proof.
By a straightforward computation, it follows from Definition 3 that(20)LA1,A2ℍfv=∫ℜ2fz12πμb112πμb2e1/2μa1/b1z12−2/b1z1v1+d1/b1v12e1/2μa2/b2z22−2/b2z2v2+d2/b2v22dz=∫ℜ2fzeμa1/2b1z12eμa2/2b2z22e−μz1v1/b1e−μz2v2/b2dz12πμb112πμb2eμd1/2b1v12eμd2/2b2v22=ℱqfzeμa1/2b1z12eμa2/2b2z22v1b1,v2b212πμb112πμb2eμd1/2b1v12eμd2/2b2v22,thus proving the theorem.Theorem 2.
Iff∈L1ℜ2;ℍ and LA1,A2ℍ∈L1ℜ2;ℍ, then the inversion formula of the QLCT is given by(21)fz=∫ℜ2LA1,A2ℍfvKA1−1v1,z1KA2−1v2,z2dv=∫ℜ2LA1,A2ℍfvKA1z1,v1¯KA2z2,v2¯dv,which is equivalent to(22)fz=∫ℜ2LA1,A2ℍfv1−2πμb1e−1/2μa1/b1z12−2/b1z1v1+d1/b1v121−2πμb2×e−1/2μa2/b2z22−2/b2z2v2+d2/b2v22dv,where As−1=ds−bs−csas, s=1,2.Proof.
A direct calculation gives(23)fy=∫ℜ2∫ℜ2fy1−2πμb11−2πμb2e−1/2μa1/b1y12−2/b1y1v1+d1/b1v12e−1/2μa2/b2y22−2/b2y2v2+d2/b2v22×12πμb112πμb2e1/2μa1/b1z12−2/b1z1v1+d1/b1v12e1/2μa2/b2z22−2/b2z2v2+d2/b2v22dvdy=∫ℜ2∫ℜ2fy12πb1e−a1/2b1μy12−z12eμ/b1y1−z1v1×12πb2e−a2/2b2μy22−z22eμ/b2y2−z2v2dvdy=∫ℜ2fye−a1/2b1μy12−z12δy1−z1e−a2/2b2μy22−z22δy2−z2dy,thus proving the theorem.
The following proposition will be very useful when proving the uncertainty principle for the QLCT.Proposition 1.
Let the kernel functions
K
A
s
z
s
,
v
s, s=1,2, be defined by (16). Then, we have(24)∫ℜvskKAszs,vsKAsys,vs¯dvs=bsk−μ−kδkzs−ys,k∈ℕ∪0,s=1,2,where δk denotes the k-th derivative of the delta function δ.Proof.
Fork∈ℕ∪0, we have(25)∫ℜvskKAszs,vsKAsys,vs¯dvs=12πbs∫ℜvskeμas/2bszs2−1/bszsvs+ds/2bsvs2e−μas/2bsys2−1/bsysvs+ds/2bsvs2dvs=12πbs∫ℜvskeμas/2bszs2−ys2bskvsk−μ−k∂∂zskeμ1/bsvsys−zsdvs=eμas/2bszs2−ys2bsk−μ−k∂∂zsk12πbs∫ℜeμ1/bsvsys−zsdvs=eμas/2bszs2−ys2bsk−μ−k∂∂zskδzs−ys=bsk−μ−keμas/2bszs2−ys2δkzs−ys.
Since the distributional support ofδkzs−ys as a distribution of zs is ys, we have(26)eμas/2bszs2−ys2δkzs−ys=eμas/2bszs2−zs2zs=ysδkzs−ys=δkzs−ys,in the distributional sense. With (25) and (26), we have (24).


### 3.2. Useful Properties of the QLCT

Parseval’s theorem for Fourier transform can be generalized to the QLCT as well. Let us now formulate Parseval’s theorem in the QLCT domain.Theorem 3 (QLCT Parseval).
Letf, g∈L2ℜ2;ℍ. Then, we have(27)f,g=LA1,A2ℍf,LA1,A2ℍg.Proof.
Forf, g∈L2ℜ2,ℍ, we have(28)f,g=∫ℜ2fzgz¯dz=∫ℜ2∫ℜ2LA1,A2ℍfvKA1−1v1,z1KA2−1v2,z2dvgz¯dz=∫ℜ2LA1,A2ℍfv∫ℜ2KA1z1,v1¯KA2z2,v2¯gz¯dzdv=∫ℜ2LA1,A2ℍfv∫ℜ2gzKA1z1,v1KA2z2,v2dz¯dv=∫ℜ2LA1,A2ℍfvLA1,A2ℍgv¯dv=LA1,A2ℍf,LA1,A2ℍg.
In the second equality in Theorem 3, we have replaced the quaternion functionf with its inverse QLCT expression in Theorem 2. In the third equality, we have interchanged the order of integration. In the fourth equality, we have applied the quaternion conjugation rule pq¯=q¯p¯ for p, q∈ℍ, thus proving the theorem.
Due to the noncommutativity of the kernel of the QLCT, we only have a left linearity property with quaternion constants; that is,(29)LA1,A2ℍαf+βgv=αLA1,A2ℍfv+βLA1,A2ℍgv.
In the following, we will summarize the important properties of the QLCT. We will see that the results are generalizations of the basic properties of the LCT. Letf∈L2ℜ2;ℍ. The translation operator τk by k∈ℜ2 is defined by(30)τkfz=fz−k.
The modulation operatorMv0 by v0=u0,v0∈ℜ2 is defined by(31)Mv0fz=fzeμz⋅v0,z⋅v0=z1u0+z2v0.
We now begin with the shift property of the QLCT.Theorem 4 (shift property).
Forf∈L2ℜ2;ℍ, we have(32)LA1,A2ℍτkfv=LA1,A2ℍfv1−a1k1,v2−a2k2e−1/2μa1c1k12+a2c2k22eμc1k1v1+c2k2v2.Next, we are concerned about the behavior of the QLCT under modulation.Theorem 5 (modulation property).
Forf∈L2ℜ2;ℍ, we have(33)LA1,A2ℍMv0fv=LA1,A2ℍfzeμz⋅v0v=LA1,A2ℍfv−bv0e−1/2μb1d1u02+b2d2v02eμd1u0v1+d2v0v2.Theorem 6 (time-frequency shift).
Forf∈L2ℜ2;ℍ, we have(34)LA1,A2ℍMv0τkfv=LA1,A2ℍfz−keμz⋅v0v=LA1,A2ℍfv−ak−bv0e−1/2μa1c1k12+a2c2k22+b1d1u02+b2d2v02×eμc1k1+d1u0v1eμc2k2+d2v0v2e−μb1c1k1u0+b2c2k2v0.We present the following result, which is needed for deriving the uncertainty principle related to the QLCT.Theorem 7.
LetAs=asbscsds∈SL2,ℜ and bs≠0 for s=1,2. Assume that f∈C1ℜ2;ℍ, that f,∂f/∂zi∈L2ℜ2;ℍ, and that viLA1,A2ℍf∈L2ℜ2;ℍ, i=1,2. Then, we have(35)∫ℜ2vi2LA1,A2ℍfv2dv=bi2∫ℜ2∂∂zifz2dz,i=1,2.Proof.
Recall the quaternion conjugation rulepq¯=q¯p¯ for p, q∈ℍ. The distributional calculation gives(36)∫ℜ2v12LA1,A2ℍfv2dv=∫ℜ2v12∫ℜ2fzKA1z1,v1KA2z2,v2dz∫ℜ2fyKA1y1,v1KA2y2,v2¯dydv=∫ℜ4∫ℜfzv12KA1z1,v1∫ℜKA2z2,v2KA2y2,v2¯dv2KA1y1,v1¯dv1fy¯dydz.
Usingk=0 in Proposition 1, we get(37)∫ℜ2v12LA1,A2ℍfv2dv=∫ℜ4fz∫ℜv12KA1z1,v1δz2−y2KA1y1,v1¯dv1fy¯dy2dy1dz=∫ℜ4fz∫ℜv12fzKA1z1,v1KA1y1,v1¯dv1fy1,z2¯dy1dz.
Again takingk=2 in Proposition 1 gives(38)∫ℜ2v12LA1,A2ℍfv2dv=b12−μ−2∫ℜ3fzδ2z1−y1fy1,z2¯dy1dz.
Since the delta function is even function and−μ−2=−1, we have(39)−μ−2δ2z1−y1=−∂∂z1−∂∂y1δz1−y1=−∂∂z1−∂∂y1δy1−z1.
Then, integration by parts in the sense of distribution implies(40)−μ−2∫ℜ3fzδ2z1−y1fy1,z2¯dy1dz=∫ℜ3fz−∂∂z1−∂∂y1δy1−z1fy1,z2¯dy1dz=∫ℜ2∂∂z1fz∫ℜδy1−z1∂∂y1fy1,z2¯dy1dz=∫ℜ2∂∂z1fz∂∂y1fy1,z2¯y1=z1dz=∫ℜ2∂∂z1fz∂∂z1fz¯dz,which implies the desired result.The above properties of the QLCT are summarized in Table1.Table 1
Useful properties of the QLCT.
Property
Quaternion func.
QLCT
Left linearity
α
f
+
β
g
α
L
A
1
,
A
2
ℍ
f
v
+
β
L
A
1
,
A
2
ℍ
g
v
Shift
f
z
−
k
L
A
1
,
A
2
ℍ
f
v
1
−
a
1
k
1
,
v
2
−
a
2
k
2
×
e
−
1
/
2
μ
a
1
c
1
k
1
2
+
a
2
c
2
k
2
2
e
μ
c
1
k
1
v
1
+
c
2
k
2
v
2
Modulation
f
z
e
μ
z
⋅
v
0
L
A
1
,
A
2
ℍ
f
v
−
b
v
0
e
−
1
/
2
μ
b
1
d
1
u
0
2
+
b
2
d
2
v
0
2
×
e
μ
d
1
u
0
v
1
+
d
2
v
0
v
2
Scaling
f
z
/
α
α
2
L
A
1
′
,
A
2
′
ℍ
f
v
/
α
,
A
i
′
=
a
i
b
i
/
α
2
c
i
α
2
d
i
,
i
=
1,2
Time-frequency shift
f
z
−
k
e
μ
z
⋅
v
0
L
A
1
,
A
2
ℍ
f
v
−
ak
−
b
v
0
×
e
−
1
/
2
μ
a
1
c
1
k
1
2
+
a
2
c
2
k
2
2
+
b
1
d
1
u
0
2
+
b
2
d
2
v
0
2
×
e
μ
c
1
k
1
+
d
1
u
0
v
1
e
μ
c
2
k
2
+
d
2
v
0
v
2
×
e
−
μ
b
1
c
1
k
1
u
0
+
b
2
c
2
k
2
v
0
Formula
Parseval’s theorem
f
=
L
A
1
,
A
2
ℍ
f
Plancherel’s theorem
f
,
g
=
L
A
1
,
A
2
ℍ
f
,
L
A
1
,
A
2
ℍ
g
Reconstruction
f
z
=
∫
ℜ
2
K
A
1
−
1
v
1
,
z
1
L
A
1
,
A
2
ℍ
f
v
K
A
2
−
1
v
2
,
z
2
d
v
∫
ℜ
v
s
k
K
A
s
z
s
,
v
s
K
A
s
y
s
,
v
s
¯
d
v
s
=
b
s
k
−
μ
−
k
δ
k
z
s
−
y
s
,
k
∈
ℕ
∪
0
∫
ℜ
2
v
i
2
L
A
1
,
A
2
ℍ
f
v
2
d
v
=
b
i
2
∫
ℜ
2
∂
/
∂
z
i
f
z
2
d
z
,
i
=
1,2


## 3.1. Definition of QLCT

Based on the definition of the type II QFT, we obtain a definition of the QLCT by replacing the kernel of the FT with the kernel of the type II QFT in the classical LCT definition. The special linear group of degree 2 overℜ, that is, the group of all real 2×2 matrices with determinant one, is denoted by SL2,ℜ. Let(15)As=asbscsds∈SL2,ℜ,s=1,2.Whenb1b2≠0, we define the kernels KAs, s=1,2, of the QLCT by(16)KAszs,vs=12πμbse1/2μas/bszs2−2/bszsvs+ds/bsvs2,s=1,2.Definition 3.
The QLCT off∈L1ℜ2;ℍ is defined by(17)LA1,A2ℍfv=∫ℜ2fzKA1z1,v1KA2z2,v2dz,b1b2≠0,d1d2fd1v1,d2v2eμc1d1/2v12eμc2d2/2v22,b1b2=0,where eμcsds/2vs2, s=1,2, are called chirp signals in signal processing. Here after, we will deal with the case when b1b2≠0 because LA1,A2ℍfv is trivial for b1b2=0. When A1=A2=01−10, the QLCT reduces to the QFT, that is,(18)LA1,A2ℍfv=∫ℜ2fz12πμe−μv1z112πμe−μv2z2dz=∫ℜ2fze−μv⋅zdz12πμ=ℱqfv12πμ.
The polar form of the QLCT (17) can be found in [37].
The following describes the general relationship between the QLCT and the type II QFT of a signalf.Lemma 1.
The QLCT of a signalf with As=asbscsds∈SL2,ℜ, s=1,2, can be seen as the QFT of a signal f in the form(19)LA1,A2ℍfv=ℱqfzeμa1/2b1z12eμa2/2b2z22v1b1,v2b212πμb112πμb2eμd1/2b1v12eμd2/2b2v22.Proof.
By a straightforward computation, it follows from Definition 3 that(20)LA1,A2ℍfv=∫ℜ2fz12πμb112πμb2e1/2μa1/b1z12−2/b1z1v1+d1/b1v12e1/2μa2/b2z22−2/b2z2v2+d2/b2v22dz=∫ℜ2fzeμa1/2b1z12eμa2/2b2z22e−μz1v1/b1e−μz2v2/b2dz12πμb112πμb2eμd1/2b1v12eμd2/2b2v22=ℱqfzeμa1/2b1z12eμa2/2b2z22v1b1,v2b212πμb112πμb2eμd1/2b1v12eμd2/2b2v22,thus proving the theorem.Theorem 2.
Iff∈L1ℜ2;ℍ and LA1,A2ℍ∈L1ℜ2;ℍ, then the inversion formula of the QLCT is given by(21)fz=∫ℜ2LA1,A2ℍfvKA1−1v1,z1KA2−1v2,z2dv=∫ℜ2LA1,A2ℍfvKA1z1,v1¯KA2z2,v2¯dv,which is equivalent to(22)fz=∫ℜ2LA1,A2ℍfv1−2πμb1e−1/2μa1/b1z12−2/b1z1v1+d1/b1v121−2πμb2×e−1/2μa2/b2z22−2/b2z2v2+d2/b2v22dv,where As−1=ds−bs−csas, s=1,2.Proof.
A direct calculation gives(23)fy=∫ℜ2∫ℜ2fy1−2πμb11−2πμb2e−1/2μa1/b1y12−2/b1y1v1+d1/b1v12e−1/2μa2/b2y22−2/b2y2v2+d2/b2v22×12πμb112πμb2e1/2μa1/b1z12−2/b1z1v1+d1/b1v12e1/2μa2/b2z22−2/b2z2v2+d2/b2v22dvdy=∫ℜ2∫ℜ2fy12πb1e−a1/2b1μy12−z12eμ/b1y1−z1v1×12πb2e−a2/2b2μy22−z22eμ/b2y2−z2v2dvdy=∫ℜ2fye−a1/2b1μy12−z12δy1−z1e−a2/2b2μy22−z22δy2−z2dy,thus proving the theorem.
The following proposition will be very useful when proving the uncertainty principle for the QLCT.Proposition 1.
Let the kernel functions
K
A
s
z
s
,
v
s, s=1,2, be defined by (16). Then, we have(24)∫ℜvskKAszs,vsKAsys,vs¯dvs=bsk−μ−kδkzs−ys,k∈ℕ∪0,s=1,2,where δk denotes the k-th derivative of the delta function δ.Proof.
Fork∈ℕ∪0, we have(25)∫ℜvskKAszs,vsKAsys,vs¯dvs=12πbs∫ℜvskeμas/2bszs2−1/bszsvs+ds/2bsvs2e−μas/2bsys2−1/bsysvs+ds/2bsvs2dvs=12πbs∫ℜvskeμas/2bszs2−ys2bskvsk−μ−k∂∂zskeμ1/bsvsys−zsdvs=eμas/2bszs2−ys2bsk−μ−k∂∂zsk12πbs∫ℜeμ1/bsvsys−zsdvs=eμas/2bszs2−ys2bsk−μ−k∂∂zskδzs−ys=bsk−μ−keμas/2bszs2−ys2δkzs−ys.
Since the distributional support ofδkzs−ys as a distribution of zs is ys, we have(26)eμas/2bszs2−ys2δkzs−ys=eμas/2bszs2−zs2zs=ysδkzs−ys=δkzs−ys,in the distributional sense. With (25) and (26), we have (24).


## 3.2. Useful Properties of the QLCT

Parseval’s theorem for Fourier transform can be generalized to the QLCT as well. Let us now formulate Parseval’s theorem in the QLCT domain.Theorem 3 (QLCT Parseval).
Letf, g∈L2ℜ2;ℍ. Then, we have(27)f,g=LA1,A2ℍf,LA1,A2ℍg.Proof.
Forf, g∈L2ℜ2,ℍ, we have(28)f,g=∫ℜ2fzgz¯dz=∫ℜ2∫ℜ2LA1,A2ℍfvKA1−1v1,z1KA2−1v2,z2dvgz¯dz=∫ℜ2LA1,A2ℍfv∫ℜ2KA1z1,v1¯KA2z2,v2¯gz¯dzdv=∫ℜ2LA1,A2ℍfv∫ℜ2gzKA1z1,v1KA2z2,v2dz¯dv=∫ℜ2LA1,A2ℍfvLA1,A2ℍgv¯dv=LA1,A2ℍf,LA1,A2ℍg.
In the second equality in Theorem 3, we have replaced the quaternion functionf with its inverse QLCT expression in Theorem 2. In the third equality, we have interchanged the order of integration. In the fourth equality, we have applied the quaternion conjugation rule pq¯=q¯p¯ for p, q∈ℍ, thus proving the theorem.
Due to the noncommutativity of the kernel of the QLCT, we only have a left linearity property with quaternion constants; that is,(29)LA1,A2ℍαf+βgv=αLA1,A2ℍfv+βLA1,A2ℍgv.
In the following, we will summarize the important properties of the QLCT. We will see that the results are generalizations of the basic properties of the LCT. Letf∈L2ℜ2;ℍ. The translation operator τk by k∈ℜ2 is defined by(30)τkfz=fz−k.
The modulation operatorMv0 by v0=u0,v0∈ℜ2 is defined by(31)Mv0fz=fzeμz⋅v0,z⋅v0=z1u0+z2v0.
We now begin with the shift property of the QLCT.Theorem 4 (shift property).
Forf∈L2ℜ2;ℍ, we have(32)LA1,A2ℍτkfv=LA1,A2ℍfv1−a1k1,v2−a2k2e−1/2μa1c1k12+a2c2k22eμc1k1v1+c2k2v2.Next, we are concerned about the behavior of the QLCT under modulation.Theorem 5 (modulation property).
Forf∈L2ℜ2;ℍ, we have(33)LA1,A2ℍMv0fv=LA1,A2ℍfzeμz⋅v0v=LA1,A2ℍfv−bv0e−1/2μb1d1u02+b2d2v02eμd1u0v1+d2v0v2.Theorem 6 (time-frequency shift).
Forf∈L2ℜ2;ℍ, we have(34)LA1,A2ℍMv0τkfv=LA1,A2ℍfz−keμz⋅v0v=LA1,A2ℍfv−ak−bv0e−1/2μa1c1k12+a2c2k22+b1d1u02+b2d2v02×eμc1k1+d1u0v1eμc2k2+d2v0v2e−μb1c1k1u0+b2c2k2v0.We present the following result, which is needed for deriving the uncertainty principle related to the QLCT.Theorem 7.
LetAs=asbscsds∈SL2,ℜ and bs≠0 for s=1,2. Assume that f∈C1ℜ2;ℍ, that f,∂f/∂zi∈L2ℜ2;ℍ, and that viLA1,A2ℍf∈L2ℜ2;ℍ, i=1,2. Then, we have(35)∫ℜ2vi2LA1,A2ℍfv2dv=bi2∫ℜ2∂∂zifz2dz,i=1,2.Proof.
Recall the quaternion conjugation rulepq¯=q¯p¯ for p, q∈ℍ. The distributional calculation gives(36)∫ℜ2v12LA1,A2ℍfv2dv=∫ℜ2v12∫ℜ2fzKA1z1,v1KA2z2,v2dz∫ℜ2fyKA1y1,v1KA2y2,v2¯dydv=∫ℜ4∫ℜfzv12KA1z1,v1∫ℜKA2z2,v2KA2y2,v2¯dv2KA1y1,v1¯dv1fy¯dydz.
Usingk=0 in Proposition 1, we get(37)∫ℜ2v12LA1,A2ℍfv2dv=∫ℜ4fz∫ℜv12KA1z1,v1δz2−y2KA1y1,v1¯dv1fy¯dy2dy1dz=∫ℜ4fz∫ℜv12fzKA1z1,v1KA1y1,v1¯dv1fy1,z2¯dy1dz.
Again takingk=2 in Proposition 1 gives(38)∫ℜ2v12LA1,A2ℍfv2dv=b12−μ−2∫ℜ3fzδ2z1−y1fy1,z2¯dy1dz.
Since the delta function is even function and−μ−2=−1, we have(39)−μ−2δ2z1−y1=−∂∂z1−∂∂y1δz1−y1=−∂∂z1−∂∂y1δy1−z1.
Then, integration by parts in the sense of distribution implies(40)−μ−2∫ℜ3fzδ2z1−y1fy1,z2¯dy1dz=∫ℜ3fz−∂∂z1−∂∂y1δy1−z1fy1,z2¯dy1dz=∫ℜ2∂∂z1fz∫ℜδy1−z1∂∂y1fy1,z2¯dy1dz=∫ℜ2∂∂z1fz∂∂y1fy1,z2¯y1=z1dz=∫ℜ2∂∂z1fz∂∂z1fz¯dz,which implies the desired result.The above properties of the QLCT are summarized in Table1.Table 1
Useful properties of the QLCT.
Property
Quaternion func.
QLCT
Left linearity
α
f
+
β
g
α
L
A
1
,
A
2
ℍ
f
v
+
β
L
A
1
,
A
2
ℍ
g
v
Shift
f
z
−
k
L
A
1
,
A
2
ℍ
f
v
1
−
a
1
k
1
,
v
2
−
a
2
k
2
×
e
−
1
/
2
μ
a
1
c
1
k
1
2
+
a
2
c
2
k
2
2
e
μ
c
1
k
1
v
1
+
c
2
k
2
v
2
Modulation
f
z
e
μ
z
⋅
v
0
L
A
1
,
A
2
ℍ
f
v
−
b
v
0
e
−
1
/
2
μ
b
1
d
1
u
0
2
+
b
2
d
2
v
0
2
×
e
μ
d
1
u
0
v
1
+
d
2
v
0
v
2
Scaling
f
z
/
α
α
2
L
A
1
′
,
A
2
′
ℍ
f
v
/
α
,
A
i
′
=
a
i
b
i
/
α
2
c
i
α
2
d
i
,
i
=
1,2
Time-frequency shift
f
z
−
k
e
μ
z
⋅
v
0
L
A
1
,
A
2
ℍ
f
v
−
ak
−
b
v
0
×
e
−
1
/
2
μ
a
1
c
1
k
1
2
+
a
2
c
2
k
2
2
+
b
1
d
1
u
0
2
+
b
2
d
2
v
0
2
×
e
μ
c
1
k
1
+
d
1
u
0
v
1
e
μ
c
2
k
2
+
d
2
v
0
v
2
×
e
−
μ
b
1
c
1
k
1
u
0
+
b
2
c
2
k
2
v
0
Formula
Parseval’s theorem
f
=
L
A
1
,
A
2
ℍ
f
Plancherel’s theorem
f
,
g
=
L
A
1
,
A
2
ℍ
f
,
L
A
1
,
A
2
ℍ
g
Reconstruction
f
z
=
∫
ℜ
2
K
A
1
−
1
v
1
,
z
1
L
A
1
,
A
2
ℍ
f
v
K
A
2
−
1
v
2
,
z
2
d
v
∫
ℜ
v
s
k
K
A
s
z
s
,
v
s
K
A
s
y
s
,
v
s
¯
d
v
s
=
b
s
k
−
μ
−
k
δ
k
z
s
−
y
s
,
k
∈
ℕ
∪
0
∫
ℜ
2
v
i
2
L
A
1
,
A
2
ℍ
f
v
2
d
v
=
b
i
2
∫
ℜ
2
∂
/
∂
z
i
f
z
2
d
z
,
i
=
1,2


## 4. Convolution Associated with QLCT

As we know, convolution is one of the fundamental results in the Fourier transform and LCT. Because the QLCT is a generalization of the LCT using the quaternion algebra, it makes possible to build the convolution theorem in the QLCT domain. For this reason, we introduce the following definition.Definition 4.
Forf, g∈L2ℜ2;ℍ, we define the convolution operator of the QLCT by(41)f⋆gz=∫ℜ2ftgz−teμa1/b1t1t1−z1eμa2/b2t2t2−z2dt.
The above definition implies the following important theorem, which describes how the convolution of two quaternion-valued functions interacts with their QLCTs.Theorem 8.
Let(42)fz=f0z+if1z+jf2z+kf3z,gz=g0z+ig1z+jg2z+kg3z,belong to L2ℜ2;ℍ. Then, the QLCTs of the convolution of f and g are given by(43)LA1,A2ℍf⋆gv=LA1,A2ℍgvLA1,A2ℍf0v+iLA1,A2ℍgvLA1,A2ℍf1v+jLA1,A2ℍgvLA1,A2ℍ+kLA1,A2ℍgvLA1,A2ℍf3v2πb1μ2πb2μe−μd1v12/2b1e−μd2v22/2b2.Proof.
LetLA1,A2ℍf and LA1,A2ℍg denote the QLCTs of f and g, respectively. Expanding the QLCT of the left-hand side of the above identity, we obtain(44)LA1,A2ℍf⋆gv=7∫ℜ2f⋆gz12πμb112πμb2e1/2μa1/b1z12−2/b1z1v1+d1/b1v12×e1/2μa2/b2z22−2/b2z2v2+d2/b2v22dz=∫ℜ2∫ℜ2ftgz−teμa1/b1t1t1−z1eμa2/b2t2t2−z2dt×12πμb112πμb2e1/2μa1/b1z12−2/b1z1v1+d1/b1v12e1/2μa2/b2z22−2/b2z2v2+d2/b2v22dz.
Changing variablesy=z−t in the above expression, we have(45)LA1,A2ℍf⋆gv=∫ℜ2∫ℜ2ftgyeμa1/b1−t1y1eμa2/b2−t2y2dt×12πμb112πμb2e1/2μa1/b1y1+t12−2/b1y1+t1v1+d1/b1v12×e1/2μa2/b2y2+t22−2/b2y2+t2v2+d2/b2v22dy=∫ℜ2∫ℜ2ftgy12πμb112πμb2eμa1/2b1y12eμa1/2b1t12e−μ1/b1y1v1e−μ1/b1t1v1eμd1/2b1v12×eμa2/2b2y22eμa2/2b2t22e−μ1/b2y2v2e−μ1/b2t2v2eμd2/2b2v22dydt=∫ℜ2∫ℜ2ftgy12πμb112πμb2e1/2μa1/b1y12−2/b1y1v1+d1/b1v12e1/2μa2/b2y22−2/b2y2v2+d2/b2v22×eμa1/2b1t12eμa2/2b2t22e−μt1v1/b1e−μt2v2/b2dydt.
Applying the QLCT definition yields(46)LA1,A2ℍf⋆gv=∫ℜ2ftLA1,A2ℍgveμa1/2b1t12eμa2/2b2t22e−μt1v1/b1e−μt2v2/b2dt.
The noncommutativity of the quaternion multiplication requires us to decomposeft into f0t+if1t+jf2t+kf3t. This gives(47)LA1,A2ℍf⋆gv=∫ℜ2f0t+if1t+jf2t+kf3tLA1,A2ℍgv×eμa1/2b1t12eμa2/2b2t22e−μt1v1/b1e−μt2v2/b2dt=∫ℜ2LA1,A2ℍgvf0t+iLA1,A2ℍfvf1t+jLA1,A2ℍfvf2t+kLA1,A2ℍgvf3teμa1/2b1t12eμa2/2b2t22e−μt1v1/b1e−μt2v2/b2dt.
Multiplying both sides of the above identity by1/2πμb1eμd1v12/2b1 and 1/2πμb2eμd2v22/2b2, we obtain(48)LA1,A2ℍf⋆gv12πμb112πμb2eμd1v12/2b1eμd2v22/2b2=∫ℜ2LA1,A2ℍgvf0t+iLA1,A2ℍfvf1t+jLA1,A2ℍfvf2t+kLA1,A2ℍgvf3t12πμb112πμb2eμa1/2b1t12e−μt1v1/b1eμd1v12/2b1eμa2/2b2t22e−μt2v2/b2eμd2v22/2b2dt=LA1,A2ℍgv∫ℜ2f0t12πμb112πμb2eμa1/2b1t12e−μt1v1/b1eμd1v12/2b1×eμa2/2b2t22e−μt2v2/b2eμd2v22/2b2dt+iLA1,A2ℍgv∫ℜ2f1t12πμb112πμb2eμa1/2b1t12e−μt1v1/b1eμd1v12/2b1×eμa2/2b2t22e−μt2v2/b2eμd2v22/2b2dt+jLA1,A2ℍgv∫ℜ2f2t12πμb112πμb2eμa1/2b1t12e−μt1v1/b1eμd1v12/2b1×eμa2/2b2t22e−μt2v2/b2eμd2v22/2b2dt+kLA1,A2ℍgv∫ℜ2f3t12πμb112πμb2eμa1/2b1t12e−μt1v1/b1eμd1v12/2b1×eμa2/2b2t22e−μt2v2/b2eμd2v22/2b2dt,thus proving the theorem.


## 5. Correlation Theorem for QLCT

In this section, we define the correlation of the QLCT. Then, we investigate the relationship between the QLCT and the correlation of two quaternion functions.Definition 5.
Forf, g∈L1ℜ2;ℍ, we define the correlation operator of the QLCT by(49)f⋄gz=∫ℜ2ft¯gt+zeμa1/b1t1t1+z1eμa2/b2t2t2+z2dt.
Then, we reap a consequence of the definition above.Theorem 9.
Forf, g∈L2ℜ2;ℍ, we have the QLCTs of correlation of f and g by(50)LA1,A2ℍf⋄gv=LA1,A2ℍgvLA1,A2ℍf0v−iLA1,A2ℍgvLA1,A2ℍf1−v−jLA1,A2ℍgvLA1,A2ℍf2−v−kLA1,A2ℍgvLA1,A2ℍf3−v.
WhenA1=A2=01−10, the above expression reduces to(51)LA1,A2ℍf⋄gv=LA1,A2ℍgvLA1,A2ℍf¯−v2πμb12πμb2.Proof.
From the definitions of the QLCT (17) and the convolution, we get(52)LA1,A2ℍf⋄gv=∫ℜ2∫ℜ2ft¯gz+teμa1/b1t1t1+z1eμa2/b2t2t2+z2dt×12πμb112πμb2e1/2μa1/b1z12−2/b1z1v1+d1/b1v12e1/2μa2/b2z22−2/b2z2v2+d2/b2v22dz.
Settingz+t=v, we get(53)LA1,A2ℍf⋄gv=∫ℜ2∫ℜ2ft¯gveμa1/b1t1v1eμa2/b2t2v2dt×12πμb112πμb2e1/2μa1/b1v1−t12−2/b1v1−t1v1+d1/b1v12×e1/2μa2/b2v2−t22−2/b2v2−t2v2+d2/b2v22dtdv=∫ℜ2∫ℜ2ft¯gv12πμb112πμb2eμa1/2b1v12eμa1/2b1t12e−μa1/b1v1t1e−μ1/b1v1v1eμ1/b1t1v1×eμa2/2b2v22eμa2/2b2t22e−μa2/b2v2t2e−μ1/b2v2v2eμ1/b2t2v2×eμd1/2b1v12eμd2/2b2v22eμa1/b1v1t1eμa2/b2v2t2dtdv=∫ℜ2∫ℜ2ft¯gv12πμb112πμb2eμa1/2b1v12e−μ1/b1v1v1eμd1/2b1v12eμa1/2b2v22e−μ1/b2v2v2×eμd2/2b2v22eμa2/2b1t12eμ1/b1t1v1eμa2/2b2t22eμ1/b2t2v2dtdv.
Using the QLCT definition, we obtain(54)LA1,A2ℍf⋄gv=∫ℜ2ft¯LA1,A2ℍgveμa1/2b1t12eμt1v1/b1eμa2/2b2t22eμt2v2/b2dt.
Substitutingft¯=f0t−if1t−jf2t−kf3t, we have(55)LA1,A2ℍf⋄gv=∫ℜ2f0t−if1t−jf2t−kf3tLA1,A2ℍgv×eμa1/2b1t12eμt1v1/b1eμa2/2b2t22eμt2v2/b2dt=∫ℜ2LA1,A2ℍgvf0t−iLA1,A2ℍfvf1t−jLA1,A2ℍfvf2t−kLA1,A2ℍgvf3teμa1/2b1t12eμt1v1/b1eμa2/2b2t22eμt2v2/b2dt.
Multiplying both sides of the above identity by1/2πμb1eμd1v12/2b1 and 1/2πμb2eμd2v22/2b2, we obtain(56)LA1,A2ℍf⋄gv12πμb112πμb2eμd1v12/2b1eμd2v22/2b2=∫ℜ2LA1,A2ℍgvf0t−iLA1,A2ℍfvf1t−jLA1,A2ℍfvf2t−kLA1,A2ℍgvf3t12πμb112πμb2eμa1/2b1t12eμt1v1/b1eμd1v12/2b1×eμa2/2b2t22e−μt2v2/b2eμd2v22/2b2dt=LA1,A2ℍgv∫ℜ2f0t12πμb112πμb2eμa1/2b1t12eμt1v1/b1eμd1v12/2b1×eμa2/2b2t22e−μt2v2/b2eμd2v22/2b2dt−iLA1,A2ℍgv∫ℜ2f1t12πμb112πμb2eμa1/2b1t12eμt1v1/b1eμd1v12/2b1×eμa2/2b2t22e−μt2v2/b2eμd2v22/2b2dt−jLA1,A2ℍgv∫ℜ2f2t12πμb112πμb2eμa1/2b1t12eμt1v1/b1eμd1v12/2b1×eμa2/2b2t22e−μt2v2/b2eμd2v22/2b2dt−kLA1,A2ℍgv∫ℜ2f3t12πμb112πμb2eμa1/2b1t12eμt1v1/b1eμd1v12/2b1×eμa2/2b2t22eμt2v2/b2eμd2v22/2b2dt.
We finally obtain(57)LA1,A2ℍf⋄gv=LA1,A2ℍgvLA1,A2ℍf0−v−iLA1,A2ℍgvLA1,A2ℍf1−v−jLA1,A2ℍgvLA1,A2ℍf2−v−kLA1,A2ℍgvLA1,A2ℍf3−v,thus proving the theorem.


## 6. Uncertainty Principle for QLCT

As in the case of the LCT [38, 39], the uncertainty principle is an important result of the QLCT. In [18, 19, 21], the authors have established the uncertainty principle for the two-sided QLCT. Based on the definition of our proposed QLCT in this article, we investigate an uncertainty principle related to the QLCT. The proof of the uncertainty differs from ones mentioned above. This uncertainty principle describes that the spread of a quaternion-valued function and its QLCT are inversely proportional. It is found that modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.Theorem 10.
Letf∈C1ℜ2;ℍ∩L2ℜ2;ℍ, ∂f/∂zi∈L2ℜ2;ℍ, and LA1,A2ℍf, viLA1,A2ℍf∈L2ℜ2;ℍ, i=1,2, and F=∫ℜ2fz2dz<∞. Then, the following inequalities hold:(58)∫ℜ2zi2fz2dz∫ℜ2vi2LA1,A2ℍfv2dv≥bi24F2,i=1,2.
Moreover, we have equality in the above expression fori=1,2 if and only if f is a modulated and shifted two-dimensional Gaussian function; that is, it has the form(59)fz=K0eμv0⋅ze−z1−k12/2α1+z2−k22/2α2,where K0 is a quaternion constant and α1, α2 are positive real constants.Proof.
We will prove this theorem in two parts. Firstly, we seti=1. By inserting Theorem 7 into the left-hand side of (58), we obtain(60)∫ℜ2z12fz2dz∫ℜ2v12LA1,A2ℍfv2dv=∫ℜ2z1fz2dz∫ℜ2b1∂f∂z1z2dz≥b124∫ℜ2∂fz∂z1z1fz¯+z1fz∂fz∂z1¯dz2=b124∫ℜ2z1∂∂z1fzfz¯dz2=b124∫ℜ2z1∂∂z1fz2dz2,where the third equality follows from the quaternion Cauchy–Schwarz inequality.
Secondly, fori=2, we can take similar steps as above and have(61)∫ℜ2z22fz2dz∫ℜ2v22LA1,A2ℍfv2dv≥b224∫ℜ2z2∂∂z2fz¯fzdz2.
Now notice that, fori=1,2, using integration by parts, we have(62)∫ℜ2zi2fz2dz∫ℜ2vi2LA1,A2ℍfv2dv≥bi24∫ℜ2zi∂∂zifz2dz2=bi24∫ℜzifz2dzl,l≠izi=−∞zi=∞−∫ℜ2fz2dz2=bi240−∫ℜ2fz2dz2=bi24F2.
Note that the equality in equation (58) only holds for a two-dimensional Gaussian function:(63)fz=K0e−z12/2α1+z22/2α2.
It is known that modulation and shift properties are valid for the type II QFT (compare to [40]). Moreover, equality in (58) holds if and only if f is a modulated and shifted two-dimensional Gaussian function, thus proving the theorem.



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*Source: 1062979-2019-09-09.xml*