General convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented. It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. We describe some useful properties of generalized convolutions and compare them with the convolution theorems of the classical Fourier transform. We finally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework.
1. Introduction
Convolution is a mathematical operation with several applications in pure and applied mathematics such as numerical analysis, numerical linear algebra, and the design and implementation of finite impulse response filters in signal processing. In [1–3], the authors introduced the Clifford convolution. It is found that some properties of convolution, when generalized to the Clifford Fourier transform (CFT), are very similar to the classical ones.
On the other hand, the quaternion Fourier transform (QFT) is a nontrivial generalization of the classical Fourier transform (FT) using quaternion algebra. The QFT has been shown to be related to the other quaternion signal analysis tools such as quaternion wavelet transform, fractional quaternion Fourier transform, quaternionic windowed Fourier transform, and quaternion Wigner transform [4–9]. A number of already known and useful properties of this extended transform are generalizations of the corresponding properties of the FT with some modifications, but the generalization of convolution theorems of the QFT is still an open problem. In the recent past, several authors [10–13] tried to formulate convolution theorems for the QFT. But they only treated them for real-valued functions which is quite similar to the classical case. In [14], the authors briefly introduced, without proof, the QFT of the convolution of two-dimensional quaternion signals.
In this paper, we establish general convolutions for QFT. Because quaternion multiplication is not commutative, we find new properties of the QFT of convolution of two quaternion-valued functions. These properties describe closely the relationship between the quaternion convolution and its QFT. The generalization of the convolution theorems of the QFT is mainly motivated by the Clifford convolution of general geometric Fourier transform, which has been recently studied in [15, 16]. We further establish the inverse QFT of the product of the QFT, which is very useful in solving partial differential equations in quaternion algebra framework.
This paper consists of the following sections. Section 2 deals with some results on the real quaternion algebra and the definition of the QFT and its basic properties. We also review some basic properties of QFT, which will be necessary in the next section. Section 3 establishes convolution theorems of QFT and some of their consequences. Section 4 presents an application of QFT to study hypoellipticity and to solve the heat equation in quaternion algebra. Some conclusions are drawn in Section 5.
2. Quaternion Algebra
For convenience, we specify the notation used in this paper. The quaternion algebra over ℝ, denoted by ℍ, is an associative noncommutative four-dimensional algebra,
(1)ℍ={q=q0+iq1+jq2+kq3;q0,q1,q2,q3∈ℝ},
which obeys the following multiplication rules:
(2)ij=-ji=k,jk=-kj=i,ki=-ik=j,i2=j2=k2=ijk=-1.
For a quaternion q=q0+iq1+jq2+kq3∈ℍ, q0 is called the scalar part of q denoted by Sc(q) and iq1+jq2+kq3 is called the vector (or pure) part of q. The vector part of q is conventionally denoted by q. Let p, q∈ℍ and let p, q be their vector parts, respectively. It is common to write for short
(3)q·p=q1p1+q2p2+q3p3,q×p=i(q2p3-q3p2)+j(q3p1-q1p3)+k(q1p2-q2p1).
Then, (2) yields the quaternionic multiplication qp as
(4)qp=q0p0-q·p+q0p+p0q+q×p.
The quaternion conjugate of q, given by
(5)q-=q0-iq1-jq2-kq3,q0,q1,q2,q3∈ℝ,
is an anti-involution; that is,
(6)qp¯=p-q-.
From (5) we obtain the norm or modulus of q∈ℍ defined as
(7)|q|=qq-=q02+q12+q22+q32.
It is not difficult to see that
(8)|qp|=|q||p|,∀p,q∈ℍ.
Using the conjugate (5) and the modulus of q, we can define the inverse of q∈ℍ∖{0} as
(9)q-1=q-|q|2,
which shows that ℍ is a normed division algebra. As in the algebra of complex numbers, we can define three nontrivial quaternion involutions [10]:
(10)α(q)=-iqi=-i(q0+iq1+jq2+kq3)i=q0+iq1-jq2-kq3,β(q)=-jqj=-j(q0+iq1+jq2+kq3)j=q0-iq1+jq2-kq3,γ(q)=-kqk=-k(q0+iq1+jq2+kq3)k=q0-iq1-jq2+kq3.
Hereinafter, besides the quaternion units i, j, and k and the vector part q of a quaternion q∈ℍ, we will use the real vector notation:
(11)x=(x1,x2)∈ℝ2,|x|2=x12+x22,x·y=x1y1+x2y2,f(x)=f(x1,x2),
and so on when there is no confusion. This gives the following definition.
Definition 1 (see [<xref ref-type="bibr" rid="B10">10</xref>]).
A function f:ℝ2→ℍ is called quaternionic Hermitian if, for the involutions α and β,
(12)f(-x1,x2)=β(f(x)),f(x1,-x2)=α(f(x)),
for each x∈ℝ2.
For any unit quaternion
(13)q=q0+q=cos(θ2)+sin(θ2),
and for any vector v∈ℝ3 the action of the operator
(14)Lq(v)=qvq-
on v is equivalent to a rotation of the vector v through an angle θ about u as the axis of rotation.
It is convenient to introduce an inner product for two functions f,g:ℝ2→ℍ as follows:
(15)(f,g)L2(ℝ2;ℍ)=∫ℝ2f(x)g(x)¯d2x.
In particular, for f=g, we obtain the scalar product of the above inner product (15) given by
(16)〈f,f〉=∥f∥L2(ℝ2;ℍ)=(∫ℝ2|f(x)|2d2x)1/2.
2.1. Multiindices and Derivatives
A couple α=(α1,α2) of nonnegative integers is called a multiindex. We denote
(17)|α|=α1+α2,α!=α1!α2!,
and for x∈ℝ2,
(18)xα=x1α1x2α2.
Derivatives are conveniently expressed by multiindices:
(19)∂α=∂|α|∂x1α1∂x2α2.
Denote by {e1,e2} the standard basis of ℝ2. The vector differentiala·∇ along the direction a is defined by
(20)a·∇=a1∂1+a2∂2,
where ∇=e1∂1+e2∂2.
2.2. QFT and Its PropertiesDefinition 2.
The QFT of f∈L2(ℝ2;ℍ) is the transform ℱq{f}∈L2(ℝ2,ℍ) given by the integral
(21)ℱq{f}(ω)=∫ℝ2e-iω1x1f(x)e-jω2x2dx,
where ℱq is called the quaternion Fourier transform operator or the quaternion Fourier transformation.
Using the Euler formula for the quaternion Fourier kernel e-iω1x1e-jω2x2, we can rewrite (21) in the following form:
(22)ℱq{f}(ω)=∫ℝ2f(x)cos(ω1x1)cos(ω2x2)dx-∫ℝ2if(x)sin(ω1x1)cos(ω2x2)dx-∫ℝ2f(x)jcos(ω1x1)sin(ω2x2)dx+∫ℝ2if(x)jsin(ω1x1)sin(ω2x2)dx.
Definition 3.
The inverse QFT of g∈L2(ℝ2;ℍ) is the transform ℱq-1{g}∈L2(ℝ2,ℍ) given by the integral
(23)ℱq-1[g(ω)](x)=1(2π)2∫ℝ2eiω1x1g(ω)ejω2x2dω.
Some important properties of the QFT are stated in the following lemmas proved in [12, 17].
Lemma 4.
Let f∈L1(ℝ2;ℍ)∩L2(ℝ2;ℍ). If ℱq{∂αf}∈L1(ℝ2;ℍ), then
(24)ℱq{∂αf}(ω)=(iω1)α1ℱq{f}(ω)(jω2)α2.
In particular, if ℱq{∂(2,0)f}∈L1(ℝ2;ℍ), then
(25)ℱq{∂(2,0)f}(ω)=(iω1)2ℱq{f}(ω).
And if ℱq{∂(0,2)f}∈L1(ℝ2;ℍ), then
(26)ℱq{∂(0,2)f}(ω)=ℱq{f}(ω)(jω2)2.
Lemma 5 (scalar QFT Parseval).
The scalar product of f,g∈L2(ℝ2;ℍ) and its QFT are related by
(27)〈f,g〉L2(ℝ2;ℍ)=1(2π)2〈ℱq{f},ℱq{g}〉L2(ℝ2;ℍ).
And in particular, with f=g, the Plancherel theorem indicates that
(28)∥f∥L2(ℝ2;ℍ)2=1(2π)2∥ℱq{f}∥L2(ℝ2;ℍ)2.
This shows that the total signal energy computed in the spatial domain is equal to the total signal energy computed in the quaternion domain.
3. Convolution of QFT
In this section, we establish the quaternion convolution of the QFT which extends the classical convolution to quaternion fields. Let us first define the convolution of two quaternion-valued functions.
Definition 6.
The convolution of f∈L2(ℝ2;ℍ) and g∈L2(ℝ2;ℍ), denoted by f⋆g, is defined by
(29)(f⋆g)(x)=∫ℝ2f(y)g(x-y)dy.
Example 7.
To illustrate the general noncommutativity (f⋆g)≠(g⋆f), let us compute the convolution of f(x)=ke-|x|2 and g(x)=(i+j)eiω1x1ejω2x2. Although g∈L∞(ℝ2;ℍ), we can still define the convolution of f and g, because f decays rapidly at infinity. A simple calculation gives
(30)(f⋆g)(x)=∫ℝ2ke-|y|2(i+j)eiω1(x1-y1)ejω2(x2-y2)dy=∫ℝ2k(i+j)eiω1x1e-iω1y1e-|y|2e-jω2y2dyejω2x2=k(i+j)eiω1x1e-|ω|2/4ejω2x2=(j-i)eiω1x1e-|ω|2/4ejω2x2.
On the other hand, we have
(31)(g⋆f)(x)=∫ℝ2(i+j)eiω1y1ejω2y2ke-|x-y|2dy=∫ℝ2(i+j)ke-iω1y1e-|x-y|2e-jω2y2dy=∫ℝ2(i+j)ke-iω1y1e-|y-x|2e-jω2y2dy=(i-j)e-iω1x1e-|ω|2/4e-jω2x2.
In the following, we summarize the elementary properties of quaternion convolution as shown in Table 1 (compared to Folland [18]).
Basic properties of quaternion convolution.
Basic property
Quaternion convolution
Linearity
(κ1f+κ2g)⋆h=κ1(f⋆h)+κ2(f⋆h),κ1,κ2∈ℍ,
h⋆(κ1f+κ2g)=κ1(h⋆f)+κ2(h⋆g),κ1,κ2∈ℝ
Shifting
(τaf⋆g)=τa(f⋆g),
(f⋆τag)=τa(f⋆g)
Conjugation
(f⋆g)¯=(g-⋆f-)
Associativity
(f⋆g)⋆h=f⋆(g⋆h)
Distributivity
f⋆(g+h)=(f⋆g)+(f⋆h)
Vector differential
a·∇(f⋆g)=(a·∇f)⋆g=f⋆(a·∇g)
Impulse convolution
f⋆δ=f
Lemma 8 (linearity).
For quaternion functions f,g, and h and quaternion constants κ1 and κ2 one gets
(32)(κ1f+κ2g)⋆h=κ1(f⋆h)+κ2(g⋆h).
One also gets for real constants κ1 and κ2 (due to the noncommutativity of the quaternion multiplication, (33) does not hold for quaternion constants κ1 and κ2)
(33)h⋆(κ1f+κ2g)=κ1(h⋆f)+κ2(h⋆g).
Lemma 9 (shifting).
Given a quaternion function f∈L2(ℝ2;ℍ), let τaf(x) denote the shifted (translated) function defined by τaf(x)=f(x-a), where a∈ℝ2. Then one gets
(34)(τaf⋆g)(x)=τa(f⋆g)(x),(35)(f⋆τag)(x)=τa(f⋆g)(x).
Proof.
For (34), a direct calculation gives
(36)(τaf⋆g)(x)=∫ℝ2τaf(y)g(x-y)dy=∫ℝ2f(y-a)g(x-y)dy=∫ℝ2f(y-a)g((x-a)-(y-a))dy=∫ℝ2f(s)g((x-a)-s)ds=(f⋆g)(x-a)=τa(f⋆g)(x),
which finishes the proof.
Remark 10.
From (34) and (35), it is not difficult to see that (τaf⋆g)(x)=(f⋆τag)(x) and (τag⋆f)(x)=(g⋆τaf)(x).
Lemma 11 (conjugation).
For all quaternion functions f,g∈L2(ℝ2;ℍ) one has
(37)(f⋆g)¯(x)=(g-⋆f-)(x).
Proof.
A straightforward computation gives
(38)(f⋆g)¯(x)=∫ℝ2f(y)g(x-y)¯dy=∫ℝ2g-((x-y))f-(y)dy=∫ℝ2g-(z)f-((x-z))dz=(g-⋆f-)(x).
This finishes the proof.
For all quaternion functions f,g∈L2(ℝ2;ℍ) one has
(39)a·∇(f⋆g)=(a·∇f)⋆g=f⋆(a·∇g).
Ell and Sangwine [19] distinguish between right and left discrete quaternion convolution due to the non-commutative property of the quaternion multiplication. Here, we only consider one kind of quaternion convolutions. We come now to the main theorem (generalization of the QFT of the quaternion convolution in general geometric Fourier transform is investigated in [15]. It can easily be seen that the result is closely related to equation (4.30) of [15]) of this paper. This theorem describes the relationship between the convolution of two quaternion functions and its QFT.
Theorem 13.
Let f(x)=f0(x)+if1(x)+jf2(x)+kf3(x) and g(x)=g0(x)+ig1(x)+jg2(x)+kg3(x) be two quaternion-valued functions, then the QFT of the convolution of f∈L2(ℝ2;ℍ) and g∈L2(ℝ2;ℍ) is given by
(40)ℱq{f⋆g}(ω)=(ℱq{f0}(ω)+iℱq{f1}(ω))×(ℱq{g0}(ω)+jℱq{g2}(-ω1,ω2))+(ℱq{f0}(ω1,-ω2)+iℱq{f1}(ω1,-ω2))×(iℱq{g1}(ω)+kℱq{g3}(-ω1,ω2))+(jℱq{f2}(-ω1,ω2)+kℱq{f3}(-ω1,ω2))×(ℱq{g0}(-ω1,ω2)+jℱq{g2}(ω))+(jℱq{f2}(-ω)+kℱq{f3}(-ω))×(iℱq{g1}(-ω1,ω2)+kℱq{g3}(ω)).
Proof.
In this proof we will use the decomposition of quaternion functions and their QFTs. Let ℱq{f}(ω) and ℱq{g}(ω) denote the QFT of f∈L2(ℝ2;ℍ) and g∈L2(ℝ2;ℍ), respectively. Expanding the QFT of the left-hand side of (40), we immediately get
(41)ℱq{f⋆g}(ω)=(21)∫ℝ2e-iω1x1(f⋆g)(x)e-jω2x2dx=(29)∫ℝ2e-iω1x1[∫ℝ2f(y)g(x-y)dy]e-jω2x2dx=∫ℝ2e-iω1x1f(y)[∫ℝ2g(x-y)e-jω2x2dx]dy.
By the change of variables z=x-y, the above transform can be written as
(42)ℱq{f⋆g}(ω)=∬ℝ2e-iω1(y1+z1)f(y)g(z)e-jω2(y2+z2)dzdy=∫ℝ2e-iω1(y1+z1)({f0(y)+if1(y)}+jf2(y)+kf3(y))×∫ℝ2(+kg3(z){g0(z)+jg2(z)}+ig1(z)+kg3(z))g(z)e-jω2(y2+z2)dzdy=∬ℝ2e-iω1(y1+z1)(f0(y)+if1(y))×(g0(z)+jg2(z))e-jω2(y2+z2)dzdy+∬ℝ2e-iω1(y1+z1)(f0(y)+if1(y))×(ig1(z)+kg3(z))e-jω2(y2+z2)dzdy+∬ℝ2e-iω1(y1+z1)(jf2(y)+kf3(y))×(g0(z)+jg2(z))e-jω2(y2+z2)dzdy+∬ℝ2e-iω1(y1+z1)(jf2(y)+kf3(y))×(ig1(z)+kg3(z))e-jω2(y2+z2)dzdy=∬ℝ2e-iω1y1(f0(y)+if1(y))×(ℱq{g0}(ω)+jℱq{g2}(-ω1,ω2))×e-jω2y2dzdy+∬ℝ2e-iω1y1(f0(y)+if1(y))×(iℱq{g1}(ω)+kℱq{g3}(-ω1,ω2))×e-jω2y2dzdy+∬ℝ2e-iω1y1(jf2(y)+kf3(y))×(ℱq{g0}(-ω1,ω2)+jℱq{g2}(ω))×e-jω2y2dzdy+∬ℝ2e-iω1y1(jf2(y)+kf3(y))×(iℱq{g1}(-ω1,ω2)+kℱq{g3}(ω))×e-jω2y2dzdy=∫ℝ2e-iω1y1(f0(y)+if1(y))e-jω2y2dy×(ℱq{g0}(ω)+jℱq{g2}(-ω1,ω2))+∫ℝ2e-iω1y1(f0(y)+if1(y))ejω2y2dy×(iℱq{g1}(ω)+kℱq{g3}(-ω1,ω2))+∫ℝ2e-iω1y1(jf2(y)+kf3(y))e-jω2y2dy×(ℱq{g0}(-ω1,ω2)+jℱq{g2}(ω))+∫ℝ2e-iω1y1(jf2(y)+kf3(y))ejω2y2dy×(iℱq{g1}(-ω1,ω2)+kℱq{g3}(ω)),
where the assumption ℱq{gi}∈L2(ℝ2;ℝ) for i=1,2,3 is used in the fourth line. This gives the desired result.
The following lemmas are special cases of Theorem 13.
Lemma 14.
Let f,g∈L2(ℝ2;ℍ), where
(43)f=f0+if1+jf2+kf3,g=g0+ig1+jg2+kg3.
If ℱq{g}∈L2(ℝ2;ℝ), then (40) takes the form
(44)ℱq{f⋆g}(ω)=(ℱq{f0}(ω)+iℱq{f1}(ω))ℱq{g}(ω)+(jℱq{f2}(-ω1,ω2)+kℱq{f3}(-ω1,ω2))ℱq{g}(-ω1,ω2).
On the other hand, if ℱq{f}∈L2(ℝ2;ℝ), then
(45)ℱq{f⋆g}(ω)=ℱq{f}(ω)(ℱq{g0}(ω)+jℱq{g2}(-ω1,ω2))+ℱq{f}(ω1,-ω2)(iℱq{g1}(ω)+kℱq{g3}(-ω1,ω2)).
Proof.
We only prove expression (44) of Lemma 14, with the other being similar. Following the steps of (41) we immediately get
(46)ℱq{f⋆g}(ω)=∬ℝ2e-iω1(y1+z1)f(y)g(z)e-jω2(y2+z2)dzdy=∬ℝ2e-iω1(y1+z1)×({f0(y)+if1(y)}+jf2(y)+kf3(y))×g(z)e-jω2(y2+z2)dzdy=∬ℝ2e-iω1(y1+z1)(f0(y)+if1(y))×g(z)e-jω2(y2+z2)dzdy+∬ℝ2e-iω1(y1+z1)(jf2(y)+kf3(y))×g(z)e-jω2(y2+z2)dzdy=∫ℝ2e-iω1y1(f0(y)+if1(y))ℱq{g}(ω)e-jω2y2dy+∫ℝ2e-iω1y1(jf2(y)+kf3(y))×ℱq{g}(-ω1,ω2)e-jω2y2dy=(ℱq{f0}(ω)+iℱq{f1}(ω))ℱq{g}(ω)+(jℱq{f2}(-ω1,ω2)+kℱq{f3}(-ω1,ω2))×ℱq{g}(-ω1,ω2),
which was to be proven.
Lemma 15.
Let f,g∈L2(ℝ2;ℍ), where
(47)f=f0+if1,g=g0+ig1+jg2+kg3.
If ℱq{f},ℱq{g}∈L2(ℝ2;ℝ), then
(48)ℱq{f⋆g}(ω)=ℱq{f}(ω)ℱq{g}(ω)=ℱq{g}(ω)ℱq{f}(ω),
which is of the same form as a convolution of the classical Fourier transform [20].
Remark 16.
It is important to notice that, if f,g∈L2(ℝ2;ℍ), where
(49)f=jf2+kf3,g=g0+ig1+jg2+kg3,
then Lemma 15 reduces to
(50)ℱq{f⋆g}(ω)=ℱq{f}(ω)ℱq{g}(-ω1,ω2)=ℱq{g}(-ω1,ω2)ℱq{f}(ω),
where ℱq{f},ℱq{g}∈L2(ℝ2;ℝ).
Table 2 compares convolution theorems of the QFT and classical FT for f,g∈L2(ℝ2;ℍ).
Comparison of convolution theorems of the QFT and classical FT for f,g∈L2(ℝ2;ℍ).
Assumptions on quaternion functions
QFT of convolution
ℱq{f},ℱq{g}∈L2(ℝ2;ℍ)
QFT ≠ classical FT
ℱq{f}∈L2(ℝ2;ℝ) and ℱq{g}∈L2(ℝ2;ℍ)
QFT ≠ classical FT
ℱq{f}∈L2(ℝ2;ℍ) and ℱq{g}∈L2(ℝ2;ℝ)
QFT ≠ classical FT
f=f0+if1 and ℱq{f},ℱq{g}∈L2(ℝ2;ℝ)
QFT = classical FT
f=jf2+kf3 and ℱq{f},ℱq{g}∈L2(ℝ2;ℝ)
QFT = classical FT
The following theorem is useful for solving the heat equation in quaternion algebra.
Theorem 17.
If f,g∈L2(ℝ2;ℍ) and ℱq{g}∈L2(ℝ2;ℝ), then
(51)ℱq-1[ℱq{f}ℱq{g}](x)=(f0⋆g)(x)+(if1⋆g)(x)+(jf2⋆g)(-x1,x2)+(kf3⋆g)(-x1,x2).
Proof.
By the QFT inversion, we get, after some simplification,
(52)ℱq-1[ℱq{f}ℱq{g}](x)=(23)1(2π)2∬ℝ2eiω1x1e-iω1y1f(y)e-jω2y2dy×ℱq{g}(ω)ejω2x2dω=1(2π)2∬ℝ2eiω1(x1-y1)f(y)×ℱq{g}(ω)ejω2(x2-y2)dydω=1(2π)2∬ℝ2eiω1(x1-y1)×(f0(y)+if1(y)+jf2(y)+kf3(y))×ℱq{g}(ω)ejω2(x2-y2)dydω=1(2π)2∬ℝ2eiω1(x1-y1)(f0(y)+if1(y))+eiω1(x1-y1)(jf2(y)+kf3(y))×ℱq{g}(ω)ejω2(x2-y2)dydω=∫ℝ2(f0(y)+if1(y))g(x-y)dy+∫ℝ2(jf2(y)+kf3(y))g(-(x1-y1),x2-y2)dy=(f0⋆g)(x)+(if1⋆g)(x)+(jf2⋆g)(-x1,x2)+(kf3⋆g)(-x1,x2),
where, in the second line, we have used the assumption ℱq{g}∈L2(ℝ2;ℝ). This completes the proof of (51).
As an immediate consequence of Theorem 17, we get the following corollaries.
Corollary 18.
If f,g∈L2(ℝ2;ℍ) and ℱq{g}∈L2(ℝ2;ℝ), where
(53)f=f0+if1,g=g0+ig1+jg2+kg3,
then (51) reduces to
(54)ℱq-1[ℱq{f}ℱq{g}](x)=(f⋆g)(x).
Corollary 19.
Let
(55)f(x)={e-(x1+x2),ifx1>0,x2>0,0,otherwise.
And consider the quaternionic Gabor filter
(56)g(x)=eiu0x1ejv0x2e-(1/2)|x|2.
The QFT of f is given by
(58)ℱq{f}(ω)=1-iω1-jω2-kω1ω2(2π)2(1+ω12+ω22+ω12ω22),
and the QFT of g is given by
(59)ℱq{g}(ω)=e-(1/2)((ω1-u0)2+(ω2-v0)2).
Therefore, using Corollary 18, we obtain (57).
4. Applications of QFT
In [19], the authors proposed to use quaternions in order to define a Fourier transform applicable to color images. Their framework makes it possible to compute a single, holistic, Fourier transform which treats a color image as a vector field. In image processing, taking a given image as the initial value, the forward solution to the heat equation or a diffusion equation in general, produces blurred images and the backward solution produces sharpen images for example, see [21, pages 342–350].
In this section, we present two applications of QFT to partial differential equations in quaternion algebra.
4.1. Hypoellipticity
In this paper, since we only deal with QFT in the L2(ℝ2;ℍ) framework, we will discuss the hypoellipticity in this framework; that is, we will only deal with L2(ℝ2;ℍ) solutions for linear partial differential operators with constant quaternion coefficients:
(60)P(∂)=∑0≤|α|≤naα∂αbα,aα,bα∈ℍ.
The noncommutativity of quaternion gives different aspects of P(∂) with constant complex coefficients aα,bα∈ℂ.
Example 20.
Let P(∂)=∂x1b, b∈ℍ, and f(x),g(x)∈C1(ℝ2;ℍ).
Since ∂x1(f(x)g(x))=(∂x1f(x))g(x)+f(x)(∂x1g(x)), we have P(∂)(f(x)g(x))=(P(∂)f(x))g(x)+f(x)(P(∂)g(x)) when b∈ℂ. But, when b∈ℍ∖ℂ, as bf(x)≠f(x)b in general, we cannot have P(∂)(f(x)g(x))=(P(∂)f(x))g(x)+f(x)(P(∂)g(x)) in general.
We have P(∂)f⋆g=f⋆P(∂)g when b∈ℂ. But, by the same reason as (i), when b∈ℍ∖ℂ, we cannot have P(∂)f⋆g=f⋆P(∂)g in general.
Let us start with the definition of our L2(ℝ2;ℍ) version of hypoellipticity (compared to [22, page 110]).
Definition 21.
The linear partial differential operator P(∂) in ℝ2 is said to be L2(ℝ2;ℍ)-hypoelliptic if, given any subset U of ℝ2 and any solution u in L2(ℝ2;ℍ) such that P(∂)u is a C∞ function in U, then all its components u(ui,i=0,1,2,3) are a C∞ function in U.
Definition 22.
Given a linear partial differential operator P(∂) of (60) with the quaternion constant coefficients. One says that a solution E(x) of P(∂)u=δ, where δ is the delta function, is called a fundamental solution of P(∂).
Let A and B be subsets of ℝ2. Define the sum A+B by A+B={x+y∈ℝ2;x∈A,y∈B}.
Theorem 23.
Assume that there is one fundamental solution E(x) of P(∂) which is a C∞ function in ℝ2∖{0}, and the identities
(61)P(∂)(f⋆g)=P(∂)f⋆g=f⋆P(∂)g
are satisfied for arbitrary sufficiently smooth quaternion-valued functions f and g such that gf is a compactly supported C∞ quaternion function with aα of P(∂) being quaternion constant coefficients and bα of P(∂) being real constant coefficients. Then, the linear partial differential operator P(∂) is L2(ℝ2;ℍ)-hypoelliptic in ℝ2.
Proof.
Firstly, let U be an arbitrary open subset of ℝ2 and u a solution in U with values in ℍ such that f=P(∂)u is a C∞ function in U. Let x0 be an arbitrary point in U. It will suffice to show that u is a C∞ function in some open neighborhood of x0. Take an open disc Dη(x0)={x∈ℝ2;|x-x0|<η} such that Dη(x0)¯⊂U. There exists a function g∈C∞(U;ℝ) such that suppg⊂U and g=1 in Dη(x0). Then, we have
(62)P(∂)(gu)=gP(∂)u+v=gf+v,
where every term of v contains a derivative of g of nonzero order; therefore v=0, where the derivatives of g vanish, especially in Dη(x0) and outside of suppg. For the fundamental solution E(x), the hypothesis (61) implies
(63)E⋆P(∂)(gu)={P(∂)E}⋆(gu)=gu.
Hence
(64)gu=E⋆(gf)+E⋆v.
But gf is a compactly supported C∞ function and the convolution of any function with any compactly supported C∞ function is a C∞ function. Therefore, it suffices to show that E⋆v is a C∞ function in an open neighborhood of x0, because gu is also a C∞ function in an open neighborhood of x0 and gu=u in Dη(x0).
Finally, we will show that E⋆v is a C∞ function in an open neighborhood of x0. Let us select ε>0 such that ε<1/2η. Then, the open disc Dε(x0) is a neighborhood of x0. Let ζε(x)∈C∞(ℝ2;ℝ), another cutoff function, be equal to one for |x|<ε/2 and to zero for |x-x0|>ε. We have
(65)E⋆v=(ζεE)⋆v+{(1-ζε)E}⋆v.
The hypothesis implies that (1-ζε)E∈C∞(ℝ2;ℍ), and therefore (1-ζε)E⋆v∈C∞(ℝ2;ℍ). Since
(66)supp{(ζεE)⋆v}⊂supp(ζεE)+suppv,supp{(ζεE)⋆v} is contained in the ε-neighborhood of suppv. We have already seen that v=0∈Dη(x0). Hence, (ζεE)⋆v vanishes in Dε(x0), and, therefore, E⋆v is a C∞ function in Dε(x0).
4.2. Parabolic Initial Value Problem
Let us consider the parabolic initial value problem
(67)∂∂tu-∇2u=0,onℝ2×(0,∞),
with
(68)u(x,0)=f(x),f∈𝒮(ℝ2;ℍ),
where 𝒮(ℝ2;ℍ) is the quaternion Schwartz space. Applying the QFT, we easily obtain
(69)ℱq{ut}=(iω1)2ℱq{u}(ω)+ℱq{u}(ω)(jω2)2=-|ω|2ℱq{u}(ω).
The general solution of (69) is given by
(70)ℱq{u}(ω,t)=Ce-|ω|2t,
where C is a quaternion constant. We impose the initial condition ℱq{u}(ω,0)=ℱq{f}(ω) to obtain
(71)ℱq{u}(ω,t)=e-|ω|2tℱq{f}(ω).
Notice that the QFT of a Gaussian quaternion function is also a Gaussian quaternion function (compared to Bahri et al. [13]). Hence
(72)14πtℱq{(e-|x|2/(4t))}=e-|ω|2t.
Applying the inverse QFT, we have
(73)u(x,t)=ℱq-1[e-|ω|2tℱq{f}](x)=14πtℱq-1[ℱq{f}ℱq{e-|x|2/(4t)}](x).
Since
(74)ℱq{e-|x|2/(4t)}(ω)=4πte-|ω|2t∈L2(ℝ2;ℝ),
then we can apply the convolution theorem of (51) to get
(75)u(x,t)=(f0⋆Kt)(x)+(if1⋆Kt)(x)+(jf2⋆Kt)(-x1,x2)+(kf3⋆Kt)(-x1,x2),
where Kt=(1/4πt)e-|x|2/(4t), and fi∈L2(ℝ2;ℝ), i=0,1,2,3. By Definition 6 of the convolution, we finally obtain
(76)u(x,t)=14πt∫ℝ2f0(y)e-((x1-y1)2+(x2-y2)2)/(4t)dy+i4πt∫ℝ2f1(y)e-((x1-y1)2+(x2-y2)2)/(4t)dy+j4πt∫ℝ2f2(y)e-((-x1-y1)2+(x2-y2)2)/(4t)dy+k4πt∫ℝ2f3(y)e-((-x1-y1)2+(x2-y2)2)/(4t)dy.
In an actual application, one often takes the quaternionic Gabor filter (see [6, 10]) as
(77)f(x)=eiu0x1e-(x12+x22)ejv0x2.
Therefore, the above identity will reduce to
(78)u(x,t)=14πt∫ℝ2cos(u0x1)cos(v0x2)e-(x12+x22)×e-((x1-y1)2+(x2-y2)2)/(4t)dy,+i4πt∫ℝ2sin(u0x1)cos(v0x2)e-(x12+x22)×e-((x1-y1)2+(x2-y2)2)/(4t)dy,+j4πt∫ℝ2cos(u0x1)sin(v0x2)e-(x12+x22)×e-((-x1-y1)2+(x2-y2)2)/(4t)dy,+k4πt∫ℝ2sin(u0x1)sin(v0x2)e-(x12+x22)×e-((-x1-y1)2+(x2-y2)2)/(4t)dy.
5. Conclusion
Due to the non-commutative property of quaternion multiplication, there are three different types of two-dimensional QFTs. These three QFTs are the so-called left-sided QFT, right-sided QFT, and a double-sided QFT, respectively. In this work, we have established convolution theorem of the double-sided QFT applied to real fields f:ℝ2→ℝ and quaternion fields f:ℝ2→ℍ. Some important properties of the QFT convolution are investigated. We have shown that the QFT convolution is useful to study hypoellipticity and to solve the heat equation in quaternion algebra framework. It can easily be seen that the solution of generalized heat equation is extension of solution of the classical heat equation.
The future work will establish the convolution theorems of the right-sided QFT. We compare some properties of the convolution theorems of the two types of QFTs. We will apply the properties to find the solution of partial differential equations in quaternion algebra framework. The solutions of generalized partial differential equations using the properties of the three types of two-dimensional QFTs will be compared too.
Acknowledgments
The authors would like to thank the reviewers whose deep and extensive comments greatly contributed to improve this paper. The first author is partially supported by Hibah Penelitian Kompetisi Internal Tahun 2013 (no. 110/UN4-.42/LK.26/SP-UH/2013) from the Hasanuddin University, Indonesia. The second author is partially supported by JSPS.KAKENHI (C)22540130 and (C)25400202 of Japan and the third author is partially supported by NSERC of Canada.
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