Convolution Theorems for Clifford Fourier Transform and Properties

The non-commutativity of the Clifford multiplication gives different aspects from the classical Fourier analysis.We establish main properties of convolution theorems for the Clifford Fourier transform. Some properties of these generalized convolutionsare extensions of the corresponding convolution theorems of the classical Fourier transform.DOI : http://dx.doi.org/10.22342/jims.20.2.143.125-140


Introduction
Recently, several attempts have been made to generalize the classical Fourier transform in the framework of Clifford algebra, so-called the Clifford Fourier transform (CFT). It was first introduced from the mathematical aspect by Brackx et al. [5,6]. The CFT was recently used in signal processing [9,14] and in other fields of mathematics and applications. Many generalized transforms, such as the Clifford wavelet transform, fractional Clifford Fourier transform, and Clifford windowed Fourier transform (see, for example, [2,7,8,10,12,15]) are closely related to the CFT. One of the most fundamental and important properties of the CFT is the convolution theorem.
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 [3,13], authors generalized convolution to the quaternion Fourier transform (QFT). They found that the QFT of the real-valued signals are very similar to the classical ones. In this paper, we establish convolution theorems for the CFT. Here we adopt the definition of the CFT suggested by several authors [1,4,9]. Because the Clifford multiplication is not commutative, we find important properties of the relationship between the convolution theorems and the CFT. We finally establish the inverse CFT of the product of two CFTs, which is very useful in solving partial differential equations in the Clifford algebra.
This paper is organized as follows. In section 2, we provide some basic knowledge of real Clifford algebra used in the paper. Subsequently, in section 3, we define the CFT and discuss its important properties, which are used to construct the properties of generalized convolution. Next in section 4, we introduce convolution on Clifford algebra Cl n,0 and derive its useful properties. Finally, in section 5, we investigate the important properties of the CFT of convolution of Clifford-valued functions and derive the inverse CFT of the product of two CFTs.
Obviously, for n = 2(mod 4), the pseudoscalar i n = e 1 e 2 · · · e n anti-commutes with each basis of the Clifford algebra while i 2 n = −1. The noncommutative multiplication of the basis vectors satisfies the rules e i e j + e j e i = 2δ ij (δ ij denotes the Dirac distribution whose support is {i, j}).
An element of the Clifford algebra is called a multivector and has the following A multivector f ∈ Cl n,0 , n = 2 (mod 4) can be decomposed as a sum of its even grade part, f even , and its odd grade part, f odd . Thus, we have where The reversef of a multivector f is an anti-automorphism given by and hence f g =gf for arbitrary f, g ∈ Cl n,0 .
The Clifford product of two vectors splits up into a scalar part (the inner product) and a so-called bivector part (the wedge product): Observe that the square of a vector x is scalar-valued and x 2 We introduce a first order vector differential operator by This operator is the so-called Dirac operator, which may be looked upon as the square root of the Laplacian operator in R n : △ n = ∂ 2 x . Let us consider L 2 (R n ; Cl n,0 ) as a left module. For f , g ∈ L 2 (R n ; Cl n,0 ), an inner product is defined by In particular, if f = g, then the scalar part of the above inner product gives the Hereinafter, if not otherwise stated, n is assumed to be n = 2 (mod 4).

Clifford Fourier Transform (CFT)
3.1. Fundamental Operators. Before we define the CFT, we need to introduce some notation, which will be used in the next section. For f ∈ L 2 (R n ; Cl n,0 ), we define the translation and modulation as follows: and their composition, which is called the time-frequency shift, Just as in the classical case, we obtain the canonical commutation relations The following lemma describes the behavior of translation, modulation, and time-frequency shift in the Clifford algebra Cl n,0 . Lemma 3.1. If a, ω 0 ∈ R n and f, g ∈ L 2 (R n ; Cl n,0 ), then we have the following: (iii) For f ∈ L 2 (R n ; Cl n,0 ), n = 3 (mod 4), we get (M ω0 f, g) L 2 (R n ;Cln,0) = (f, M ω0 g odd + M −ω0 g even ) L 2 (R n ;Cln,0) .
(iv) Under the assumption stated in (iii), we obtain Proof. Proof of (i). It follows from (7) that Proof of (ii). By equations (7) and (9), we easily obtain Here, in the second equality of (12), we have used the assumption that ensures to interchange the position. Proof of (iii). The proof is similar to (ii) and is left to the reader. Proof of (iv). By simple computations, we get Here, in the second and third equalities of (13), we have used the assumption and properties of the decomposition of multivector g.

Definition of CFT.
The Cl n,0 Clifford Fourier transform (CFT) is a generalization of the FT in Clifford algebra obtained by replacing the FT kernel with the Clifford Fourier kernel. For detailed discussions of the properties of the CFT and their proofs, see, e.g., [4,9].
with x, ω ∈ R n .
Decomposing the multivector f into f even and f odd , equation (14) can be rewritten as The Clifford exponential e −inω·x is often called the Clifford Fourier kernel. For dimension n = 3 (mod 4), this kernel commutes with all elements of the Clifford algebra Cl n,0 , but for n = 2 (mod 4) it does not. Notice that the different commutation rules of the pseudoscalar i n play a crucial rule in establishing the properties of the convolution theorems of the CFT.
In the following, we collect the fundamental properties of the CFT.

. Then F{f } is invertible and its inverse is calculated by the formula
Proof. Substituting (14) into (16) yields Equation (16) is called the Clifford Fourier integral theorem. It describes how to get from the transform F{f } back to the original function f .
It is straightforward to see that the inverse CFT and the CFT share the same properties. One may check the properties of the inverse CFT analogous to those in Lemma 3.3. For an example,

Clifford Convolution And Its Properties
In this section, we introduce the Clifford convolution and establish its important properties. Ebling and Scheuermann [9] distinguish between right and left Clifford convolutions due to the non-commutative property of the Clifford multiplication. Here, we only consider one kind of Clifford convolution. Let us first define the convolution of two Clifford-valued functions.
Definition 4.1. The Clifford convolution f ⋆ g of f and g belong to L 2 (R n ; Cl n,0 ) is defined by Since, in general, the basis vectors e A e B ̸ = e B e A , the Clifford convolution is not commutative, i.e., (f ⋆ g) ̸ = (g ⋆ f ). It is clear that the Clifford convolution of f and g is a binary operation, which combines shifting, geometric product and integration.
If we perform the change of variables z = x − y and relabel z back to y, then equation (17) can be written as Lemma 4.2 (Linearity). Let f , g, h ∈ L 2 (R n ; Cl n,0 ) and α, β ∈ Cl n,0 . Then, we have .

Lemma 4.3 (Shifting).
Let f ∈ L 2 (R n ; Cl n,0 ). Then we have Proof. We only prove (20), the proof of (21) being similar. A direct calculation yields On the other hand, by the change of variables, z = x − y − a, we easily get This completes the proof.
Equations (20) and (21) tell us that the Clifford convolutions commute with translations.

Lemma 4.5 (Reversion).
Let f , g ∈ L 2 (R n ; Cl n,0 ). Then, we have Proof. A straightforward computation gives which was to be proved.

Main Results
In this section, we investigate some important properties of the CFT of convolution of two Clifford-valued functions. We find that most of these properties are extensions of the classical case. The following theorem gives the relationship between the reversion of Clifford convolution and its CFT. Theorem 5.1. Let f, g ∈ L 2 (R n ; Cl n,0 ). Denote by g odd (resp. g even ), the odd (resp. even) grade part of g. Then Proof. An application of the CFT definition combined with the Clifford convolution property of Lemma 4.5 gives where the last equality follows from the change of variables z = x−y. By splitting f into its even grade and odd grade parts, the above identity may be rewritten as Again we decompose the multivector g into its even grade and odd grade parts to get ( g odd (y) + g even (y)) e inω·y d n y F{f odd }(ω) This concludes the proof.
As an immediate consequence of Theorem 5.1, we get the following corollaries.
Proof. An alternative proof of Corollary 5.2 uses [4, (4.59) of Theorem 4.33], i.e., It is worth noting here that a similar argument cannot be applied to prove Corollary 5.3. Its proof follows directly from Theorem 5.1.
We next establish the shift property of the convolution theorem of the CFT. The proof of this property uses the shift property of the CFT and the decomposition of a multivector f . Theorem 5.4. Let f , g ∈ L 2 (R n ; Cl n,0 ). Then Proof. By the definition of CFT, we easily get For the third equality, we have used the substitution of variable z = x − y. For the last equality, we have used the shift property of the CFT. This completes the proof of (26).
We have the following simple corollary to Theorem 5.4.
Corollary 5.5. When g ∈ L 2 (R n ; Cl n,0 ) with n = 3 (mod 4), Theorem 5.4 takes the form Or, equivalently, Proof. Using a similar argument as the proof of equation (24), we immediately get where, in the last line of (29), we have used the first equation in Lemma 3.3.
Now we establish the modulation property of the convolution theorem of the CFT.
Theorem 5.6. Let f , g ∈ L 2 (R n ; Cl n,0 ). Then and Proof. We only sketch the proof of (30), the other being similar. A direct computation gives which was to be proved.

Remark 5.7.
Note that if g ∈ L 2 (R n ; Cl n,0 ) with n = 3 (mod 4), then Theorem 5.6 has the form which is of the same form as the modulation property of the convolution of the FT (see [11]).
We further establish the time-frequency shift of the convolution theorem of the CFT. Theorem 5.8. Let f , g ∈ L 2 (R n ; Cl n,0 ). Then Proof. Applying equations (10) and (14), we immediately obtain We decompose f and g into f odd + f even and g odd + g even , respectively. Then, we obtain F{g even }(ω).
which was to be proved.
Proof. By the definition of the CFT (14) and Clifford convolution (17), we have which was to be proved.
The following corollary is a special case of Theorem 5.12.