1 Introduction
There are two aspects of the generalization of the classical Fourier transform: one is to the high dimensional space, the other is to the fractional Fourier transform. The quaternion Fourier transform (QFT) is one of the generalized forms of the classical Fourier transform in high dimensional space and has been proved to be very useful in signal processing, non-marginal color image processing, electromagnetism, multi-channel processing, quantum mechanics, and partial differential systems. Many scholars have done a lot of research on the QFT and got many excellent results. In recent years, some properties of the QFT and the two-sided QFT have been studied [
4‐
11].
In 2007, Hitzer [
6] researched the QFT properties useful for applications to differential equations, image processing and optimized numerical implementations and studied the general linear transformation behavior of the QFT with matrices. In 2010, Hitzer [
7] derived a directional uncertainty principle for quaternion-valued functions subject to the QFT. In 2016, Hitzer [
4] defined the FT on the quaternion domain and analyzed its main properties, including quaternion dilation, modulation, shift properties and Parseval identities. In 2017, Haoui and Fahlaoui [
2] presented the Heisenberg inequality and Hardy’s theorem for the two-sided QFT. In [
11], Yang et al. studied uncertainty principles of the QFT under the polar coordinate form. In [
1], Bahri proposed the uncertainty principle for the two-sided QFT. That uncertainty principle described that the spread of a quaternion-valued function and its two-sided QFT was inversely proportional.
On the basis of the above work, we give some properties of the two-sided FrQFT and its application. The structure of the article is as follows: in the second section, we introduce the basic knowledge related to the quaternion Fourier analysis. In the third section, we first give a definition of the two-sided FrQFT. Then based on the nature of the two-sided QFT, we study the relationship between the two-sided QFT and the two-sided FrQFT. We give some differential properties, shift properties of the two-sided FrQFT and Parseval identity. Finally, we give an example to illustrate the application of the two-sided FrQFT and its inverse transform in solving partial differential equations.
2 Preliminaries
Let
\(\mathbb{R}^{2}\) be a real linear space with basis
\(\{e_{1}, e_{2}\}\), the quaternion algebra
\(\mathbb{H}\) which is an associative and noncommutative algebra structure spanned by
$$ \{1, e_{1}, e_{2}, e_{1}e_{2}\}. $$
And basis elements satisfy the following multiplication laws:
$$ \textstyle\begin{cases} e_{i}^{2}=-1,\quad i=1, 2; \\ {e_{1}e_{2}=-e_{2}e_{1}=e_{12}}; \\ {{e_{12}}^{2}=e_{12}e_{12}=-1}; \\ {e_{2}e_{12}=-e_{12}e_{2}=e_{1}}; \\ {e_{12}e_{1}=-e_{1}e_{12}=e_{2}}. \end{cases} $$
Every quaternion
$$ q=q_{0}+q_{1}e_{1}+q_{2}e_{2}+q_{3}e_{12} \in \mathbb{H},\quad q_{0}, q_{1}, q_{2}, q_{3}\in \mathbb{R,} $$
has a quaternion conjugate
\(\overline{q}=q_{0}-q_{1}e_{1}-q_{2}e_{2}-q_{3}e_{12}\), where
\((q)_{0}=q_{0}\).
For arbitrary \(p, q\in \mathbb{H}\), \(\overline{pq}=\overline{q} \overline{p}\).
For quaternion-valued functions
\(f,g:\mathbb{R}^{2}\rightarrow \mathbb{H}\), the quaternion-valued inner product is defined by
$$ (f, g)= \int _{\mathbb{R}^{2}}f(\mathbf{x})\overline{g(\mathbf{x})}\,d\mathbf{x}, $$
and the real scalar part is
$$ \langle f, g\rangle =\frac{1}{2}\bigl[(f,g)+(g,f)\bigr]= \int _{\mathbb{R}^{2}}\bigl(f(\mathbf{x})g( \mathbf{x}) \bigr)_{0}\,d\mathbf{x}, $$
where
\(d\mathbf{x}=dx_{1}\,dx_{2}\).
In particular, when
\(f=g\), this leads to
$$ \Vert f \Vert ^{2}_{L^{2}(\mathbb{R}^{2};\mathbb{H})}=(f, f)=\langle f, f\rangle = \int _{ \mathbb{R}^{2}} \bigl\vert f(\mathbf{x}) \bigr\vert ^{2}\,d\mathbf{x}. $$
Here are some properties of δ function as described below.
For convenience, we divide \(f\in \mathbb{H}\) into two parts as follows.
3 Some properties of the two-sided FrQFT
In this section we state some properties of the two-sided FrQFT. We first give a definition of the two-sided FrQFT and its inverse transformation. Then we get the relationship between two-sided QFT and the two-sided FrQFT. Finally we study the properties of this transformation, such as the shift property, differential properties of functions and their image functions, and differential properties of kernel functions.
When \(p_{1}=p_{2}=1\), the two-sided FrQFT becomes the two-sided QFT.
We know that the two-sided QFT is defined as (see [
1])
$$ \mathcal{F}\{f\}(\mathbf{w})= \int _{\mathbb{R}^{2}}e^{-e_{1}x_{1}w_{1}}f( \mathbf{x})e^{-e_{2}x_{2}w_{2}}\,d\mathbf{x}. $$
(3.3)
Its inverse transformation is defined as
$$ \mathcal{F}^{-1}\{f\}(\mathbf{x})=\frac{1}{(2\pi )^{2}} \int _{ \mathbb{R}^{2}}e^{e_{1}x_{1}w_{1}}f(\mathbf{w})e^{e_{2}x_{2}w_{2}}\,d\mathbf{w}. $$
(3.4)
The Plancherel identity is given by
$$ ( f_{1},f_{2})=\frac{1}{(2\pi )^{2}}\bigl( \mathcal{F} \{f_{1}\}( \mathbf{w}),\mathcal{F}\{f_{2}\}(\mathbf{w})\bigr). $$
(3.5)
Next, we give some important properties of the two-sided FrQFT; we begin with the shift property.
In the following theorem we give the differential properties of the two-sided FrQFT. These conclusions are similar in nature to those of the classical FT, although they have different forms.
Theorem
3.4 describes the relationship between the two-sided FrQFT of the derivative of a function
f and the two-sided FrQFT of the function
f itself.
Theorem
3.5 describes the relationship between the derivative of the two-sided FrQFT of a function
f and the two-sided FrQFT of the function
f itself.
Some important differential properties of kernel functions \(K_{\theta _{1}}(x_{1},w_{1})\) and \(K_{\theta _{2}}(x_{2},w_{2})\) are stated in the following theorems.
Lemma
3.1 describes the relationship between the two-sided FrQFT of the derivative of a function
f and the two-sided FrQFT of the function itself.
Lemma
3.2 describes the relationship between the derivative of the two-sided FrQFT of a function
f and the two-sided FrQFT of
\((-e_{1}x_{1}\csc \theta _{1})^{m_{1}}f\) and
\((-x_{1}\csc \theta _{1})^{m_{2}}f e_{2}^{m_{2}}\).
We will give the properties of inner and scalar products.
In particular, when
\(f_{1}=f_{2}=f\), by Theorem
3.9, we immediately arrive at the following conclusion. Of course, the following equality can also be proved to be true by the definition of the norm.
From this conclusion, we can see that the two-sided FrQFT has norm-preserving properties.
4 The application of the two-sided fractional QFT
Next, we give an application of differential properties of the two-sided FrQFT in solving partial differential equations.
Example. Find solutions to the following partial differential equations.
$$ \biggl(\frac{\partial }{\partial x_{1}}+e_{1}x_{1}\cot \theta _{1}\biggr)^{4}y( \mathbf{x}) \biggl(\frac{\partial }{\partial x_{2}}+e_{2}x_{2} \cot \theta _{2}\biggr)^{5} \overline{e_{2}}^{5}-y( \mathbf{x}) =f(\mathbf{x}) , $$
(4.1)
where
\(f(x)\) is a known quaternion-valued function and
\(y(x)\) is an unknown quaternion-valued function.
Solution. Using differential properties of the two-sided FrQFT, we take the two-sided FrQFT at both sides of differential equation (
4.1). Then, by Theorem
3.4, we have
$$\begin{aligned}& (e_{1}w_{1}\csc \theta _{1})^{4} \mathcal{F}_{\theta _{1},\theta _{2}} \bigl\{ y(\mathbf{x})\bigr\} (\mathbf{w}) (e_{2}w_{2} \csc \theta _{2})^{5} \overline{e_{2}}^{5}-\mathcal{F}_{\theta _{1},\theta _{2}}\bigl\{ y( \mathbf{x})\bigr\} (\mathbf{w}) \\& \quad =\mathcal{F}_{\theta _{1},\theta _{2}}\bigl\{ f(\mathbf{x})\bigr\} (\mathbf{w}). \end{aligned}$$
Then
$$ \bigl[(w_{1}\csc \theta _{1})^{4}(w_{2} \csc \theta _{2})^{5}-1\bigr] \mathcal{F}_{\theta _{1},\theta _{2}} \bigl\{ y(\mathbf{x})\bigr\} (\mathbf{w}) = \mathcal{F}_{\theta _{1},\theta _{2}}\bigl\{ f( \mathbf{x})\bigr\} (\mathbf{w}). $$
That is,
$$ \mathcal{F}_{\theta _{1},\theta _{2}}\bigl\{ y(\mathbf{x})\bigr\} (\mathbf{w}) = \frac{\mathcal{F}_{\theta _{1}\theta _{2}}\{f(\mathbf{x})\}(\mathbf{w})}{(w_{1}\csc \theta _{1})^{4}(w_{2}\csc \theta _{2})^{5}-1}. $$
According to the Fourier inverse transform of the two-sided FrQFT, we can get
$$\begin{aligned}& y(\mathbf{x}) \\& \quad = \mathcal{F}_{\theta _{1},\theta _{2}}^{-1}\biggl\{ \frac{\mathcal{F}_{\theta _{1},\theta _{2}}\{y(\mathbf{x})\}(\mathbf{w})}{(w_{1}\csc \theta _{1})^{4}(w_{2}\csc \theta _{2})^{5}-1} \biggr\} \\& \quad = \int _{\mathbb{R}^{2}}K_{-\theta _{1}}(x_{1},w_{1}) \frac{\mathcal{F}_{\theta _{1},\theta _{2}}\{y(\mathbf{x})\} (\mathbf{w})}{(w_{1}\csc \theta _{1})^{4}(w_{2}\csc \theta _{2})^{5}-1}K_{- \theta _{2}}(x_{2},w_{2})\,d\mathbf{x} \\& \quad = \int _{\mathbb{R}^{2}}A(\mathbf{w}) \mathcal{F}_{\theta _{1}, \theta _{2}}\bigl\{ y( \mathbf{x})\bigr\} (\mathbf{w})K_{-\theta _{2}}(x_{2},w_{2})\,d\mathbf{x}, \end{aligned}$$
where
\(A(\mathbf{w})= \frac{K_{-\theta _{1}}(x_{1},w_{1})}{(w_{1}\csc \theta _{1})^{4}(w_{2}\csc \theta _{2})^{5}-1}\).
5 Conclusion
Using the basic concepts of quaternion algebra we introduced a two-sided FrQFT. Important properties of the two-sided FrQFT such as shift, differential properties, Parseval identities were demonstrated.
But so far there are still some problems to be studied. Firstly, we mention the relationship between the integral expression of the two-sided FrQFT of f when \(\theta _{i}=n\pi \) and that when \(\theta _{i}\neq n\pi \). Secondly, we mention that applications of the two-sided FrQFT in signal processing, non-marginal color image processing and electromagnetism etc. are not given.
Acknowledgements
We thank the editors and all reviewers for taking time out of their busy schedules to read our papers and thank them for valuable suggestions.
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