Split quaternion Fourier transforms for two-dimensional real invariant field

Ji Eun Kim

doi:10.1515/dema-2025-0151

Article Open Access

Split quaternion Fourier transforms for two-dimensional real invariant field

Ji Eun Kim

Published/Copyright: July 31, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

The article gives the corresponding split quaternions in Clifford analysis and the split Fourier transform (FT). Also, we investigate some properties of the split FT and apply to generalizations of the quaternion FT.

Keywords: split quaternion; polar coordinate; quaternion Fourier transforms; Clifford analysis

MSC 2010: 32A99; 32W50; 30G35; 11E88

1 Introduction

The split quaternion algebra is a noncommutative extension of the complex numbers, which has the conjugate of a split-quaternion restricting to the subalgebra of complex numbers. It is a linear combination of the form q = a + b i + c j + d k , where a , b , c , and d are real numbers and the imaginary units are such that

i 2 = − 1 , j 2 = k 2 = i j k = 1 .

Cockle [1] introduced the split quaternion algebra, and the suggestion that split quaternionic analysis and representation theory has been developed into four-dimensional physics by Frenkel and Libine [2]. Libine [3] has proved two different analogues of the Cauchy-Fueter formula for regular functions and has applied the formulas of quaternionic analysis to solve the problems of harmonic analysis. Kim et al. [4,5] have obtained some results for regular functions defined by the ternary quaternion and reduced quaternion field in Clifford analysis. Also, they have researched for the regularity of functions on the form of dual split quaternions in Clifford analysis, and researched corresponding Cauchy-Riemann systems and properties of functions with values in special quaternions by substituting the differential operators of quaternions number systems [6–8].

The classical Fourier transform (FT) is a fundamental and widely-utilized tool in the fields of signal and image processing, with various applications that range from audio signal analysis to image enhancement and reconstruction (see references [9,10] for further details). In a parallel development, the quaternion Fourier transform (QFT) has emerged as a significant extension of the classical FT, specifically designed to handle quaternion-valued signals and images. This transformation holds particular importance in image processing tasks that involve color and other quaternion-based data (refer to works [11–17] for detailed studies on the QFT).

The QFT inherits and modifies several essential properties from the classical FT, making it a versatile tool for various applications in quaternion signal processing. Key properties demonstrated for the QFT include convolution, which describes how two signals can be combined; correlation, which helps in assessing the similarity between signals; energy conservation, ensuring that the total energy is preserved through the transformation; and various mathematical inequalities that govern the behavior of these signals. Each of these properties serves to adapt and extend the corresponding attributes present in the classical FT, thereby enriching the toolset available for signal processing (for a comprehensive discussion, refer to [18,19]).

Moreover, the QFT not only provides practical transformation capabilities but also serves as a foundation for addressing various complex mathematical problems framed within the context of quaternion algebras. Notable applications include solving quaternion linear systems, referenced in [20], and analyzing linear quaternion differential equations, as discussed in [21]. These advancements illustrate the growing importance of quaternion-based approaches in both theoretical and practical aspects of signal processing, further establishing the QFT as an essential framework in the analysis of quaternion signals and images.

The FT is a fundamental tool in probability theory, recognized for its pivotal role in understanding random variables. Specifically, it relates to the characteristic function of any real-valued random variable, which is essential for computing various statistical moments as well as the overall distribution function. While the literature has documented the emergence of the one-dimensional QFT in several studies [22–24], and it has been applied by the authors of reference [17] to establish the one-dimensional quaternion linear canonical transform, its potential applications within the realm of probability theory have yet to be explored comprehensively [25,26].

In light of the definition and inherent properties associated with the concept of a split quaternion, we proceed to establish a formal definition for the FT. Our objective is to demonstrate that the FT can be redefined in a novel manner by incorporating the distinct properties of a split quaternion into its mathematical framework. This integration will allow us to explore the implications and applications of the FT in greater depth, enhancing our understanding of its role within the context of a split quaternion. By doing so, we aim to create a more comprehensive mathematical structure that reflects the interrelationship between a split quaternion and the FT, thereby enriching the theoretical foundation of both concepts.

In this article, we study the FT for functions with split quaternions as variables by considering the definition and properties of split quaternions. We give the corresponding split quaternions in Clifford analysis and investigate the split FT applied to split quaternions and computations for real signals. Also, we research some properties of the split FT and relate to generalizations of the QFT. Section 2 first introduces the split quaternions. We define the split quaternions and their algebraic properties and describe the product between split quaternions. Section 3 gives the FT by defining a function with split quaternions as variables. We investigate the properties and composition of the FT as it is described in split quaternions.

2 Preliminaries

Let the set of split quaternion be

S ≔ { p ∣ p = e 0 x 0 + e 1 x 1 + e 2 x 2 + e 3 x 3 , x r ∈ R ( r = 0 , 1 , 2 , 3 ) } ,

which is noncommutative and a four-dimensional field over the field of real numbers, where its four base elements 1 , e 1 , e 2 , and e 3 satisfy

e 1 2 = − 1 , e 2 2 = e 3 2 = 1 , e 2 e 3 = − e 3 e 2 = − e 1 , e 3 e 1 = − e 1 e 3 = e 2 , e 1 e 2 = − e 2 e 1 = e 3

and e 0 ≔ 1 is the identity of S . It is isomorphic with R ( 2,3 ) , and also C 2 . We give the multiplication of S for two split quaternion p = ∑ r = 0 3 e r x r and q = ∑ r = 0 3 e r y r , as follows:

p q = ( x 0 + e 1 x 1 + e 2 x 2 + e 3 x 3 ) ( y 0 + e 1 y 1 + e 2 y 2 + e 3 y 3 ) = x 0 y 0 − x 1 y 1 + x 2 y 2 + x 3 y 3 + e 1 ( x 0 y 1 + x 1 y 0 − x 2 y 3 + x 3 y 2 ) + e 2 ( x 0 y 2 − x 1 y 3 + x 2 y 0 + x 3 y 1 ) + e 3 ( x 0 y 3 + x 1 y 2 + x 3 y 0 − x 2 y 1 ) .

The split quaternion conjugation p * is

p * = x 0 − e 1 x 1 − e 2 x 2 − e 3 x 3 ,

and the modulus M p of p and the inverse p − 1 are given by:

M p ≔ p p * = x 0 2 + x 1 2 − x 2 2 − x 3 2 , p − 1 = p * M p ( x 0 2 + x 1 2 ≠ x 2 2 + x 3 2 ) .

For each element of S , we also write as the following form:

p = S c ( p ) + P u ( p ) ,

where S c ( p ) = x 0 is called the real part of p and P u ( p ) = e 1 x 1 + e 2 x 2 + e 3 x 3 is called the pure split quaternion. For two pure split quaternions P u ( p ) and P u ( q ) , their dot product and cross product are, respectively,

P u ( p ) ⋅ ˜ P u ( q ) ≔ ( x 1 , x 2 , x 3 ) ⋅ ( y 1 , − y 2 , − y 3 ) = x 1 y 1 − x 2 y 2 − x 3 y 3

and

P u ( p ) × ˜ P u ( q ) ≔ ( − x 1 , x 2 , x 3 ) × ( y 1 , − y 2 , − y 3 ) = e 1 ( − x 2 y 3 + x 3 y 2 ) − e 2 ( x 1 y 3 − x 3 y 1 ) + e 3 ( x 1 y 2 − x 2 y 1 ) ,

where ⋅ is the standard dot product and × is the standard cross product in R 3 . Then we have

P u ( p ) P u ( q ) = − P u ( p ) ⋅ ˜ P u ( q ) + P u ( p ) × ˜ P u ( q ) .

For two split quaternions p and q , if their pure parts are satisfied by the equation P u ( p ) × ˜ P u ( q ) = 0 , then it is called parallel split quaternions. Also, if their pure parts are satisfied by the equation P u ( p ) ⋅ ˜ P u ( q ) = 0 , then it is called perpendicular split quaternions.

Now, we give the representation of split quaternions in polar coordinate forms as follows:

p = M p { cos ( θ ) + ν sin ( θ ) } = M p exp ( ν θ ) ,

where

ν = e 1 x 1 + e 2 x 2 + e 3 x 3 x 1 2 − x 2 2 − x 3 2 ( x 1 2 > x 2 2 + x 3 2 ) and ( x 0 2 + x 1 2 > x 2 2 + x 3 2 ) ,

and ν 2 = − 1 and θ is satisfied by

cos ( θ ) = ∣ x 0 ∣ M p and sin ( θ ) = x 1 2 − x 2 2 − x 3 2 M p .

Let P u ( p ) be a pure split quaternion. Then the rotation through α about the unit vector ν is represented by the transformation

P u ( p ) ↦ R { P u ( p ) } R * ,

where

R = cos α 2 + ν sin α 2 and R * = cos α 2 − ν sin α 2 .

Let Ω = Ω 1 × Ω 2 be an open set of R 2 . Then a mapping F : Ω → S is said to be a split quaternion-valued function if it satisfies (Figure 1)

F [ w , u ] = ∑ r = 0 3 e r g r ( w , u ) ,

where g r ( w , u ) ( r = 0 , 1, 2, 3) are real-valued functions of two real variables. Also, the conjugate of F [ w , u ] is

F * [ w , u ] = g 0 ( w , u ) − ∑ r = 1 3 e r g r ( w , u ) .

The set L 2 ( R 2 , S ) is given by

L 2 ( R 2 , S ) = F [ w , u ] ∣ F [ w , u ] ⋅ ˜ F [ w , u ] = ∫ Ω F [ w , u ] F * [ w , u ] d w d u < ∞ ,

where

∫ Ω F [ w , u ] F * [ w , u ] = ∫ Ω 1 ∫ Ω 2 F [ w , u ] F * [ w , u ] d w d u = ∫ Ω 1 ∫ Ω 2 ( g 0 2 + g 1 2 − g 2 2 − g 3 2 ) d w d u .

$Figure 1 ν \nu is satisfied by the graph.$

Figure 1

ν is satisfied by the graph.

3 Split quaternion Fourier transformation

We introduce a split quaternion Fourier transformation. The Fourier transformation is a mapping of real-valued functions into complex-valued functions. The split quaternion Fourier transformation is a mapping of real-valued functions into split quaternion-valued functions. In particular, referring [27], we give a split quaternion FT, which is a mapping of functions of two real arguments into functions of two orthogonal complex arguments.

Definition 3.1

For f ∈ L 2 ( R 2 , R ) , let Sℱ : L 2 ( R 2 , R ) → L 2 ( R 2 , S ) . Then the transformation Sℱ is said to be a split quaternion Fourier transformation if it satisfies the following equation:

Sℱ [ f ( t , x ) ] ≔ ∫ R ∫ R exp ( − e 1 w t ) f ( t , x ) exp ( − e 2 u x ) d t d x = F [ w , u ] ,

where exp ( − e 1 w t ) = cos ( w t ) − e 1 sin ( w t ) and exp ( − e 2 u x ) = cosh ( w t ) − e 2 sinh ( w t ) , called the split quaternion Fourier kernels.

Theorem 3.2

For f ∈ L 2 ( R , R ) , let Sℱ : L 2 ( R 2 , R ) → L 2 ( R 2 , S ) be a split quaternion Fourier transformation. Then we have the inverse of a split quaternion Fourier transformation as follows:

Sℱ − 1 [ F [ w , u ] ] ≔ 1 4 π 2 ∫ R ∫ R exp ( e 1 w t ) F [ w , u ] exp ( e 2 u x ) d u d w = f ( t , x ) .

Proof

Since we have the partial representations as follows:

(1) f ( λ , x ) = 1 2 π ∫ R ∫ R f ( λ , μ ) exp ( − e 2 u μ ) d μ exp ( e 2 u x ) d u

and

(2) f ( t , x ) = 1 2 π ∫ R exp ( e 1 w t ) ∫ R exp ( − e 1 w t ) f ( λ , x ) d λ d w ,

by combining the equations (1) and (2), we obtain the result

□ f ( t , x ) = 1 2 π ∫ R exp ( e 1 w t ) ∫ R exp ( − e 1 w λ ) 1 2 π ∫ R ∫ R f ( λ , μ ) exp ( − e 2 u μ ) d μ exp ( e 2 u x ) d u d λ d w = 1 4 π 2 ∫ R ∫ R exp ( e 1 w t ) ∫ R ∫ R exp ( − e 1 w λ ) f ( λ , μ ) exp ( − e 2 u μ ) d μ d λ exp ( e 2 u x ) d u d w = 1 4 π 2 ∫ R ∫ R exp ( e 1 w t ) F [ w , u ] exp ( e 2 u x ) d u d w .

By using the aforementioned process, we give the partial transformation as follows:

Definition 3.3

For f ∈ L 2 ( R 2 , R ) , let Sℱ e 1 , Sℱ e 2 : L 2 ( R 2 , R ) → L 2 ( R 2 , S ) . Then the transformations Sℱ e 1 , Sℱ e 2 are said to be the e 1 -partial split quaternion Fourier transformation and the e 2 -partial split quaternion Fourier transformation, respectively, if it satisfies the following equations:

Sℱ e 1 [ f ( t , x ) ] ≔ ∫ R exp ( − e 1 w t ) f ( t , x ) d t = F [ w ] ( x )

and

Sℱ e 2 [ f ( t , x ) ] ≔ ∫ R f ( t , x ) exp ( − e 2 u x ) d x = F ( t ) [ u ] .

Furthermore, the transformations Sℱ e 1 − 1 , Sℱ e 2 − 1 are said to be the inverse of an e 1 -partial split quaternion Fourier transformation and the inverse of an e 2 -partial split quaternion Fourier transformation, respectively, if it satisfies the following equations:

Sℱ e 1 − 1 [ F [ w ] ( x ) ] ≔ 1 2 π ∫ R exp ( e 1 w t ) F [ w ] ( x ) d w = f ( t , x )

and

Sℱ e 2 − 1 [ F ( t ) [ u ] ] ≔ 1 2 π ∫ R F ( t ) [ u ] exp ( e 2 u x ) d u = f ( t , x ) .

Now, we consider the time shifted function f ( t − t 0 , x ) by replacing t by t − t 0 .

Proposition 3.4

For f ( t − t 0 , x ) ∈ L 2 ( R 2 , R ) , the split quaternion Fourier transformation is

Sℱ [ f ( t − t 0 , x ) ] = exp ( − e 1 w t 0 ) F [ w , u ] .

Proof

From Definition 3.1, we have

Sℱ [ f ( t − t 0 , x ) ] = ∫ R ∫ R exp ( − e 1 w t ) f ( t − t 0 , x ) exp ( − e 2 u x ) d t d x .

By putting t = λ + t 0 and d t = d λ , we have

Sℱ [ f ( t − t 0 , x ) ] = ∫ R ∫ R exp ( − e 1 w ( λ + t 0 ) ) f ( λ , x ) exp ( − e 2 u x ) d λ d x = exp ( − e 1 w t 0 ) ∫ R ∫ R exp ( − e 1 w λ ) f ( λ , x ) exp ( − e 2 u x ) d λ d x = exp ( − e 1 w t 0 ) F [ w , u ] .

Hence, we obtain the result.□

Likewise, by using properties of the integral operator and replacing suitable variables, we have the following propositions of the split quaternion Fourier transformation.

Proposition 3.5

For f ( a t , b x ) ∈ L 2 ( R 2 , R ) , where a , b ∈ R − { 0 } , the split quaternion Fourier transformation is

Sℱ [ f ( a t , b x ) ] = 1 ∣ a b ∣ F w a , u b .

Proof

From Definition 3.1, we have

Sℱ [ f ( a t , b x ) ] = ∫ R ∫ R exp ( − e 1 w t ) f ( a t , b x ) exp ( − e 2 u x ) d t d x .

We put a t = λ and b x = μ . Then we have

Sℱ [ f ( a t , b x ) ] = ∫ R ∫ R exp − e 1 w λ a f ( λ , μ ) exp − e 2 u λ a 1 ∣ a ∣ d λ 1 ∣ b ∣ d μ = 1 ∣ a ∣ 1 ∣ b ∣ ∫ R ∫ R exp − e 1 w λ a f ( λ , μ ) exp − e 2 u λ a d λ d μ = 1 ∣ a ∣ 1 ∣ b ∣ F w a , u b .

Hence, we obtain the result.□

Proposition 3.6

For partial differentials ∂ ∂ t f ( t , x ) , ∂ ∂ x f ( t , x ) ∈ L 2 ( R 2 , R ) , the split quaternion Fourier transformation is

Sℱ ∂ ∂ t f ( t , x ) = e 1 w F [ w , u ]

and

(3) Sℱ [ ∂ ∂ x f ( t , x ) ] = F [ w , u ] e 2 u .

Furthermore, for ∂ 2 ∂ t ∂ x f ( t , x ) ∈ L 2 ( R 2 , R ) , the split quaternion Fourier transformation is

(4) Sℱ [ ∂ ∂ x f ( t , x ) ] = e 1 w F [ w , u ] e 2 u .

Proof

From Definition 3.1, we have

Sℱ ∂ ∂ t f ( t , x ) = ∫ R ∫ R exp ( − e 1 w t ) ∂ ∂ t f ( t , x ) exp ( − e 2 u x ) d t d x .

By using Integration by parts, we obtain

Sℱ ∂ ∂ t f ( t , x ) = − ∫ R ( − e 1 w ) exp ( − e 1 w t ) ∫ R f ( t , x ) exp ( − e 2 u x ) d x d t = e 1 w ∫ R ∫ R exp ( − e 1 w t ) f ( t , x ) exp ( − e 2 u x ) d x d t .

Similarly, we also obtain equations (3) and (4).□

We give Parseval’s theorem generalized to the split quaternion Fourier transformation.

Theorem 3.7

For f ( t , x ) ∈ L 2 ( R 2 , R ) , if the integral

∫ R ∫ R ∣ f ( t , x ) ∣ 2 d t d x < ∞ ,

then we have

f ( t , x ) ⋅ f ( t , x ) 4 π 2 = F [ w , u ] ⋅ ˜ F [ w , u ] ,

where

F [ w , u ] ⋅ ˜ F [ w , u ] = ∫ R ∫ R F [ w , u ] F * [ w , u ] d w d u .

Proof

From the usual Parseval’s theorem [26] and by the properties of the integral for a continuous function, for a fixed x , we have

∫ R ∫ R ∣ f ( t , x ) ∣ 2 d t d x = ∫ R 2 π ∫ R F [ w ] ( x ) F * [ w ] ( x ) d w d x = 2 π ∫ R ∫ R F [ w ] ( x ) F * [ w ] ( x ) d x d w = 2 π ∫ R 2 π ∫ R F [ w , u ] F * [ w , u ] d u d w = 4 π 2 ∫ R ∫ R F [ w , u ] F * [ w , u ] d w d u .

Therefore, we obtain the result.□

Example 3.8

Consider the quaternionic distribution signal [19]

f ( t , x ) = exp ( e 1 λ t ) exp ( e 2 μ x ) .

If ω ≠ ω 0 , where ω ≔ ( t , x ) and ω 0 ≔ ( λ , μ ) , then the split quaternion Fourier transformation of f is

Sℱ [ f ( t , x ) ] = ∫ R 2 exp ( − e 2 w t ) exp ( e 1 w λ ) exp ( e 2 u μ ) exp ( − e 2 u x ) d t d x = 2 π ∫ R exp ( e 2 u μ ) exp ( − e 2 u x ) d x = ( 2 π ) 2 .

Also, if ω = ω 0 , then Sℱ [ f ( t , x ) ] = 0 . Therefore, we obtain the result as follows:

Sℱ [ f ( t , x ) ] = ( 2 π ) 2 δ ( ω − ω 0 ) .

4 Conclusion

The FT is a fundamental mathematical process that transforms a function into a different representation, revealing its frequency components. It is widely used in various fields, including signal processing, image analysis, and data compression. Over time, this process has been adapted to multiple domains and numerical representations, such as complex numbers and quaternions, to accommodate a broader range of applications. In this article, we propose a novel approach that utilizes split quaternions for the FT. Split quaternions, as an extension of traditional quaternion algebra, offer a unique set of mathematical properties and a compact symbolic representation, making them particularly advantageous for certain applications.

Although split quaternions have been successfully employed in various scientific and computational contexts due to their ability to clearly and efficiently represent multicomponent data, there has been a relative scarcity of research focused on integrating them with Fourier processes. This gap motivates the current study. We introduce the split quaternion fourier transform (SQFT), specifically designed for data represented in split quaternion form over split quaternion domains. This new transform effectively enables the analysis and manipulation of multidimensional data in ways that traditional FTs may not capture. Furthermore, we present the SQFT, which allows for the retrieval of the original split quaternion-valued data from its transformed representation. The SQFT demonstrates substantial potential in exploring hypercomplex algebra in higher dimensions, particularly in solving split quaternion partial differential equations and functional equations. This application could lead to advancements in various engineering and scientific fields, where complex multicomponent systems need to be analyzed and understood. This article lays the groundwork for future research into the integration of split quaternions with Fourier analysis, highlighting the broader implications for computational efficiency and clarity in data representation.

Split quaternions have become increasingly recognized as powerful mathematical tools for managing multicomponent data. Their unique structure allows for efficient, compact, and unified methods to manipulate, concatenate, and interpolate various types of data. In this article, we explore how the split quaternion representation can be effectively integrated with the FT, opening up new avenues for analysis and application. By combining Split quaternions with the FT, we can address several inherent limitations found in traditional data processing techniques, such as sensitivity to noise and challenges related to interpretability. While the filters generated through this combination are fundamentally linear, the flexibility of split quaternions permits the development of more intricate, nonlinear filters when desired.

This article illustrates that the SQFT proves to be highly effective in a variety of domains, particularly in adapting and filtering the spectral content of signal data. When filtering is conducted in the split quaternion frequency domain, we are able to treat the motion data as a cohesive entity rather than processing each channel in isolation. This holistic approach is advantageous because it allows for the preservation of more accurate signal information, enhancing the overall quality of the data analysis. Furthermore, we want to emphasize that the application of the split quaternion FT is not confined to the transformation of rotational and translational data. It has significant potential in other areas, including but not limited to pattern recognition and data compression. By leveraging the unique properties of split quaternions, our research complements existing interactive tools that are used for editing and constructing signals, ultimately advancing the use of split quaternions as a foundational format for various data processing tasks. Through this work, we aim to both illustrate the versatility of split quaternions and encourage further exploration of their applications in mathematical data manipulation and analysis.

Acknowledgement

This study was supported by the Dongguk University Research Fund and the National Research Foundation of Korea (NRF) (2021R1F1A1063356).

Funding information: This study was supported by the Dongguk University Research Fund and the National Research Foundation of Korea (NRF) (2021R1F1A1063356).
Author contribution: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Ethical approval: The conducted research is not related to either human or animal use.
Data availability statement: The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials.

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Received: 2024-10-26

Accepted: 2025-05-30

Published Online: 2025-07-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2025-0151

Keywords for this article

split quaternion; polar coordinate; quaternion Fourier transforms; Clifford analysis

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