Quaternion Fourier Transform on Quaternion Fields and Generalizations (1306.1023v1)

Abstract: We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear ($GL$) transformation behavior of the QFT with matrices, Clifford geometric algebra and with examples. We finally arrive at wide-ranging non-commutative multivector FT generalizations of the QFT. Examples given are new volume-time and spacetime algebra Fourier transformations.

• The paper investigates the quaternion Fourier transform (QFT) for quaternion-valued functions and establishes key properties including Plancherel theorems.
• It extends the QFT framework by analyzing its behavior under general linear transformations using Clifford algebra and invariant techniques.
• The paper introduces non-commutative multivector Fourier transform generalizations, including volume-time and spacetime algebra transforms, applicable in physics and complex domains.

Overview of Quaternion Fourier Transform on Quaternion Fields and Generalizations

The paper "Quaternion Fourier Transform on Quaternion Fields and Generalizations" by Eckhard M. S. Hitzer offers a comprehensive investigation into the quaternionic Fourier transform (QFT) tailored to quaternion fields. This paper not only revisits existing properties of the QFT when applied to quaternion-valued functions but extends its framework to support more complex and broader applications through mathematical generalizations.

Introduction to Quaternions and QFT

Quaternions, as higher-dimensional algebraic structures, extend complex numbers to four dimensions, characterized by a real part and three imaginary units. This algebra is remarkably useful in various mathematical areas such as rotation representation in three-dimensional space, which is explicitly leveraged in the presented work.

The author explores the quaternionic Fourier transform applied to quaternion fields $f: \mathbb{R}^2 \rightarrow \mathbb{H}$, where $\mathbb{H}$ represents the quaternion algebra. This encompasses the treatment of functions that produce quaternion-valued outputs as opposed to purely real or complex signals. The QFT is further scrutinized for properties that may benefit practical implementations in areas like solving partial differential equations and image processing. A particular emphasis is placed on establishing Plancherel theorems that fortify the theoretical underpinnings of QFT’s application.

General Linear Transformations and Clifford Algebra

Central to this work is the extension of QFT's behavior under general linear (GL) transformations using matrices as well as the Clifford geometric algebra approach. The Clifford algebra provides a means to work with more generalized algebraic systems, thereby furthering the scope of QFT applications. Specifically, the treatment of GL transformations is enhanced through the use of invariant techniques and explicit examples such as stretches, reflections, and rotations. This analysis leads to the realization of wide-ranging non-commutative multivector Fourier transform generalizations.

The quaternion module and its algebraic subtleties are methodically explored to depict the interaction and transformation of quaternion module functions under Fourier transforms. A detailed examination of the Plancherel and Parseval theorems for quaternion fields is provided, underscoring their role in preserving the energy of signals under QFT.

Multivector Fourier Transform Generalizations

A notable contribution of the paper is the non-commutative multivector Fourier transform (MVFT) generalizations. The author introduces the novel concept of volume-time algebra and spacetime algebra Fourier transformations to cater to functions mapping domain spaces into Clifford algebras. These transformations are particularly significant in translating the theoretical aspects of quaternions into practical computational methods applicable in physics, notably in dynamic fluid analysis, electromagnetic phenomena, and similar complex domains.

The approach facilitates a new dimension of function representation, where multivector modeling serves as a bridge between mathematical theory and real-world applications. These non-commutative Fourier transforms enable new possibilities for processing data that are intrinsically tied to three-dimensional space and time, promising advancements in accurate and efficient signal and system analysis.

Implications and Future Directions

The implications of this research extend beyond immediate applications in signal processing and PDEs. By framing quaternions within the context of Clifford algebras, it provides a scalable path forward for computational techniques that capitalize on the algebra’s robustness in higher-dimensional data representation. The methodologies and theorem proofs offer a promising foundation for future exploration into new realms of quaternion-based computations.

Future research is likely to explore the optimization of these transforms in digital computation, potentially impacting artificial intelligence and machine learning areas where high-dimensional data interpretation is critical. Furthermore, the exploration of new algebraic methods aligned with physical phenomena underscores a direction for integrating mathematical rigor with empirical insights, creating a symbiosis between theoretical mathematics and applied science.

This comprehensive paper successfully establishes a versatile framework for quaternion Fourier transforms applicable to a broader class of functions and underscores the potential of quaternion and Clifford algebras as pivotal tools in advanced computing contexts.