Quaternion Fourier Transform on Quaternion Fields and Generalizations
This work provides theoretical extensions of Fourier transforms for advanced mathematical fields like quaternion and Clifford algebra, which is incremental for researchers in signal processing and geometric algebra.
The paper investigates the quaternionic Fourier transform (QFT) for quaternion fields, deriving properties and Plancherel theorems, and generalizes it to non-commutative multivector Fourier transformations, including new examples like volume-time and spacetime algebra transforms.
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.