OPS-QFTs: A new type of quaternion Fourier transforms based on the orthogonal planes split with one or two general pure quaternions
This work provides a novel mathematical framework for quaternion Fourier transforms, which is incremental in extending existing methods to more general cases.
The authors tackled the problem of generalizing quaternionic Fourier transforms by introducing a new type based on orthogonal planes splits with one or two pure quaternions, establishing inverse transformations and commenting on their geometric meaning.
We explain the orthogonal planes split (OPS) of quaternions based on the arbitrary choice of one or two linearly independent pure unit quaternions $f,g$. Next we systematically generalize the quaternionic Fourier transform (QFT) applied to quaternion fields to conform with the OPS determined by $f,g$, or by only one pure unit quaternion $f$, comment on their geometric meaning, and establish inverse transformations. Keywords: Clifford geometric algebra, quaternion geometry, quaternion Fourier transform, inverse Fourier transform, orthogonal planes split