# Quaternion Domain Fourier Transformation

Eckhard Hitzer, International Christian University, College of Liberal Arts, Department of Material Science, Japan

So far quaternion Fourier transforms have been mainly defined over $\mathbb{R}^2$ as signal domain space. But it seems natural to define a quaternion Fourier transform for quaternion valued signals over quaternion domains. Such signals may e.g. represent vector fields in space and time. The quaternion domain Fourier transform thus developed may also be useful for solving quaternion partial differential equations or functional equations, three-dimensional time-dependent color field (space color video) processing, and in crystallography. We define the quaternion domain Fourier transform and analyze its main properties, including quaternion dilation, modulation and shift properties, Plancherel and Parseval identities, covariance under orthogonal transformations, transformations of coordinate polynomials and differential operator polynomials, transformations of derivative and Dirac derivative operators, as well as signal width related to band width uncertainty relationships.