Identities of the Fractional Fourier Transform and the Versor Transform

TL;DR

This paper introduces the Versor Transform, a novel mathematical object linked to unit quaternions, generalizing the Fourier and Laplace Transforms, and explores their identities and relationships.

Contribution

It defines the Versor Transform associated with unit quaternions, connecting it to classical transforms and providing new identities and insights.

Findings

Versor Transform generalizes Fourier and Laplace Transforms

Derived identities for both fractional Fourier and Versor Transforms

Established connections between transforms and quaternionic units

Abstract

We provide an introduction to the Fractional Fourier Transform $\mathcal{F}_{\theta}$ and draw a connection between it and the unit complex number $e^{i\theta}$. Motivated by this, we define an entirely new object associated with any unit quaternion $e^{i\xi_{1}}\cos\eta+e^{i\xi_{2}}j\sin\eta$, which we call the Versor Transform $\mathcal{V}_{(\xi_{1},\eta,\xi_{2})}$. This transform, which has both the Fourier and Laplace Transforms as special cases, encourages an alternate view of the relationship between them. We also derive several identities for both $\mathcal{F}_{\theta}$ and $\mathcal{V}_{(\xi_{1},\eta,\xi_{2})}$.