Identities of the Fractional Fourier Transform and the Versor Transform

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 ${\mathcal{V}}_{({\xi }_1,\eta ,{\xi }_2)}$.