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 $\mathcal{ V}_{ (\xi_{ 1},\eta,\xi_{ 2})}$.
Categories: math.GM