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arXiv · 2211.16668

Identities of the Fractional Fourier Transform and the Versor Transform

Maurice Pierre
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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 $\mathcal{V}_{(\xi_{1},\eta,\xi_{2})}$.

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Maurice Pierre. 2022-11-30. Identities of the Fractional Fourier Transform and the Versor Transform. https://arxiv.org/abs/2211.16668

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