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arXiv · 1003.0689

On the Clifford-Fourier transform

H. De Bie·Y. Xu
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Abstract

For functions that take values in the Clifford algebra, we study the Clifford-Fourier transform on $R^m$ defined with a kernel function $K(x,y) := e^{\frac{i π}{2} Γ_{y}}e^{-i }$, replacing the kernel $e^{i }$ of the ordinary Fourier transform, where $Γ_{y} := - \sum_{j<k} e_{j}e_{k} (y_{j} \partial_{y_{k}} - y_{k}\partial_{y_{j}})$. An explicit formula of $K(x,y)$ is derived, which can be further simplified to a finite sum of Bessel functions when $m$ is even. The closed formula of the kernel allows us to study the Clifford-Fourier transform and prove the inversion formula, for which a generalized translation operator and a convolution are defined and used.

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H. De Bie, Y. Xu. 2010-12-19. On the Clifford-Fourier transform. https://arxiv.org/abs/1003.0689

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