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arXiv 1003.0689math.CAmath.CV
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关于Clifford-傅里叶变换

On the Clifford-Fourier transform

H. De Bie, Y. Xu

备注：
Some small changes, 30 pages, accepted for publication in IMRN
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摘要：

对于取值于Clifford代数的函数，研究了$R^m$上的Clifford-傅里叶变换，该变换定义了一个核函数$K(x,y) := e^{\frac{i π}{2} Γ_{y}}e^{-i <x,y>}$，取代了普通傅里叶变换的核$e^{i <x,y>}$，其中$Γ_{y} := - \sum_{j<k} e_{j}e_{k} (y_{j} \partial_{y_{k}} - y_{k}\partial_{y_{j}})$ .导出了$K(x,y)$的显式表达式，当$m$为偶数时，该表达式可进一步简化为贝塞尔函数的有限和。核的闭公式允许我们研究Clifford-傅里叶变换并证明反演公式，为此定义并使用了广义平移算子和卷积。

英文摘要：

For functions that take values in the Clifford algebra, we study the Clifford-Fourier transform on $R^m$ defined with a kernel function $K(x,y) := e^{\frac{i π}{2} Γ_{y}}e^{-i <x,y>}$, replacing the kernel $e^{i <x,y>}$ 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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