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Article overview
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On the Clifford-Fourier transform | H. De Bie
; Y. Xu
; | Date: |
2 Mar 2010 | Abstract: | For functions that take values in the Clifford algebra, we study the
Clifford-Fourier transform on $R^m$ defined with a kernel function $K(x,y) :=
e^{frac{i pi}{2} Gamma_{y}}e^{-i <x,y>}$, replacing the kernel $e^{i <x,y>}$
of the ordinary Fourier transform, where $Gamma_{y} := - sum_{j<k} e_{j}e_{k}
(y_{j} partial_{y_{k}} - y_{k}partial_{y_{j}})$. 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. | Source: | arXiv, 1003.0689 | Services: | Forum | Review | PDF | Favorites |
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