On the Clifford-Fourier transform
By Hendrik De Bie, Yuan Xu. 2010
112 citations recorded by the source corpus.
For functions that take values in the Clifford algebra, we study the Clifford-Fourier transform on $R^m$ defined with a kernel function $K(x,y) := e^{\frac{i \pi}{2} \Gamma_{y}}e^{-i }$, replacing the kernel $e^{i }$ 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.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/d6840eff8d879899f15564d640fa7778dbe06451
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In Applied Mathematics
- Saifallah Ghobber
- Philippe Jaming
- Hendrik De Bie
- Nele De Schepper
- F. Sommen
- Irene Sabadini
- Fabrizio Colombo
- Eugenia Malinnikova
In Mathematical Physics
- Saifallah Ghobber
- Philippe Jaming
- Nele De Schepper
- F. Sommen
- Hendrik De Bie
- Irene Sabadini
- Fabrizio Colombo
Citations in the corpus
- Uncertainty principles for integral operators (2012, methodology): “We also follow as closely as possible the presentation of Clifford analysis from [8, 14].”
- The Class of Clifford-Fourier Transforms (2011, methodology): “Secondly, we want to apply these ideas to the so-called Clifford-Fourier transform (introduced in [3]).”