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On the clifford-fourier transform

Hendrik De Bie (UGent) and Yuan 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(i pi/2) Gamma Ye(-i){x,y}, replacing the kernel e(i < x,y >) of the ordinary Fourier transform, where Gamma(y) :=- Sigma(j<k) e(j)e(k)(Y(j)partial derivative(Yk) - Y(k)partial derivative(Yj)). 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.
Keywords
Clifford analysis, Fourier transform, REPRESENTATION

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Citation

Please use this url to cite or link to this publication:

Chicago
De Bie, Hendrik, and Yuan Xu. 2011. “On the Clifford-fourier Transform.” International Mathematics Research Notices (22): 5123–5163.
APA
De Bie, H., & Xu, Y. (2011). On the clifford-fourier transform. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, (22), 5123–5163.
Vancouver
1.
De Bie H, Xu Y. On the clifford-fourier transform. INTERNATIONAL MATHEMATICS RESEARCH NOTICES. 2011;(22):5123–63.
MLA
De Bie, Hendrik, and Yuan Xu. “On the Clifford-fourier Transform.” INTERNATIONAL MATHEMATICS RESEARCH NOTICES 22 (2011): 5123–5163. Print.
@article{1113155,
  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(i pi/2) Gamma Ye(-i)\{x,y\}, replacing the kernel e(i {\textlangle} x,y {\textrangle}) of the ordinary Fourier transform, where Gamma(y) :=- Sigma(j{\textlangle}k) e(j)e(k)(Y(j)partial derivative(Yk) - Y(k)partial derivative(Yj)). 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.},
  author       = {De Bie, Hendrik and Xu, Yuan},
  issn         = {1073-7928},
  journal      = {INTERNATIONAL MATHEMATICS RESEARCH NOTICES},
  keyword      = {Clifford analysis,Fourier transform,REPRESENTATION},
  language     = {eng},
  number       = {22},
  pages        = {5123--5163},
  title        = {On the clifford-fourier transform},
  url          = {http://dx.doi.org/10.1093/imrn/rnq288},
  year         = {2011},
}

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