 Author
 Hendrik De Bie (UGent) and Yuan Xu
 Organization
 Abstract
 For functions that take values in the Clifford algebra, we study the CliffordFourier 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 CliffordFourier 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: http://hdl.handle.net/1854/LU1113155
 Chicago
 De Bie, Hendrik, and Yuan Xu. 2011. “On the Cliffordfourier Transform.” International Mathematics Research Notices (22): 5123–5163.
 APA
 De Bie, H., & Xu, Y. (2011). On the cliffordfourier transform. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, (22), 5123–5163.
 Vancouver
 1.De Bie H, Xu Y. On the cliffordfourier transform. INTERNATIONAL MATHEMATICS RESEARCH NOTICES. 2011;(22):5123–63.
 MLA
 De Bie, Hendrik, and Yuan Xu. “On the Cliffordfourier 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 CliffordFourier 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 CliffordFourier 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 = {10737928}, journal = {INTERNATIONAL MATHEMATICS RESEARCH NOTICES}, keyword = {Clifford analysis,Fourier transform,REPRESENTATION}, language = {eng}, number = {22}, pages = {51235163}, title = {On the cliffordfourier transform}, url = {http://dx.doi.org/10.1093/imrn/rnq288}, year = {2011}, }
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