### On the clifford-fourier transform

(2011) INTERNATIONAL MATHEMATICS RESEARCH NOTICES. p.5123-5163- 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.

Please use this url to cite or link to this publication:
http://hdl.handle.net/1854/LU-1113155

- author
- Hendrik De Bie UGent and Yuan Xu
- organization
- year
- 2011
- type
- journalArticle (original)
- publication status
- published
- subject
- keyword
- Clifford analysis, Fourier transform, REPRESENTATION
- journal title
- INTERNATIONAL MATHEMATICS RESEARCH NOTICES
- Int. Math. Res. Notices
- issue
- 22
- pages
- 5123 - 5163
- Web of Science type
- Article
- Web of Science id
- 000297927400005
- JCR category
- MATHEMATICS
- JCR impact factor
- 1.014 (2011)
- JCR rank
- 40/288 (2011)
- JCR quartile
- 1 (2011)
- ISSN
- 1073-7928
- DOI
- 10.1093/imrn/rnq288
- language
- English
- UGent publication?
- yes
- classification
- A1
- copyright statement
*I have transferred the copyright for this publication to the publisher*- id
- 1113155
- handle
- http://hdl.handle.net/1854/LU-1113155
- date created
- 2011-01-31 18:10:43
- date last changed
- 2016-12-19 15:45:26

@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}, }

- 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.