# Ghent University Academic Bibliography

### On the clifford-fourier transform

Hendrik De Bie UGent and Yuan Xu (2011) 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:
author
organization
year
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
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.