On the Clifford-Fourier transform

Abstract

For functions that take values in the Clifford algebra, we study the Clifford-Fourier transform on $R^m$ defined with a kernel function $K(x,y):=e^{\frac{i\pi }{2}{\Gamma }_y}{e}^{-i<x,y>}$, replacing the kernel $e^{i<x,y>}$ 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.

Some small changes, 30 pages, accepted for publication in IMRN

Cited by in corpus (1)

• The class of Clifford-Fourier transforms