  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.
