On the Clifford-Fourier Transform

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.