The Clifford–Fourier integral kernel in even dimensional Euclidean space

Recently, we devised a promising new multi-dimensional integral transform within the Clifford analysis setting, the so-called Fourier–Bessel transform. In the specific case of dimension two, it coincides with the Clifford–Fourier transform introduced earlier as an operator exponential. Moreover, the L2-basis elements, consisting of generalized Clifford–Hermite functions, appear to be simultaneous eigenfunctions of both integral transforms. In the even dimensional case, this allows us to express the Clifford–Fourier transform in terms of the Fourier–Bessel transform, leading to a closed form of the Clifford–Fourier integral kernel.