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 L-2-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. (C) 2009 Elsevier Inc. All rights reserved.