The Cylindrical Fourier Transform

In this paper we devise a so-called cylindrical Fourier transform within the Clifford analysis context. The idea is the following: for a fixed vector in the image space the level surfaces of the traditional Fourier kernel are planes perpendicular to that fixed vector. For this Fourier kernel we now substitute a new Clifford-Fourier kernel such that, again for a fixed vector in the image space, its phase is constant on co-axial cylinders w.r.t. that fixed vector. The point is that when restricting to dimension two this new cylindrical Fourier transform coincides with the earlier introduced Clifford-Fourier transform. We are now faced with the following situation: in dimension greater than two we have a first Clifford-Fourier transform with elegant properties but no kernel in closed form, and a second cylindrical one with a kernel in closed form but more complicated calculation formulae. In dimension two both transforms coincide. The paper concludes with the calculation of the cylindrical Fourier spectrum of an L2-basis consisting of generalized Clifford-Hermite functions.