  Recently, there has been an increasing interest in the study of hypercomplex
signals and their Fourier transforms. This paper aims to study such integral
transforms from general principles, using 4 different yet equivalent
definitions of the classical Fourier transform. This is applied to the
so-called Clifford-Fourier transform (see [F. Brackx et al., The
Clifford-Fourier transform. J. Fourier Anal. Appl. 11 (2005), 669--681]). The
integral kernel of this transform is a particular solution of a system of PDEs
in a Clifford algebra, but is, contrary to the classical Fourier transform, not
the unique solution. Here we determine an entire class of solutions of this
system of PDEs, under certain constraints. For each solution, series
expressions in terms of Gegenbauer polynomials and Bessel functions are
obtained. This allows to compute explicitly the eigenvalues of the associated
integral transforms. In the even-dimensional case, this also yields the inverse
transform for each of the solutions. Finally, several properties of the entire
class of solutions are proven.
