  Hypercomplex Fourier transforms are increasingly used in signal processing
for the analysis of higher-dimensional signals such as color images. A main
stumbling block for further applications, in particular concerning filter
design in the Fourier domain, is the lack of a proper convolution theorem. The
present paper develops and studies two conceptually new ways to define
convolution products for such transforms. As a by-product, convolution theorems
are obtained that will enable the development and fast implementation of new
filters for quaternionic signals and systems, as well as for their higher
dimensional counterparts.
