ABSTRACT
Several of the transforms featured in this chapter have been studied in more detail in previous chapters. They are briefly characterized here once more in order to make it easier to grasp the larger picture of hypercomplex Fourier transforms and their mutual relationships. At the beginning Clifford Fourier transforms (CFT) are introduced, including the important class of quaternion Fourier transforms (QFT), with some detail, emphasize and relatively new developments. There is an alternative operator exponential Clifford Fourier transform approach. New work in this direction closely relates to the square roots of -1 approach explained. A CFT analyzes scalar, vector and multivector signals in terms of sine and cosine waves with multivector coefficients. Basically, the imaginary unit of complex numbers in the transformation kernel, exp(ip) = cos p +i sin p, is replaced by a multivector square root of -1 in Cl(p,q). This produces a host of CFTs, a still incomplete brief overview is sketched in a tree diagram. Additionally, the square roots of -1 in Cl(p,q) allow to construct further types of integral transformations, notably Clifford wavelets. [177 words]
