ABSTRACT
In applied mathematics the Fourier transform has developed into an important tool. It is a powerful method for solving partial differential equations. The Fourier transform provides also a technique for signal and image analysis where the signal from the original domain is transformed to the spectral or frequency domain. In the frequency domain many characteristics of the signal are revealed. But how to extend the Fourier transform to Clifford's geometric algebras? This central chapter begins with an introduction to Clifford Fourier transforms (CFT) in Cl(3,0), the geometric algebra of three-dimensional Euclidean space, one of the most studied and applied versions of the CFT. Next is the generalization to one-sided CFTs in n-dimensional Euclidean space Clifford algebras Cl(n,0) and quadratic space Clifford algebras Cl(p,q). After that an alternative operator exponential approach to one-sided CFTs is presented. Then attention is given to two-sided CFTs in Clifford algebras Cl(p,q), which makes sense due to the non-commutativity of Clifford algebras. Distinctions are made based on the number of square roots of -1 (and therefore kernel factors) involved. Of further interest are the uncertainty and convolution properties of CFTs. Finally, special CFTs (windowed, Fourier Mellin, and in conformal geometric algebra) are investigated. [197 words]
