Fourier Transforms in Clifford Analysis

Abstract

This chapter gives an overview of the theory of hypercomplex Fourier transforms, which are generalized Fourier transforms in the context of Clifford analysis. The emphasis lies on three different approaches that are currently receiving a lot of attention: the eigenfunction approach, the generalized roots of −1 approach, and the characters of the spin group approach. The eigenfunction approach prescribes complex eigenvalues to the L ₂ basis consisting of the Clifford–Hermite functions and is therefore strongly connected to the representation theory of the Lie superalgebra $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\vert 2)$. The roots of −1 approach consists of replacing all occurrences of the imaginary unit in the classical Fourier transform by roots of −1 belonging to a suitable Clifford algebra. The resulting transforms are often used in engineering. The third approach uses characters to generalize the classical Fourier transform to the setting of the group S pi n (4), resp. S pi n (6) for application in image processing. For each approach, precise definitions of the transforms under consideration are given, important special cases are highlighted, and a summary of the most important results is given. Also directions for further research are indicated.