Operator Exponentials for the Clifford Fourier Transform on Multivector Fields in Detail

Abstract

In this paper we study Clifford Fourier transforms (CFT) of multivector functions taking values in Clifford’s geometric algebra, hereby using techniques coming from Clifford analysis (the multivariate function theory for the Dirac operator). In these CFTs on multivector signals, the complex unit \({i \in \mathbb{C}}\) is replaced by a multivector square root of −1, which may be a pseudoscalar in the simplest case. For these integral transforms we derive an operator representation expressed as the Hamilton operator of a harmonic oscillator.