Plancherel theorem and quaternion Fourier transform for square integrable functions

The quaternion Fourier transform (QFT), a generalization of the classical 2D Fourier transform, plays an increasingly active role in particular signal and colour image processing. There tends to be an inordinate degree of interest placed on the properties of QFT. The classical convolution theorem and multiplication formula are only suitable for 2D Fourier transform of complex-valued signal, and do not hold for QFT of quaternion-valued signal. The purpose of this paper is to overcome these problems and establish the Plancherel and inversion theorems of QFT in the square integrable signals space L2. First, we investigate the behaviours of QFT in the integrable signals space L1. Next, we deduce the energy preservation property which extends functions from L1 to L2 space. Moreover, some other important properties such as modified multiplication formula are also analyzed for QFT.