  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.
