Plancherel theorem and quaternion Fourier transform for square integrable functions

TL;DR

This paper extends the quaternion Fourier transform's theoretical framework by establishing Plancherel and inversion theorems for square integrable functions, addressing limitations of classical properties for quaternion signals.

Contribution

It introduces the Plancherel and inversion theorems for QFT in L2 space, overcoming previous limitations of convolution and multiplication properties for quaternion-valued signals.

Findings

Established Plancherel theorem for QFT in L2 space

Proved energy preservation property for QFT

Analyzed modified multiplication formula for quaternion signals

Abstract

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.