The purpose of this article is to further develop the theory of octonion Fourier transformations (OFT), but from a different perspective than before. It follows the earlier work by Błaszczyk and Snopek, where they proved a few essential properties of the OFT of real-valued functions of three continuous variables. The research described in this article applies to discrete transformations, i.e. discrete-space octonion Fourier transform (DSOFT) and discrete octonion Fourier transform (DOFT). The described results combine the theory of Fourier transform with the analysis of solutions for difference equations, using for this purpose previous research on algebra of quadruple-complex numbers. This hypercomplex generalization of the discrete Fourier transformation provides an excellent tool for the analysis of 3-D discrete linear time-invariant (LTI) systems and 3-D discrete data.

本文的目的是进一步发展八度傅立叶变换（OFT）的理论，但与以往不同。它遵循Błaszczyk和Snopek的早期工作，他们证明了三个连续变量的实值函数的OFT的一些基本属性。本文描述的研究适用于离散变换，即离散空间八重傅里叶变换（DSOFT）和离散八重傅里叶变换（DOFT）。所描述的结果将傅里叶变换的理论与差分方程解的分析相结合，为此目的使用了先前关于四复数代数的研究。