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The paper is devoted to the development of the octonion Fourier transform(OFT) theory initiated in 2011 in articles by Hahn and Snopek. It is also acontinuation and generalization of earlier work by Blaszczyk and Snopek, wherethey proved few essential properties of the OFT of real-valued functions, e.g.symmetry properties. The results of this article focus on proving that the OFTis well-defined for octonion-valued functions and almost all well-knownproperties of classical (complex) Fourier transform (e.g. argument scaling,modulation and shift theorems) have their direct equivalents in octonion setup.Those theorems, illustrated with some examples, lead to the generalization ofanother result presented in earlier work, i.e. Parseval and PlancherelTheorems, important from the signal and system processing point of view.Moreover, results presented in this paper associate the OFT with 3-D LTIsystems of linear PDEs with constant coefficients. Properties of the OFT incontext of signal-domain operations such as derivation and convolution of$\mathbb{R}$-valued functions will be stated. There are known results for QFT,but they use the notion of other hypercomplex algebra, i.e. double-complexnumbers. Considerations presented here require defining other higher-orderhypercomplex structure, i.e. quadruple-complex numbers. This hypercomplexgeneralization of the Fourier transformation provides an excellent tool for theanalysis of 3-D LTI systems.

A Generalization of the Octonion Fourier Transform to 3-D  Octonion-Valued Signals -- Properties and Possible Applications to 3-D LTI  Partial Differential Systems