Octonion Fourier Transform of real-valued functions of three variables - selected properties and examples

The paper is devoted to properties of the Octonion Fourier Transform (OFT) defined in 2011 by Hahn and Snopek, i.e. symmetry properties, Parseval-Plancherel and Wiener-Khintchine Theorems. This work has been inspired by the Hermitian symmetry of the Complex Fourier Transform and known symmetry relations of the Quaternion Fourier Transform that were defined by Bülow. Similar symmetry relations for the OFT are derived using the notion of octonion involutions. The proof of the corresponding theorem is presented and the result is illustrated with multiple examples. Also the octonion analogues of Parseval and Plancherel Theorems are derived. Those results, along with the shift property of OFT, lead to the proof of the octonion version of Wiener-Khintchine Theorem and the octonion definitions of autocorrelation function and power spectral density of a signal.