The offset octonion linear canonical transform: Differential properties and uncertainty principle

As information technology continues to advance rapidly, the field of signal processing is confronted with increasingly complex high-dimensional data processing. Traditional linear canonical transforms (LCT) show certain limitations when dealing with high-dimensional data. Therefore, researchers have begun to explore hypercomplex number fields, which can represent rotations in three-dimensional space more naturally and accurately, and have demonstrated significant advantages in signal processing tasks. Transformations based on quaternions and octonions, such as the quaternion Fourier transform (QFT) and the octonion Fourier transform (OFT), have been widely applied in areas such as signal detection, pattern recognition, and time-frequency analysis. Against this backdrop, this paper investigated the quaternion offset linear canonical transform (QOLCT) and its extension in the octonion domain—the offset octonion linear canonical transform (OOCLCT). First, the paper established the modulatory and differentiation properties of the QOLCT, which were missing in existing literature, laying a solid theoretical foundation for subsequent research. Then, based on the relationship between the QOLCT and the OOCLCT, multiple classical uncertainty principles within the framework of the OOCLCT were derived. These principles included the Heisenberg uncertainty principle, the sharp Hausdorff-Young inequality, the Matolcsi-Szucs uncertainty principle, and the Benedicks-Amrein-Berthier uncertainty principle. Through the derivation and verification of these uncertainty principles, this paper not only deepened the understanding of the OOCLCT theory but also provided new mathematical tools and methods for high-dimensional signal processing. Finally, we discussed the future research directions of the OOCLCT, providing a reference for subsequent studies.