Octonionic Hyperbolic Fourier Transform

The three-dimensional octonionic hyperbolic Fourier transform (OHFT) is introduced as a hypercomplex integral Fourier transform associated with three-dimensional octonion-valued signals defined within an open rectangular prism $(-t_1,t_1)\times (-t_2,t_2)\times (-t_3,t_3)$, where $t_1,t_2,t_3\in {ℝ}^{+}$, in Euclidean space ${ℝ}^3$, equipped with a hyperbolic measure. This new transform reduces to the octonion Fourier transform when $t_1,t_2,$ and $t_3$ tend to $+\infty$. The purpose of this paper is to define the OHFT and establish its fundamental properties, including linearity, hyperbolic modulation, scaling, reflection, the inversion formula, Plancherel's theorem, Parseval's formula, the Hausdorff–Young inequality, and partial derivatives for three-dimensional hyperbolic octonion functions. I also demonstrate the application of the OHFT in solving two linear partial differential equations within the octonionic hyperbolic framework and extend the Heisenberg–Weyl uncertainty principle to the OHFT domain.