We introduce the Cayley–Dickson Fourier transform (CDFT), a novel framework for harmonic analysis of functions valued in the non-associative Cayley–Dickson algebras \( \mathcal {C}_m \) . The central challenge lies in the failure of associativity and alternativity for \( m \geqslant 4 \) , which obstructs classical Fourier analytic methods. To overcome this, we develop a two-stage approach: first, we construct the transform on real-valued Schwartz-type spaces, establishing continuity, inversion, and isometric properties; second, we extend the theory to fully \( \mathcal {C}_m \) -valued functions by leveraging intrinsic algebraic structures, such as slice-wise multiplicativity and weak commutativity. Key innovations include a modified duality between differentiation and multiplication, governed by twisted sign involutions that precisely compensate for non-associative distortions, and a restricted convolution theorem for Gaussian-type functions that exploits the real scalar structure of their transforms. We prove that the CDFT admits an explicit inverse via symmetrization over coordinate reflections, acts isometrically on \( L^2(\mathbb {R}^m, \mathcal {C}_m) \) , and exhibits a period-four symmetry that generalizes classical Fourier periodicity. These results collectively establish the CDFT as a rigorous and structurally faithful extension of Fourier analysis to the full Cayley–Dickson hierarchy.
