Octonionic wavelet transform and uncertainly principle

Abstract

This article centers around the octonion wavelet transform, exploring its transformation function ${\psi }^{a,b,S}\left(x\right)$ derived from the admissible octonionic mother wavelet ψ, incorporating translation, rotation, and dilation components. We establish the inverse transform and the Plancherel formula, unveiling the inner product relationship of transformed functions. The Uncertainty Principle for the octonion wavelet transform reveals inherent bounds in wavelet analysis within the octonionic framework. However, it is essential to note that these discoveries are specific to the alternative properties of octonions and cannot be extended to general Cayley-Dickson algebras, where the sedenion wavelet transform lacks the isometry property observed in the octonionic setting.