Octonion Beurling theorem: Uncertainty principles for Hermite functions on ℝ3

Abstract

We generalize Beurling's theorem to the octonion Fourier transform for octonion‐valued functions. Despite challenges such as the failure of the octonion Fourier transform to change differentiation into multiplication and the absence of the octonionic Plancherel's theorem, we establish the octonionic Beurling theorem for Hermite functions on by introducing an integral condition involving the octonion Fourier transform. This extension of uncertainty principles by Hardy, Gelfand–Shilov, and Cowling–Price to the octonionic setting demonstrates that the function and its Fourier transform cannot have arbitrary Gaussian decay simultaneously.