The uncertainty principle for the octonion Fourier transform. (26th August 2022)

• Record Type: Journal Article
• Authors: Zayed, Mohra; El Haoui, Youssef
• Is Part Of: Mathematical methods in the applied sciences. Volume 46:Number 2(2023)
• Page Start: 2651
• Page End: 2666
• Publication Date: 2022-08-26
• Subjects: clifford algebra -- fourier transform -- octonion algebra -- uncertainty principle; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.8667
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : The octonion Fourier transform (OFT) is a hypercomplex Fourier transform that generalizes the quaternion Fourier transform. However, in octonion algebra, there are two major obstacles that are presented in the loss of associativity and commutativity. Researchers have been trying to extend the results of the Euclidean Fourier transform to quaternion‐valued signals using special techniques to overcome these two problems. In this context, we intend to generalize the Heisenberg uncertainty principles associated with covariance and Hardy's uncertainty principle for octonion multivector valued signals over ℝ 3 $$ {\mathbb{R}}^3 $$ using the polar form of an octonion, the quaternion decomposition, and the relationship between the OFT and the three‐dimensional (3D)‐Clifford‐Fourier transform.