Octonion Beurling theorem: Uncertainty principles for Hermite functions on ℝ3

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...

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Bibliographic Details
Published inMathematical methods in the applied sciences Vol. 47; no. 6; pp. 4199 - 4235
Main Authors Fan, Shihao, Ren, Guangbin
Format Journal Article
LanguageEnglish
Published Freiburg Wiley Subscription Services, Inc 01.04.2024
Subjects
Beurling theorem
Fourier transforms
octonion
Principles
Theorems
Uncertainty principles
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Summary: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 ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$ 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.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.9811