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Octonionic dunkl transform: definition, properties, and uncertainty principles
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DOI：10.1007/s13324-026-01207-5.png)
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Abstract
En 中文
The Dunkl transform generalizes the classical Fourier transform in the context of finite reflection groups, introducing a family of operators parameterized by a multiplicity function k >= 0 Hypercomplex extensions of integral transforms, particularly based on quaternion and octonion algebras, have gained prominence for processing multi-dimensional signals. In this paper, we introduce the Octonionic Dunkl Transform (ODT), which unifies the algebraic structure of the octonions-the largest normed division algebra-with the analytic framework of Dunkl operators. We provide its explicit definition, accounting for the non-associative nature of octonion multiplication through a fixed left-nested parenthesization convention. Fundamental properties are established, including linearity, a Bessel function series representation, and a detailed parity decomposition that separates the transform into eight real-valued components. We prove a sharp inversion formula and an isometry (Plancherel theorem) in the associated weighted L-2 space. Furthermore, we establish two uncertainty principles: a Heisenberg-type inequality relating the weighted dispersions of a signal and its transform, and a Donoho-Stark-type concentration principle. These results extend earlier work on the quaternion Dunkl transform and the octonion Fourier transform, providing a new theoretical tool for the analysis of octonion-valued signals in settings with reflection symmetry.
Keywords:
Dunkl transform
Octonion algebra
Hypercomplex signal processing
Uncertainty principle
Non-associative algebra
Harmonic analysis
Journal
A
IF:
1.6
Papers:
76
Citations:
0
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