The Two-Sided Clifford Dunkl Transform and Miyachi's Theorem
Mohamed Essenhajy, Said Fahlaoui
TL;DR
This work extends Dunkl harmonic analysis to Clifford algebras by introducing the two-sided Clifford Dunkl transform (CDT) built from two square roots of $-1$ in $Cl_{p,q}$. It develops the CDT's foundational properties, including inversion, Plancherel identities, a convolution framework via a generalized translation, and explicit translation formulas, while illustrating reductions to classical Fourier, Dunkl, and quaternionic transforms in special cases. The authors also prove a Miyachi-type uncertainty principle for the CDT, thereby broadening a central Fourier-analytic result to the Clifford-Dunkl setting. Collectively, these results establish a versatile Clifford-valued transform with potential applications in hypercomplex signal processing and Clifford-Dunkl analysis, unifying several known transforms under a common framework.
Abstract
Recent advances have extended the Dunkl transform to the setting of Clifford algebras. In particular, the two-sided quaternionic Dunkl transform has been introduced as a Dunkl analogue of the two-dimensional quaternionic Fourier transform. In this paper, we develop the two-sided Clifford Dunkl transform, defined using two square roots of -1 in Cl_{p,q}. We establish its fundamental properties, including the inversion and Plancherel formulas, and provide two explicit expressions for the associated translation operator. Moreover, we prove an analogue of Miyachi's theorem for this transform, thereby extending a classical result in harmonic analysis to the Clifford-Dunkl framework.