The Fractional Two-Sided Quaternionic Dunkl Transform and Heisenberg-Type Inequalities

TL;DR

This work introduces the two-dimensional fractional two-sided quaternionic Dunkl transform (FrQDT), unifying fractional Fourier analysis, Dunkl operators, and quaternionic signals. It develops the operator framework with explicit kernels $E^{\textbf{a}}_{\chi_1,\theta_1}$ and $E^{\textbf{b}}_{\chi_2,\theta_2}$, establishes inversion and Plancherel results, and reveals a tensor-product spectral structure via generalized Hermite functions with eigenvalues $e^{\textbf{a} n \theta_1} e^{\textbf{b} m \theta_2}$. The main result is a sharp Heisenberg-type uncertainty principle for the FrQDT, with equality characterized by quaternionic Gaussians, alongside a higher-order moment inequality for any $p \ge 1$ and a corresponding uncertainty principle for the FrQFT. This framework enriches the harmonic-analysis toolkit in settings with quaternionic and Dunkl symmetries and suggests impactful applications in quaternionic signal processing and spectral theory.

Abstract

This report investigates the main definitions and fundamental properties of the fractional two-sided quaternionic Dunkl transform in two dimensions. We present key results concerning its structure and emphasize its connections to classical harmonic analysis. Special attention is given to inversion, boundedness, spectral behavior, and explicit formulas for structured functions such as radial or harmonic functions. Within this framework, we establish a generalized form of the classical Heisenberg-type uncertainty principle. Building on this foundation, we further extend the result by proving a higher-order Heisenberg-type inequality valid for arbitrary moments $p \geq 1$, with sharp constants characterized through generalized Hermite functions. Finally, by analyzing the interplay between the two-sided fractional quaternionic Dunkl transform and the two-sided fractional quaternionic Fourier transform, we derive a corresponding uncertainty principle for the latter.

Theorems & Definitions (32)

• Proposition 2.1
• Definition 2.2
• Remark 2.3
• Proposition 2.4
• Theorem 2.5
• Theorem 2.6: Bochner-type identity
• Definition 3.1
• Remark 3.2: Connections to Classical and Fractional Quaternionic Transforms
• Remark 3.3: Tensor factorization
• Proposition 3.4: Boundedness