[Paper Review] $r$-extension of Dunkl operator in one variable and Bessel functions of vector index

Ahmed Fitouhi, Lazhar Dhaouadi+1 more|arXiv (Cornell University)|Sep 24, 2012

Mathematical Analysis and Transform Methods8 references4 citations

TL;DR

This paper introduces an $r$-extension of the Dunkl operator in one variable using the cyclic group $C_r$ and $r$th roots of unity, constructing a differential-difference operator $D_{ u}$ whose $r$th power equals a generalized Bessel-type differential operator $\Delta_{\nu}$. The key contribution is that $D_{\nu}$ has eigenfunctions given by Bessel functions of vector index $\mu = (\alpha_0, \dots, \alpha_{r-1})$, generalizing the classical Dunkl operator when $r=2$. This framework enables a new harmonic analysis based on $r$-symmetric function spaces and transmutation operators.

ABSTRACT

In this work we present an operator $D_μ$ constructed with the help of the cyclic group set of the $r^{\small th}$ roots of unity. This operator constitute an $r$-extension of the Dunkl operator in one variable because when $r=2$ it reduces to the classical one and admits as eigenfunctions the Bessel functions of vector index early deeply studied by Klyuchantsev. This paper is argued by specific examples and contains some interesting results which are the prelude of harmonic analysis related to this operator.

Motivation & Objective

• To generalize the classical Dunkl operator in one variable to an $r$-order extension using the cyclic group $C_r$ and $r$th roots of unity.
• To define a differential-difference operator $D_{\mu}$ whose $r$th power yields a generalized Bessel-type operator $\Delta_{\mu}$, extending classical harmonic analysis.
• To establish the eigenfunction property of Bessel functions with vector index $\mu$ under $D_{\mu}$, generalizing the $r=2$ case.
• To develop foundational tools such as $r$-even and $r$-odd function decomposition, projectors $T_k$, and the $r$-extension of the Fourier transform and transmutation operators.

Proposed method

• Define the cyclic group $C_r = \{1, \omega, \omega^2, \dots, \omega^{r-1}\}$ with $\omega = e^{2\pi i / r}$, and use it to construct $r$-symmetric function spaces $F_k$.
• Introduce projection operators $T_k = \frac{1}{r} \sum_{n=0}^{r-1} s_k^n$, where $s_k g(x) = \omega^k g(\omega x)$, to decompose function spaces into invariant subspaces.
• Construct the $r$-extension of the Dunkl operator as $D_{\mu} = \frac{d}{dx} + \frac{1}{x} \sum_{k=0}^{r-1} a_k T_k$, with coefficients $a_k = r\alpha_k + k$.
• Define the generalized Bessel operator $\Delta_{\mu} = L_{a_{r-1}} \circ \cdots \circ L_{a_0}$, where $L_a(f) = x^{-a} \frac{d}{dx}(x^a f)$, and prove $D_{\mu}^r = \Delta_{\mu}$ on $F_k$.
• Establish the $r$-extension of the Fourier transform $\mathcal{F}_{\mu}$ and show its intertwining property with $D_{\mu}$, satisfying $\mathcal{F}_{\mu} D_{\mu} g(\lambda) = -\theta \lambda \mathcal{F}_{\mu} g(\lambda)$ under self-adjointness.
• Demonstrate that the operator $D_{\mu}$ generalizes the one-dimensional Dunkl-Opdam operator $T(\kappa)$ by showing equivalence via a linear system in Fourier coefficients, but is strictly more general due to over-determination in the inverse problem.

Experimental results

Research questions

• RQ1 How can the classical Dunkl operator in one variable be generalized to an $r$-order differential-difference operator using the cyclic group $C_r$?
• RQ2 What is the relationship between the $r$-th power of the extended operator $D_{\mu}$ and the generalized Bessel-type operator $\Delta_{\mu}$?
• RQ3 Are Bessel functions of vector index $\mu = (\alpha_0, \dots, \alpha_{r-1})$ eigenfunctions of $D_{\mu}$, and how do they generalize the classical Bessel functions?
• RQ4 Can a transmutation operator be constructed to relate $D_{\mu}$ to the standard derivative, and what is its role in the $r$-extension of the Fourier transform?
• RQ5 How does the proposed $D_{\mu}$ relate to the one-dimensional specialization of the Dunkl-Opdam operators, and in what sense is it a proper generalization?

Key findings

• The $r$-extension of the Dunkl operator $D_{\mu}$ satisfies $D_{\mu}^r = \Delta_{\mu}$ on each invariant subspace $F_k$, generalizing the classical case when $r=2$.
• The Bessel functions of vector index $\mu$ are eigenfunctions of $D_{\mu}$, with the eigenvalue equation $D_{\mu} f = \lambda f$ holding in the $F_k$ decomposition.
• The $r$-extension of the Fourier transform $\mathcal{F}_{\mu}$ is defined via a transmutation operator and satisfies $\mathcal{F}_{\mu} D_{\mu} g(\lambda) = -\theta \lambda \mathcal{F}_{\mu} g(\lambda)$ when $D_{\mu}^* = -D_{\mu}$, establishing a spectral link.
• The operator $D_{\mu}$ generalizes the one-dimensional Dunkl-Opdam operator $T(\kappa)$, but is strictly more general: while $T(\kappa)$ can be embedded into $D_{\mu}$, the reverse requires solving an over-determined system with no solution in general.
• The decomposition of functions into $r$-even and $r$-odd parts via projectors $T_k$ enables the construction of $D_{\mu}$ as a sum of derivative and weighted shift operators, with $D_{\mu}$ mapping $F_k$ to $F_{k+1}$.
• The Riemann-Liouville transform and associated integral representations are used to express the eigenfunctions and support the transmutation framework, providing a foundation for future harmonic analysis.