Sonine Transform Associated to the Dunkl Kernel on the Real Line

Abstract

Abstract. We consider the Dunkl intertwining operator V α and its dual t V α , we define and study the Dunkl Sonine operator and its dual on R. Next, we introduce complex powers of the Dunkl Laplacian ∆ α and establish inversion formulas for the Dunkl Sonine operator S α,β and its dual t S α,β . Also, we give a Plancherel formula for the operator t S α,β .

Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets

“…Throughout this section assume β > α > −1/2. Definition 3.1 (see [14]). The dual Dunkl-Sonine integral transform t X α,β is defined for smooth functions on R by…”

“…Definition 3.6 (see [14]). The Dunkl-Sonine integral transform X α,β is defined for a locally bounded function f on R by…”

“…Besides its mathematical interest, the Dunkl operator Λ α has quantum-mechanical applications; it is naturally involved in the study of onedimensional harmonic oscillators governed by Wigner's commutation rules [6,11,16]. It is known, see for example [14,15], that the Dunkl kernels on R possess the following Sonine type integral representation e β (λx) = |x| −|x| K α,β (x, y) e α (λy) |y| 2α+1 dy, λ ∈ C, with β > α > −1/2, and a α,β := Γ(β + 1) Γ(α + 1) Γ(β − α) .…”

“…Soltani has showed in [14] that the dual Dunkl-Sonine integral transform t X α,β is a transmutation operator between Λ α and Λ β on the Schwartz space S(R), i.e., it is an automorphism of S(R) satisfying the intertwining relation…”

“…The same author [14] has obtained inversion formulas for the transform t X α,β involving pseudodifferential-difference operators and only valid on a restricted subspace of S(R). The purpose of this paper is to investigate the use of Dunkl wavelets (see [5]) to derive a new inversion of the dual Dunkl-Sonine transform on some Lebesgue spaces.…”