Sonine Transform Associated to the Dunkl Kernel on the Real Line

TL;DR

This paper introduces and analyzes the Dunkl Sonine operator and its dual on the real line, establishing inversion formulas and a Plancherel formula for these operators within the Dunkl analysis framework.

Contribution

It defines the Dunkl Sonine operator and its dual, introduces complex powers of the Dunkl Laplacian, and derives inversion and Plancherel formulas for these operators.

Findings

Inversion formulas for Dunkl Sonine operators and their duals.

A Plancherel formula for the dual Dunkl Sonine operator.

Extension of Dunkl analysis with complex powers of the Dunkl Laplacian.

Abstract

We consider the Dunkl intertwining operator $V_\alpha$ and its dual ${}^tV_\alpha$, we define and study the Dunkl Sonine operator and its dual on $\mathbb{R}$. Next, we introduce complex powers of the Dunkl Laplacian $\Delta_\alpha$ and establish inversion formulas for the Dunkl Sonine operator $S_{\alpha,\beta}$ and its dual ${}^tS_{\alpha,\beta}$. Also, we give a Plancherel formula for the operator ${}^tS_{\alpha,\beta}$.