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

TL;DR

This paper develops a Calderón reproducing formula for the Dunkl wavelet transform on the real line and uses it to establish new inversion formulas for the dual Dunkl-Sonine integral transform, advancing harmonic analysis in Dunkl settings.

Contribution

It introduces a Calderón reproducing formula for Dunkl wavelets and derives novel inversion formulas for the dual Dunkl-Sonine transform, expanding analytical tools in Dunkl analysis.

Findings

Established a Calderón reproducing formula for Dunkl wavelet transform.

Derived new inversion formulas for the dual Dunkl-Sonine integral transform.

Enhanced methods for harmonic analysis involving Dunkl operators.

Abstract

We prove a Calder\'on reproducing formula for the Dunkl continuous wavelet transform on $\mathbb{R}$. We apply this result to derive new inversion formulas for the dual Dunkl-Sonine integral transform.