Inversion of the Dunkl intertwining operator and its dual using Dunkl wavelets.
In \S1 of the paper, the author gives results on the Dunkl operators and on eigenfunction called the Dunkl kernel and then he defines the Dunkl intertwining operator \(V_k\) and its dual \({}^tV_k\) and mentions their main properties. In Sections 3 and 4, the harmonic analysis associated with Dunkl operators (Dunkl transform, Dunkl translation operators and Dunkl convolution product) are studied, while in the fifth section the author gives spaces other than \({\mathcal E}(\mathbb{R}^d)\) and \({\mathcal D}(\mathbb{R}^d)\) on which the Dunkl intertwining operator \(V_k\) and its dual \({}^tV_k\) are bijective and establishes inversion formulas for these operators. Next he defines and studies Dunkl wavelets and Dunkl continuous wavelet transform and then Plancherel and inversion formulas are given for this transform. Finally using Dunkl wavelets, formulas which give the inverse operators of the operators \(V_k\) and \({}^tV_k\) are obtained.
scientific article; zbMATH DE number 2012121
scientific article