Dunkl wavelets and applications to inversion of the Dunkl intertwining operator and its dual (Q1774659)

The author considers the Dunkl operator \(\Lambda_\alpha\) on \(\mathbb{R}\) of index \((\alpha+{1\over 2})\), \(\alpha>-{1\over 2}\), associated with the reflection group \(\mathbb{Z}_2\) given by \[ \Lambda_\alpha u(x)={d\over dx} u(x)+ {\alpha+{1\over 2}\over x} [u(x)+ u(- x)]. \] He studies the Dunkl wavelet and the corresponding Dunkl wavelet transforms, and he proves for these transforms Plancherel and reconstruction formulas. As applications of these results he deduces the relations which give the inverse operators of the Dunkl intertwining operator and of its dual.