Dunkl Transform

Abstract

The Dunkl transform is a generalization of the Fourier transform and is an isometry in ${L}^{2}(\mathbb{R}^{d},{{h}^{2}_{\kappa}})$ with ${{h}_{\kappa}}$ being a reflection invariant weight function. In this chapter we study the Dunkl transform from the point of view of harmonic analysis. In Section 6.1 we show that the Dunkl transform is an isometry in ${L}^{2}$ space with respect to the measure ${{h}^{2}_{\kappa}}(x)dx$ on $\mathbb{R}^{d}$ and it preserves Schwartz class of functions.