Finite Reflection Invariant Measure Of The Harmonic Analysis

The purpose of this dissertation is to study the harmonic analysis associated with measures which are invariant under finite reflection groups. This subject began with a series of papers of C. Dunkl, which built up the framework for a theory of special functions and integral transforms in several variables related with finite reflection groups. We select some basic and important problems in harmonic analysis associated with Dunkl's theory, which are(1) The Bochner-Riesz means of the Dunkl transform;(2) The Poisson integral and Riesz Transforms of the Dunkl transform;(3) The Hermite expansions associate with measures invariant under finite reflection groups;(4) Uncertainty principles for Sturm-Liouville operators.