Plancherel and inversion formulas for the Dunkl-type Segal-Bargmann transform

Abstract

In 1961, Bargmann introduced the classical Segal-Bargmann transform and in 1984, Cholewinsky introduced the generalized Segal-Bargmann transform. These two transforms are the aim of many works, and have many applications in mathematics. In this paper we introduce the Dunkl-type Segal-Bargmann transform \(\mathscr {B}_{\alpha }\) associated with the Coxeter group \(\mathbb {Z}^d_2\). Next, we investigate for this transform the main theorems of harmonic analysis (Plancherel and inversion formulas). Finally, we study some local uncertainty principles associated with the transform \(\mathscr {B}_{\alpha }\).