The Clifford Deformation of the Hermite Semigroup

Abstract

This paper is a continuation of the paper [H. De~Bie et al., Dunkl operators and a family of realizations of $\mathfrak{osp}(1|2)$, arXiv:0911.4725], investigating a natural radial deformation of the Fourier transform in the setting of Clifford analysis.

At the same time, it gives extensions of many results obtained in [S. Ben Said et al., Laguerre semigroup and Dunkl operators, arXiv:0907.3749].

We establish the analogues of Bochner's formula and the Heisenberg uncertainty relation in the framework of the (holomorphic) Hermite semigroup, and also give a detailed analytic treatment of the series expansion of the associated integral transform.

Keywords

• Dunkl operators
• Clif ford analysis
• generalized Fourier transform
• Laguerre polynomials
• Kelvin transform
• holomorphic semigroup