The Clifford Deformation of the Hermite Semigroup

TL;DR

This paper explores a radial deformation of the Fourier transform within Clifford analysis, extending previous results and establishing analogues of classical formulas like Bochner's and the Heisenberg uncertainty relation for the Hermite semigroup.

Contribution

It introduces a new radial deformation of the Fourier transform in Clifford analysis and extends key classical results to this framework.

Findings

Established analogues of Bochner's formula and the Heisenberg uncertainty relation.

Provided a detailed analysis of the series expansion of the associated integral transform.

Extended previous results in Clifford analysis related to the Hermite semigroup.

Abstract

This paper is a continuation of the paper [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 [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.