# Slice monogenic functions: algebraic approach and associated Clifford-Fourier transform

Presenter: **Lander Cnudde**, Ghent University, Belgium

Joint work with **Hendrik De Bie, Guangbin Ren**

In recent years, the study of slice monogenic functions has attracted more and more attention in the literature. In this talk, an extension of the well-known Dirac operator is defined which allows to establish the Lie superalgebra structure behind the theory of slice monogenic functions. According to this slice Dirac operator, an inner product is defined on a corresponding function space. Based on the polynomial null-solutions of this differential operator, an analogue to the classical Hermite polynomials is constructed. Together with the inner product, these polynomials allow for the construction of orthogonal and normalised Clifford-Hermite functions. Being analogues of the classical Hermite functions, they can be seen as eigenfunctions of a Clifford-Fourier transform. Corresponding to the Mehler formula in classical Fourier transform theory, a Mehler kernel is found for this Clifford-Fourier transform and its differential properties are investigated. The similarities between these properties and the differential properties of the classical Fourier kernel evoke a closer study of the solutions of the corresponding partial differential equation.