Discrete Weierstrass Transform: Generalisations

The classical Weierstrass transform is an isometric operator mapping elements of the weighted L 2 - space L 2 ( R , exp ( - x 2 / 2 ) ) to the Fock space. We defined an analogue version of this transform in discrete Hermitian Clifford analysis in one dimension, where functions are defined on a discrete line rather than the continuous space. This transform is based on the classical definition, in combination with a discrete version of the Gaussian function and discrete counterparts of the classical Hermite polynomials. The aim of this paper is to extend the definition to higher dimensions, where we must take into account the anticommutativity of the basic Clifford elements and use the generalised discrete Hermite polynomials. Furthermore, we investigate, back in dimension 1, the asymptotic behaviour if the mesh width approaches 0.