Rotations in discrete Clifford analysis.

The Laplace and Dirac operators are rotation invariant operators which can be neatly expressed in (continuous) Euclidean Clifford analysis. In this paper, we consider the discrete counterparts of these operators, i.e. the discrete Laplacian Δ or star-Laplacian and the discrete Dirac operator ∂. We explicitly construct rotations operators for both of these differential operators (denoted by Ω a, b and dR ( e a, b ) respectively) in the discrete Clifford analysis setting. The operators Ω a, b satisfy the defining relations for so ( m , C ) and they are endomorphisms of the space H k of k -homogeneous (discrete) harmonic polynomials, hence expressing H k as a finite-dimensional so ( m , C ) -representation. Furthermore, the space M k of (discrete) k -homogeneous monogenic polynomials can likewise be expressed as so ( m , C ) -representation by means of the operators dR ( e a, b ). We will also consider rotations of discrete harmonic (resp. monogenic) distributions, in particular point-distributions, which will allow us to evaluate functions in a rotated point. To make the discrete rotations more visual, we explicitly calculate the rotation of general point-distributions in two dimensions, showing the behavior of such discrete rotations in relation to the continuous case. [ABSTRACT FROM AUTHOR]