On the Harmonic and Monogenic Decomposition of Polynomials

The decomposition of polynomials in terms of spherical harmonics is widely used in various branches of analysis. In this paper we describe a set of REDUCE procedures generating this decomposition and its more general, monogenic, counterpart in Clifford analysis. We then illustrate their use by inverting the Laplacian and the Dirac operator on both Euclidean and Minkowski spaces. Let ~k denote the space of real-valued homogeneous polynomials of degree k in the real variables (xl, Xz. .. .. x,,) and ~ the subspace of harmonic polynomials of degree k. As is well known (Vilenkin, 1969), a n y f k e N k may be uniquely decomposed as fk = hk + r2hk_ 2 + " " + r2tk/2Jhk-2tk/21, (1) where h j ~ and r 2 stands for Ixl2=x~+ ,,, +x~. This result can be elegantly proved following an idea of Stein & Weiss (1971). With the inner product