Explicit Formulae for Monogenic Projections

Spherical monogenics were studied from the very beginning of Clifford analysis (see [2]) and a lot of their properties are well understood. In particular, the monogenic projection pi(M) (i.e., the projection from the space of homogeneous polynomials of order k to the space of spherical monogenics of order k) plays a key role in many different investigations. There is a standard integral formula for the projection (see e.g. [4]). Recently, an explicit,differential formula was given ([7, 3]) for the projection, analogous to the classical formula for spherical harmonics. The aim of the article is to describe some other differential formulae useful in further applications.Spherical monogenics were studied from the very beginning of Clifford analysis (see [2]) and a lot of their properties are well understood. In particular, the monogenic projection pi(M) (i.e., the projection from the space of homogeneous polynomials of order k to the space of spherical monogenics of order k) plays a key role in many different investigations. There is a standard integral formula for the projection (see e.g. [4]). Recently, an explicit,differential formula was given ([7, 3]) for the projection, analogous to the classical formula for spherical harmonics. The aim of the article is to describe some other differential formulae useful in further applications.P