ON CONVERGENCE PROPERTIES OF 3D SPHEROIDAL MONOGENICS

Abstract

Morais has recently introduced certain complete orthogonal sets of monogenic polynomials over 3D prolate spheroids with remarkable properties. The underlying functions take on either values in the reduced and full quaternions (identified, respectively, with ℝ3and ℝ4), and are generally assumed to be nullsolutions of the well known Riesz and Moisil–Théodoresco systems in ℝ3. In continuation of these studies, we recall some fundamental properties of the polynomials, and prove some recursive formulae between them. As a consequence, we obtain a two-term type recurrence relation satisfied by those basis polynomials. These results are then employed to investigate a rather wide class of approximation properties for monogenic functions over 3D prolate spheroids in terms of spheroidal monogenics.