On 3D orthogonal prolate spheroidal monogenics

S. G. Georgiev, Complete orthogonal systems of monogenic polynomials over 3D prolate spheroids have recently experienced an upsurge of interest because of their many remarkable properties. These generalized polynomials and their applications to the theory of quasi‐conformal mappings and approximation theory have played a major role in this development. In particular, the underlying functions of three real variables take on values in the reduced quaternions (identified with R3) and are generally assumed to be null‐solutions of the well‐known Riesz system in R3. The present paper introduces and explores a new complete orthogonal system of monogenic functions as solutions to this system for the space exterior of a 3D prolate spheroid. This will be made in the linear spaces of square integrable functions over R. The representations of these functions are explicitly given. Some important properties of the system are briefly discussed, from which several recurrence formulae for fast computer implementations can be derived. Copyright © 2015 John Wiley & Sons, Ltd.