An orthogonal set of weighted quaternionic Zernike spherical functions

Abstract

In this work, we give a brief description of the theory and properties of the three-dimensional quaternionic Zernike spherical polynomials (QZSPs). A refinement of the QZSPs to functions vanishing over the unit sphere leads to the computation of the weighted quaternionic Zernike spherical functions (WQZSFs). In particular, the underlying functions are of three real variables and take on values in the quaternions (identified with ℝ4). Also, in this work, we prove that the WQZSFs are orthonormal in the unit ball with respect to a suitable weight function. The representation of these functions are given explicitly, and a summary of their fundamental properties is also discussed. To the best of our knowledge, this does not appear to have been done in literature before. © 2014 Springer International Publishing.