On Invariance, Equivariance, Correlation and Convolution of Spherical
  Harmonic Representations for Scalar and Vectorial Data
Janis Keuper
The mathematical representations of data in the Spherical Harmonic (SH)
domain has recently regained increasing interest in the machine learning
community. This technical report gives an in-depth introduction to the
theoretical foundation and practical implementation of SH representations,
summarizing works on rotation invariant and equivariant features, as well as
convolutions and exact correlations of signals on spheres. In extension, these
methods are then generalized from scalar SH representations to Vectorial
Harmonics (VH), providing the same capabilities for 3d vector fields on spheres