In addition to what Carlo Beenakker stated, you can find a detailed introduction to Fourier Transforms, Spherical Harmonics and Wigner-D matrices  [here at p. 51 ff.][1]. 

It focuses on applications for visual robot navigation (and therefore especially on real-valued implementations), but a brief summary of the general idea is given. For a purely math based focus, you can check the references.

  [1]: https://pub.uni-bielefeld.de/download/2913219/2913220/diss_ddiffert.pdf