# Spherical Harmonics: band-limited representations of a vector field on a sphere

I have used pyShtools in the past to expand scalar functions to spherical harmonics and to synthesize band-limited representations of them. However, I am not too sure how to achieve this for a vector field on the surface of a sphere, where $u(\theta, \phi)$ and $v(\theta, \phi)$ are the co-latitudinal and longitudinal components, respectively.

I have also used spherepack in the past for decomposing vector fields into poloidal/toroidal components; but unfortunately I couldn't find what I am looking for.

I would very much appreciate any help in pointing me in the right direction.

• It's not clear to me why spherepack does not do what you're looking for. Analysis, filter, then synthesis? right? Aug 12 '16 at 19:43
• Hi Spencer, thanks a lot for your reply. I can't find a synthesis function that allows me to set the maximum order lmax. Will it be a correct workaround if I zeroed all coefficients for orders > lmax before the synthesis step? Please excuse my ignorance; will this approach lead to ringing artefacts? Many thanks. Aug 13 '16 at 0:31
• That approach would likely lead to ringing artifacts, yes, so some care should be taken. You can use special filters for this, instead of an abrupt cut off as you suggest (Gaussian filters are typically used, I think). What approach you take is application-dependent, though there is some discussion of this here: ppsloan.org/publications/StupidSH36.pdf Aug 13 '16 at 1:26