Any function $f(\theta,\phi)$ which obeys:
$\int_0^{2\pi} \int_0^{\pi} \sin \theta |f(\theta,\phi)|^2 d\phi d\theta < \infty$
can be written in terms of spherical harmonics $Y_{lm}(\theta,\phi)$ as
$f(\theta,\phi) = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} a_{lm} Y_{lm}(\theta,\phi)$.
Up to an additive constant can $f(\theta,\phi)$ also be expanded in terms of the functions $\partial Y_{lm}(\theta,\phi)/\partial \theta$?