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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 $\partial Y_{lm}(\theta,\phi)/\partial \theta$?

D_J_S
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  • Why would anyone want to do that? This set don't form a basis of an algebra. – user10354138 Jun 17 '19 at 19:28
  • @user10354138. Some of the components of vector spherical harmonics depend upon the derivative. I want to know whether such components can be any scalar function on $L^2$. If the derivatives form a complete set, up to an additive constant, then my question is answered. – D_J_S Jun 17 '19 at 20:24

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