The spherical harmonics form a complete set of the Hilbert space of square integrable functions on the sphere. However, looking at them, I can't see how they could ever be summed to equal a function which has a complex value for $\varphi =0$, because looking at the definition of the spherical harmonics, they're always real when $\varphi =0$:
$Y_\ell^m (\theta, \varphi ) = N \, e^{i m \varphi } \, P_\ell^m (\cos{\theta} )$
(where N is a real normalization factor and $P_\ell^m()$ are the also real-valued associated Legendre Polynomials)
Surely there are square integrable functions on the sphere that have a complex component at some point of the 0° meridian?
I'm sure I'm making some stupid mistake - so I would like to apologize for the question beforehand