When are spherical harmonic expansions valid?

Question

It is known that a square integrable function on the sphere can be expanded in a basis of spherical harmonics, $$ f(\theta,\phi) = \sum_{l=0}^{\infty} \sum_{m=-l}^l c_l^m Y_l^m(\theta,\phi) $$ where $\theta,\phi$ are the polar and azimuthal angles defined with respect to a Cartesian coordinate system. Suppose now that I define two new angular coordinates, $$ \alpha = \alpha(r,\theta,\phi) $$ and $$ \beta = \beta(r,\theta,\phi) $$ where $\alpha,\beta$ chart the sphere, however they may be a nonlinear transformation of the usual Cartesian coordinates. Is it then true that I can expand any square integrable function in terms of spherical harmonics evaluated at the new angular coordinates, $$ g(\alpha,\beta) = \sum_{l=0}^{\infty} \sum_{m=-l}^l d_l^m Y_l^m(\alpha,\beta) $$ ? In the case that $\alpha,\beta$ are a rigid rotation of the usual coordinate system, I know this is allowed, since spherical harmonics are well studied under rotations. But what about the general case when $\alpha,\beta$ are a nonlinear transformation of the usual Cartesian coordinates?