It is well-known that the spherical harmonics form a complete orthogonal basis over the unit sphere $S^2$. But what about the spatial spherical harmonics (with the additional $r^n$ part), do they form a complete basis over ${\rm I\!R}^3$?
If not, I have a follow-up question. Is it then possible to find a solution to the Laplace equation that cannot be represented by a series expansion of the spatial spherical harmonics?