If one considers the Laplace (or Helmholtz) equation in two dimensions, then through separation of variables in plane polar coordinates, the azimuthal dependence is seen to be of the form of a harmonic $e^{in\phi}$, where $n$ is an integer. The set of harmonics form a basis for any (reasonable) function on a circle i.e. any function with $2\pi$ periodicity (the Fourier series).
Similarly, if one considers the Laplace (or Helmholtz) equation in three dimensions, then through separation of variables in spherical polar coordinates, the angular dependence is seen to be of the form of a spherical harmonic $Y_{l}^{m}(\theta,\phi)$, where $l$ and $m$ are integers with $|m|<l$. The set of spherical harmonics form a basis for any (reasonable) function on a sphere.
Is there a reason why solutions to Laplace's equation form a basis for functions on manifolds of the space? I suspect there is a theorem for this but I've either never come across it or it has completely slipped my mind.
Thanks in advance for any help.
I would not claim to understand a more general set-up, in a context of "geometric analysis" (and separation of variables by a subtle arrangement of vector fields), but I have thought about the $\mathbb R^n$ case.
Namely, first, it is an odd thing that restrictions of harmonic (annihilated by the Laplacian of $\mathbb R^n$) homogeneous polynomials from the ambient $\mathbb R^n$ to the sphere $S^{n-1}$ give eigenfunctions for the Laplacian on the sphere.
One can readily do a computation showing that homogeneous functions on $\mathbb R^n$, more generally, have an understandable behavior under the Laplacian on the sphere. Choosing a good homogeneity degree by adjustments like $f(x)=|x|^c\cdot F(x)$ for given homogeneous $F$... is then seen to produce Laplacian eigenfunctions on the sphere from homogeneous harmonic functions on the ambient Euclidean space. Right, we already knew this. :)
My own understanding of this (probably special?) case is that it's about the interactions of the dilation group, the rotation group, the Euclidean Laplacian, and the spherical Laplacian.
I suspect it would be hard to generalize this sort of relationship very far, but maybe there are generalizations that do not depend so much on group-theoretic good fortune.