TITLE: Klein Bottle discrete harmonics?
QUESTION [6 upvotes]: Studying discrete representations of  a function, on a $2$ Dimensional compact surface, brings to the use of spherical harmonics for the 2-sphere, and discrete Fourier transformations for the 2-torus.
But how does it work with the Klein Bottle?
That is, once a function $f$ on a Klein Bottle given, how discrete coefficients $f_{lm}$ are calculated from $f$, and reciprocally  ($f$ from $f_{lm}$  )?

REPLY [7 votes]: Trimok's answer is essentially correct and in fact can be strengthened: the eigenfunctions on the (flat) Klein bottle are exactly 
$$e^{i(2n+1)x}\sin(my),\ e^{i(2n)x}\cos(my).$$
This is a consequence of the following theorem, which is not so hard to prove using the fact that Riemannian covers are local isometries:

If $\hat{M}\to M$ is a regular cover of closed Riemannian manifolds with deck group $\pi$ acting by isometries on $\hat{M}$, the eigenfunctions of $M$ are precisely the $\pi$-invariant eigenfunctions of $\hat{M}$.

If we are interested in any (complete) metric at all on the Klein bottle, then I am not sure what one can say. There may be a rigidity theorem that implies eigenvalues are extremized on the flat Klein bottle.