Is there a version of the sampling theorem for mesh geometry? For instance, in a suitably defined Fourier transform, can we guarantee information-preserving downsampling?
@aloo A function (e.g., textures) defined over the surface of a sphere can be represented as a weighted sum of orthogonal functions known as spherical harmonics. The spherical harmonics are a special case of the standard Fourier series, and an analysis on samples and aliasing can be done here as well.
More generally, as mentioned here, there is a way to generalize the Fourier series to other surfaces (i.e., compact Riemannian manifolds) as well. The basis used in Euclidean space (Fourier transform) and over the sphere (spherical harmonics) are just a couple of examples.
As for your specific question on "mesh geometry", you might be interested in taking a look at this paper titled Spectral Geometry Processing with Manifold Harmonics which "presents an explicit method to compute a generalization of the Fourier Transform on a mesh".