I asked this question a couple of days ago on math.stackexchange, but have yet to receive a response, so I have decided to post this here.

This question is also vaguely related (both questions arose from the same thing I was working on) to this question I just asked last night.

The question is simple: on a general manifold $M$, can one generalize the space of Schwartz functions on $\mathbb{R}^n$ to a space of smooth functions $\mathcal{S}(M)$ on $M$ that obeys similar properties? I would like to be able to define the convolution of two Schwartz functions, so I guess I better require that $M$ at least be a (unimodular) Lie group. Is it then possible to define $\mathcal{S}(M)$? What about the Fourier transform? Is there a natural definition of the Fourier Transform on $\mathcal{S}(M)$?