Can one construct for all/any compact Surfaces in $\mathbb{R}^3$ a Schwartz space like object in a way so that the space can be given a structure analogous to the convolution algebra structure it is typically equipped in vector spaces (over $\mathbb{R}$ and $\mathbb{C}$)?
Does anything analogous to the convolution theorem hold in such a setting?
The Schwartz space itself, $\mathscr S(\mathbb R^n)$, can be viewed as a subspace of smooth functions defined on the sphere $\mathbb S^n$, the unit sphere of $\mathbb R^{n+1}.$ We recall that $$ \mathscr S(\mathbb R^n)=\{\phi\in C^\infty(\mathbb R^n),\forall \alpha, \beta\in \mathbb N^n, x^\alpha\partial_x^\beta \phi\in L^\infty(\mathbb R^n)\}. $$ It is easy to see that $\mathscr S(\mathbb R^n)$ can be identified with the space $\Sigma_N(\mathbb S^n)$ of smooth functions on the sphere $\mathbb S^n$ which are flat (i.e. vanish as well as all their derivatives) at the North pole: taking such a function $\Phi$, with $\pi$ standing for the stereographic projection ($\pi:\mathbb S^n\backslash\{north pole\}\rightarrow \mathbb R^n$, we define for $x\in \mathbb R^n$ $$ \phi(x)=\Phi(\pi^{-1}(x)). $$ The convolution of functions on $\mathbb R^n$ can be obviously transferred to the space $\Sigma_N(\mathbb S^n)$.
As a footnote to the previous comments, it is interesting to know that the notation $\mathscr S(\mathbb R^n)$ was introduced by Laurent Schwartz as a reminiscence of the so-called spherical functions, which are exactly $\Sigma_N(\mathbb S^n)$. So originally that letter $\mathscr S$ was for "spherical", although it became soon linked to the name Schwartz.
