I am trying to see what is known about uniform density of function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$ (bounded continuous functions on $\mathbb{R}^n$). By uniform density, I mean density with respect to uniform convergence on the whole of $\mathbb{R}^n$, not any compactly supported version of uniform convergence. I do not know of any version of Stone Weierstrass theorem covering such cases. But I would be happy to see certain special examples where people have proved density of certain function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$. I am just curious about this, I do not have an especially good reason for asking this. As a possible example, if someone has proved somewhere that continuous functions of bounded smooth functions are dense in $C(\mathbb{R}^n)$, I would be interested in learning how such a proof might work. Thanks!

Comment: edited after Robert Israel and Eric Wofsey's comment.

is$C(\mathbb R^n)$. That is, for any $f \in C(\mathbb R^n)$, we can say $f(x_1, \ldots, x_n) = f(X_1, \ldots, X_n)$ where $X_j = x_j$ is a real analytic function of $x_j$. $\endgroup$ – Robert Israel Aug 11 '15 at 17:46