Let $M$ be the set of smooth divergence-free vector fields $u$ on $\mathbb{R}^3$ with $$|\partial_x^{\alpha} u(x)| \leq C_{\alpha K}(1+|x|)^{-K}$$ on $\mathbb{R}^3$ for any $\alpha,K$. Further, we consider the subset of analytic functions $M_0 \subset M$. The question is now, whether for each $u \in M$ and $\varepsilon >0$, there is an $v \in M_0$, such that $$|u(x)-v(x)| < \varepsilon$$ for each $x \in \mathbb{R}^3$.

As a second question consider $M$ to be the set of smooth divergence-free vector fields $u$ on $\mathbb{R}^3$ with $$u(x+e_j) = u(x)$$ for all $x \in \mathbb{R}^3$ and $1 \leq j \leq 3$. Again, consider the subset of analytic functions $M_0 \subset M$. The question here is now again, whether for each $u \in M$ and $\varepsilon >0$, there is an $v \in M_0$, such that $$|u(x)-v(x)| < \varepsilon$$ for each $x \in \mathbb{R}^3$.

These questions are fairly similar to the Stone-Weierstrass theorem, but it seems it is not possible to derive it directly from it.