On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?)

If we assume a little more smoothness, we can use a trick similar to the one in this question.

Let me take $n=2$ for concreteness. Suppose $f$ is $C^2$ on a neighborhood $U$ of $K$. With an appropriate cutoff function we can find a $C^2$, compactly supported $g$ which agrees with $f$ on a (possibly smaller) neighborhood of $K$. We can also find a large enough square $Q = (-M,M)$ which contains the support of $g$, so that $g$ and all its derivatives vanish on the boundary $\partial Q$.

Fix $\epsilon > 0$. The second partial $\partial_x \partial_y g$ is continuous, so we can find a polynomial $p_0$ with $|p_0 - \partial_x \partial_y g| < \epsilon$ on $\bar{Q}$. Set $$p(x,y) = \int_{-M}^x \int_{-M}^y p_0(s,t)\,dt\,ds.$$ Then by the fundamental theorem of calculus and Fubini's theorem we have, for any $(x,y) \in Q$, $$\begin{align*}|p(x,y) - g(x,y)| &= \left|\int_{-M}^x \int_{-M}^y p_0(s,t) - \partial_x \partial_y g(s,t)\,ds\,dt\right|\\ & \le \int_{-M}^x \int_{-M}^y |p_0(s,t) - \partial_x \partial_y g(s,t)|\,ds\,dt \\&\le (2M)^2 \epsilon \end{align*}$$ and $$\begin{align*}|\partial_x p(x,y) - \partial_x g(x,y)| &= \left| \int_{-M}^y p_0(s,t) - \partial_x \partial_y g(s,t)\,dt\right|\\ & \le \int_{-M}^y |p_0(s,t) - \partial_x \partial_y g(s,t)|\,dt \\&\le 2M \epsilon. \end{align*}$$ A similar argument works for $\partial_y$. Since $f=g$ on a neighborhood of $K$, we have that $p$ and $f$, as well as their derivatives, are uniformly close on a neighborhood of $K$.

In $\mathbb{R}^n$, this approach requires us to assume that $f$ is $C^n$.

By using a partition of unity, we may assume that $f$ has compact support. Let $\phi_m$ be the density of the normal vector with mean $\bf 0$ and covariance matrix $I/m$. Let $q_m$ be a Maclaurin polynomial approximating $\phi_m$ of sufficiently high degree that $\lim_{m \to \infty} \|\phi_m - q_m\|_{C(K)} = 0$, where $K$ is the Minkowski difference of where you want the approximation to hold and the support of $f$. Now let $p_m := q_m * f$. These $p_m$ do the job.

Remark: Nachbin has a generalization of the Stone-Weierstrass theorem to include approximating derivatives, even on manifolds. See MR0030590: Nachbin, Leopoldo, Sur les algèbres denses de fonctions différentiables sur une variété. C. R. Acad. Sci. Paris 228, (1949), 1549–1551.