I'm looking for a version of Weierstrass's approximation theorem that works for a continuous function $f:D \to \mathbb{R}^d$.
Versions that I know of
Multivariate Weierstrass theorem? uses a generalization of Weierstrass for multivariate domain based on the Stone's generalization of the theorem.
Wikipedia gives several versions but none (as far as i read) works with a multivariate image.
What I need it for
I need it to properly justify Proof of Peano's existence theorem for ODEs or Proving Peano's Existence Theorem by approximating with $C^{\infty}$ functions using Weierstrass' Theorem.
Seen the above application that I'm seeking I think that perhaps I can do it component-wise. (????)
Statement of the (possible) theorem
Let $D \subseteq \mathbb{R} \times \mathbb{R}^d$ be an open set and $f:D \to \mathbb{R}^d$ a continuous function.
$\exists.p_n:D \to \mathbb{R}^d$ a sequence of polynomials such that $p_n \stackrel{\|\cdot\|_{\infty}}{\to} f$ on a compact set of the form $[a,b] \times \overline{B}(x_0,b) \subseteq D$.
Perhaps this simple case will help: Suppose $f=(f_1,\dots,f_k):\mathbb R^j\to \mathbb R^k$ is continuous. By Stone-Weierstrass, for each $m$ there exist polynomials $p_{m1}, \cdots, p_{mk}$ such that
$$|p_{mi}-f_i| < 1/m\,\, \text { on } \overline {B(0,m)},\, i=1,\dots, k.$$
Set $p_m = (p_{m1},\dots , p_{mk}).$ Then $p_m\to f$ uniformly on compact subsets of $\mathbb R^j.$