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 $[a,b] \times \overline{B}(x_0,b) \subseteq D$.

  • $\begingroup$ acadpubl.eu/monographs/201301/1012732acadpubl.201301.pdf seems a good place to start, althougth i would like the theorem restated in my easy conditions, perhaps theorem 4.3.1 $\endgroup$ – Javier Apr 8 '18 at 14:27
  • $\begingroup$ State the version you would like to hold. $\endgroup$ – zhw. Apr 8 '18 at 15:03
  • $\begingroup$ Why not just use Stone Weierstrass on each coordinate function of $f?$ $\endgroup$ – zhw. Apr 8 '18 at 15:31

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 $\mathbb R^j.$


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