The Stone-Weierstrass Theorem says that the polynomials are dense in C[X] under the sup norm, where X is any Hausdorff space. There have been several previous posts asking about approximations of $f \in C[X]$ with polynomials
$P_n = \{p | p $ is a polynomial of degree less than or equal to $n\}$
for a fixed degree $n$. It seems like this is a problem that has certain bounds for $\inf_{p \in P_n} ||f - p||_\infty < \epsilon(n)$, where $\epsilon(n)$ is on the order $O(1/n)$ (1,2)
I have two questions:
If we restrict the output of $f: [-a, b] \rightarrow [0,1]$, are there any tighter bounds that can be said? Clearly, $\inf_{p \in P_n} ||f - p||_\infty < 1/2$ for the trivial case $p=1/2$.
Can we prove the same bounds for $f: [-a, b]^n -> \mathbb{R}$ and a multi-variate polynomial $P(x_1, ... x_n)$ as we did in the one-dimensional case? What if we restrict the range to $f: [-a, b]^n -> [0,1]$?
I am not sure how one would go about the first question, other than to note that $f$ is 1-Lipshitz and by the above posts there are potentially some tighter bounds there for Lipschitz functions. For the second question, I showed that the polynomials are still dense in $C[X]$, which makes me think there might be an analgous way to prove the error bounds as well?
This is probably all well-understood, but I'm not well-read on approximation theory. Any guidance would be wonderful.