Let $X=\mathbb{R}$, and $\mathcal{A}:=\mathbb{R}[x]$ be the subalgebra (of $C(X)$) of univariate polynomials.
Given $\varphi\in C_b(X)$ and $K\subset X$ compact, we know from Stone-Weierstrass that
$$\tag{1} \forall\,\varepsilon>0 \, : \ \exists\, p_\varepsilon\in\mathcal{A} \quad\text{such that}\quad \sup\nolimits_{x\in K}\!|\varphi(x) – p_\varepsilon(x)|\leq\varepsilon.$$
(Here, $C_b(X)$ is the space of all bounded continuous $\mathbb{R}$-valued functions on $X$.)
Question: Can the polynomials in $(1)$ be chosen such that $p_\varepsilon \leq \varphi$ pointwise on $X$?
If the answer is affirmative, can we generalise this to the case where $X$ is a $\sigma$-compact and bounded closed subset of a Banach space and $\mathcal{A}$ is a point-separating and pointwise non-vanishing subalgebra of $C_b(X)$?
(Note for the general case that the algebra $\mathcal{A}$ is dense in $C_b(X)$ wrt. the strict topology.)
Best Answer
Yes, this works. Let's say $K\subseteq [-1,1]$. Start out by finding a polynomial $q$ such that $\varphi-\epsilon/2 \le q \le \varphi-\epsilon/4$ on $[-3,3]$. We can then take $p(x)=q(x)-(\epsilon/4)(x/2)^{2N}$. This will approximate $\varphi$ on $K$ with the desired accuracy for any choice of $N$, and also $p\le q\le\varphi$ on $[-3,3]$ for any $N$. Finally, taking $N$ large enough will make sure that also $p\le\varphi$ outside $[-3,3]$.