A strengthening of Stone–Weierstrass which applies to arbitrary closed intervals Note in this question, we concern ourselves only with the space $C([a,b])$ of continuous real-valued functions on compact intervals $[a,b]$. 
The Müntz–Szász theorem is a well-known result related to the Stone–Weierstrass theorem. It states that for $0 \leq a < b < \infty$ and given $S \subset \mathbb{N}$, $\text{span}(\{x^n\}_{n \in S})$ is dense in $C([a,b])$ iff $\sum_{n \in S} \frac{1}{n} = \infty$. In one sense, it is a stronger result than the classic Stone–Weierstrass. However, it requires the additional assumption that $[a,b]$ is a compact interval of $[0, \infty)$, whereas S-W applies for any compact interval in $\mathbb{R}$, so in that sense it is a weaker result. 

I'm wondering if there are any nontrivial strengthenings of the S-W theorem which apply to arbitrary closed intervals, including the case where $a<0$. Specifically, it would be nice to give a characterization of when $\text{span}(\{x^n\}_{n \in S})$ is dense in $C([a,b])$ where $a$ is an arbitrary real number (possibly negative). 

One observation is that the only "interesting" case is $a<0<b$, for if $a<b \leq 0$ then the transformation $x \mapsto -x$ brings us back to normal Müntz–Szász territory. 
Given that $a<0<b$, it is also clear that $S$ must contain some odd numbers, since if each $n$ is even, all of the polynomials in the span are even functions, so they certainly cannot be dense. 
 A: We may as well assume we have an interval of the form $[-b,b]$.  Now consider the operators $E,O:C([-b,b])\to C([-b,b])$ defined by $E(f)=\frac{f+\tilde{f}}{2}$ and $O(f)=\frac{f-\tilde{f}}{2}$, where $\tilde{f}(x)=f(-x)$.  Note that $E(f)$ is even and $O(f)$ is odd, and $f=E(f)+O(f)$.  Moreover, $E$ and $O$ are bounded operators, and it follows that $f_n\to f$ iff $E(f_n)\to E(f)$ and $O(f_n)\to O(f)$ (all convergence here and below is uniform).  If $p$ is a polynomial then $E(p)$ is just the even degree terms and $O(p)$ is just the odd degree terms.
It follows that $f$ is a limit of polynomials with exponents in $S$ iff $E(f)$ is a limit of polynomials with even exponents in $S$ and $O(f)$ is a limit of polynomials with odd exponents in $S$.  But a sequence of even polynomials converges to $E(f)$ iff they converge to $E(f)$ on $[0,b]$, and a sequence of odd polynomials converge to $O(f)$ iff they converge to $O(f)$ on $[0,b]$.  Whether such sequences exist is determined by the Müntz–Szász theorem.
Thus, we conclude that $\text{span}(\{x^n\}_{n \in S})$ is dense in $C([-b,b])$ iff both $\sum\limits_{n\text{ even, }n \in S} \frac{1}{n} = \infty$ and $\sum\limits_{n\text{ odd, }n \in S} \frac{1}{n} = \infty$.
