Approximating functions on the real line

Question

While it is not possible to approximate any function with polynomials on the entire real line, I am wondering if there are modified conditions under which the approximation is possible. Consider $f \in C([0, \infty))$ such that $ f(x) \rightarrow 0$ as $x\to\infty$. Is it true that for any $\epsilon >0$ we can find a polynomial $p(x)$ such that $$\left\vert f(x) - e^{-x}p(x) \right\vert < \epsilon \quad \forall x \in [0, \infty)$$

I suspect the answer is no, because $p$ effectively approximates $f$ on any compact interval but $e^{-x}p(x)$ doesn't have to be small outside said interval. However, I can't find a counter example.

Answer 1

This is the problem of weighted polynomial approximation, which has been studied extensively. Koosis, The logarithmic integral I gives an overview. The corollary of Section VI.D there states that weighted polynomial approximation works for the weight $w(x)=e^{|x|}$. This means that given $g\in C(\mathbb R)$ such that $g/w\to 0$ as $|x|\to\infty$ and $\epsilon>0$, there is a polynomial $p$ such that $|(g-p)/w|<\epsilon$.

In particular, the answer to your question is positive (restrict to $x\ge 0$, take $g=wf$).