Is the set $\{\phi+\phi':\phi\in C^1_c(\mathbb{R})\}$ dense in $L^2(\mathbb{R})$?

Question

Let $C^1_c(\mathbb{R})$ be the set of continuously differentiable functions with compact support in $\mathbb{R}$. I'm wondering whether the statement in the title is true. Does anyone know of techniques to prove similar statements? I don't have any direction right now.

Answer 1

Here's a sketch of proof.

1. $C_c^{\infty}$ is dense in $L^2$.

2. Given $f \in C_c^{\infty}$, there exists $f_n \in C_c^{\infty}$ such that $f_n \to f$ in $L^2$ and $$\int_{\mathbb{R}} f_n(x)e^x dx = 0.$$ Hint: Let $f_n = f + g_n$, where $g_n$ is supported on $[n, \infty)$.

3. Let $$\phi_n(x) = e^{-x} \int_{-\infty}^x f_n(t) e^t dt.$$ Then $\phi_n \in C_n^1$ while $\phi_n + \phi_n' = f_n$.