I am working on $\mathbb{R}^n$ for this problem. So every $L_p$ is actually $L_p(\mathbb{R}^n)$.
I know that the Schwartz space $S$ is dense in $L_p$ for $1\leq p<\infty$. This is how I prove it :
First I approximate $f\in L_p$ by a continuous function $g$ supported in (say) $B_R$. I then approximate $g$ by $p\chi_{B_R}$ where $p$ is a polynomial. I then approximate this function using a cut-off which makes it infinitely differentiable.
Now my problem is that I have a function $f\in L_1\cap L_2$, and I want to approximate it in both $L_1$ and $L_2$ norms with a single (sequence of) Schwartz function(s). When I try to replicate the above proof I run into a problem- is there a single compactly supported continuous function that approximates $f$ in both norms?
Any help would be appreciated.
