Approximation by $C^\infty$-function

Question

Given $\Delta >0$, $g \in L_1(\mathbb{R}^d) \cap L_2(\mathbb{R}^d)$, show that there exists a $\rho \in C^\infty$ with compact support, so that $\|g-\rho\|_{L_2} < \Delta$ and $\|g-\rho\|_{L_1} < \Delta$.

I dont have a real clue how to approch this, I know that C-infinity functions are dense in the $L_p$ spaces, but I dont see any relation between these two norms.

Greetings.

Answer 1

$\newcommand{\norm}[1]{\|#1\|}$ (this is only a very rough outline, comment if you need additional help)

First, consider the question on a domain of finite measure, e.g.\ $B_n(0)$ instead of $\mathbb R^n$.

There it is possible, because the inequality $$ \norm{f-g}_{L_1(B_n(0))} \leq C \norm{f-g}_{L_2(B_n(0))} $$ holds.

We also use that $g \chi_{B_n(0)} \to g$ in both the $L_1$ and the $L_2$ norm. This is a consequence of $\mathbb R^n = \bigcup_{n\in N} B_n(0)$.

Then you can find a sequence of approximations of $g \chi_{B_n(0)}$, which can be arbitrarily close to $g$ in both norms.