$\mathbf {The \ Problem \ is}:$ Let, $f\in A=L^1(\mathbb R^n) \cap L^2(\mathbb R^n).$
Does there exists $\{g_n\}_n \in \mathcal S(\mathbb R^n),$ the space of all Schwartz class functions on $\mathbb R^n,$ with $g_n {\rightarrow}^{L^1} f$ & $g_n {\rightarrow}^{L^2} f ?$
$\mathbf {My \ approach}:$ The space of all smooth compactly supported functions on $\mathbb R^n,$ $C_c^{\infty}(\mathbb R^n),$ is dense in $A$ but in both spaces, can we say whether same sequence exists which converges to $f$ under $2$ different norms ?
Any small hints . Thanks in adv.