Approximating $L^p$ functions using Schwartz functions with compact support on the Fourier side

Question

For $1\leq p<\infty$, how would you show for any $f\in L^p(\mathbb{R})$ and given $\epsilon>0$, there exists $L<\infty$ and $g\in \mathcal{S}(\mathbb{R})$ such that $\|f-g\|_p<\epsilon$ and $\operatorname{supp}(\widehat{g})\subset [-L, L]$

We know that $C^\infty_c$ is dense subset of $S$ and $L^p$ but then we cannot make sure the Fourier transform of $g$ is compactly supported.

Answer 1

Extended Hint: It suffices to assume that $f$ itself is a Schwartz function (why?). Then use that

$$ \widehat{f \ast g} = \widehat{f} \cdot \widehat{g}, $$

i.e. if $\widehat{g}$ has compact support, then so has $\widehat{f\ast g}$.

Now choose a suitable "approximation of unity" $(g_n)_n$.

I will leave the details to you.