Let $\mathcal{F}(f) := \int_{\mathbb{R}} f(x) e^{-ixy} dx \, $ denote the Fourier transform of a function $f \in L^1(\mathbb{R})$ and let $C_b^2 (\mathbb{R}) = \{ f \in \mathbb{C}^{\mathbb{R}} \mid f \text{ is twice differentiable and is bounded } \}$.
Then define the set $A$ as follows,
$$ A := \{ f \in C_b^2(\mathbb{R}) \mid \mathcal{F}(f) \in L^1(\mathbb{R}) \text{ and } f \text{ has compact support } \} $$
Prove that $A$ is dense in $L^1(\mathbb{R})$.
Not sure where to start on this. Was considering trying to prove that the condition $\mathcal{F}(f) \in L^1(\mathbb{R})$ is superflous, and maybe relating this to the fact that $C^{\infty}(\mathbb{R})$ is dense in $L^1(\mathbb{R}).$
Do this for a closed interval instead of $\mathbb{R}.$ The compact support is superfluous then, and the density is an immediate consequence of the Stone-Weierstrass theorem. Since functions of compact support are dense in $L^1$ (by the usual step function argument), you are done.