The set $\{f \in \mathcal L^2(\mathbb R) : \hat f \in \mathcal C_c(\mathbb R)\}$ is dense in $\mathcal L^2(\mathbb R)$

Question

Let $\mathcal C_c(\mathbb R)$ denotes the set of all continuous functions with compact support on $\mathbb R$. We know that $\mathcal C_c(\mathbb R)$ is dense in $\mathcal L^2(\mathbb R)$. Now consider a subset $\cal M$ of $\mathcal L^2(\mathbb R)$ as $\mathcal M =\{f \in \mathcal L^2(\mathbb R) : \hat f \in \mathcal C_c(\mathbb R)\}$, where $\hat f$ denotes the Fourier transform of $f$. Now I want to show that $\mathcal M$ is dense in $\mathcal L^2(\mathbb R)$.
Basically, if we consider the characteristic functions and construct a sequence $f_n \in \mathcal M$ such that limit of $f_n$ is the characteristic function, then we are done. Please help me with the construction. Thank you for your help and time.

Answer 1

If $f\in \mathcal L^2(\mathbb R)$ then $f=\hat g$ for some $g\in \mathcal L^2(\mathbb R)$. There is a sequence $(g_n)$ in $\mathcal C_c(\mathbb R)$ convering to $g$ in $\mathcal L^2(\mathbb R)$. Now $\|f- \hat {g_n}\|=\|\hat g- \hat {g_n}\|=\|g_n-g\| \to 0$.

Note that if $h_n=\hat {g_n}$ then $\hat {h_n}(x)=g_n(-x)$, which is a $C^{\infty}$ function with compact support.