Showing the Fourier transform of $C_c^{\infty}(\mathbb{R})$ is dense in $L^2(\mathbb{R})$

Question

I am trying to show that $\left\lbrace \widehat{\phi} : \phi \in C_c^{\infty}(\mathbb{R}) \right\rbrace$ is dense in $L^2(\mathbb{R})$, where $\widehat{\phi}$ is the Fourier transform of the function $\phi$.

I believe it may follow simply from Plancherel's Theorem and using the fact that $C_c^{\infty}(\mathbb{R})$ is dense in $L^2(\mathbb{R})$ but I can't quite seem to put the pieces together

Any pointers would be appreciated!

Answer 1

If $f \in L^{2}$ then $\hat {f} \in L^{2}$ so there exists $\{\phi_n\} \subset C_c^{\infty}$ such that $\|\phi_n -\hat {f}\|_2 \to 0$. Since $f \to \hat {f}$ is an isometry and $\hat {\hat{f}}(x)=f(-x)$ we are done.