It is well known that the class of Schwartz functions $\mathcal{S}$ in dense in all $L^p$ spaces therefore for each $f \in L^2$ there exists a sequence of Schwartz functions $(f_k)$ such that $\lVert f - f_k \rVert_{L^2} \to 0$ as $k \to \infty$.

If we suppose further that $f$ has compact support can we find a sequence of Schwartz functions $(f_k)$ such that $\lVert f - f_k \rVert_{L^2} \to 0$ as $k \to \infty$ (as above) and additionally $\operatorname{supp}(f_k) \subseteq \operatorname{supp}(f)$ for all $k$?