Every $f \in L^{1}(G)$ is product of two of $L^{2}(G)$

Question

Let be a $G$ locally compact Hausdorff abelian group. My question is:

If $h : G \rightarrow \mathbb{C}$, $h \in L^{1}(G)$, then $h = f \cdot g$ for some $f, g \in L^{2}(G)$.

If $h$ was real-valued I'll see clear, but I don't see how to proof it in the complex-valued case.

Thanks

Answer 1

$f=\sqrt {|h|}, g=\frac h {\sqrt {|h|}}$ on $\{h \neq 0\}$, $g=1$ on $\{h=0\}$.