Is a smooth function with compact support in $\mathbb{R}$ can be written as the convolution of two square integrable functions?

Question

Suppose h is smooth compact supported function on $\mathbb{R}$. How to show that there exist $f, g \in L^2(\mathbb{R}, m)$, where $m$ is the usual Lebsegue measure, such that $h = f * g $, where $*$ denotes the convolution?

Answer 1

Here is an outline:

Consider this in the Fourier domain: $\widehat{h}=\widehat{f}\widehat{g}$. $\widehat{h}$ is smooth and rapidly decaying based on the properties of $h$. Take $\widehat{f}$ to be something positive and smooth with polynomial decay. Then $\widehat{g} = \widehat{h}/\widehat{f}$.