TITLE: Is the rectangular function a convolution of $L^1$ functions?
QUESTION [28 upvotes]: Do there exist functions $f,g$ in $L^1(\mathbf{R})$ such that the convolution $f \star g$ is (almost everywhere) equal to the indicator function of the interval $[0,1]$ ?

REPLY [10 votes]: By the Cohen–Hewitt factorization theorem, every $f\in L^1(\mathbb R)$ can be written as $g\star h$ for some $g,h\in L^1(\mathbb R).$