If $f,g$ are smooth functions with support in the interval $[-r,r]$ for some $r>0$, then their convolution $f*g$ is smooth with support in $[-2r,2r]$. My question is about the converse: Given smooth $h$ with support in $[-2r,2r]$, can I always write it as $h=f*g$ with $f,g$ as above? (By Fourier transform, one can formulate this problem also as a decomposition of entire functions of exponential type $2r$ into a product of entire functions of exponential type $r$ with additional restrictions on the real line.)