Approximating a function by a convolution of given function? Let $g:\mathbb{R}\to \mathbb{R}$ be a given differentiable function of exponential decay on both sides. Now let us be given a function $f:\mathbb{R}\to \mathbb{R}$, also of exponential decay, if you wish. When can I approximate $f$ well by a convolution of $g$ with some function $h:\mathbb{R}\to \mathbb{R}$ in $L^1$ with exponential decay on at least one side?
"Approximate" here can mean "with respect to the $L^1$ norm" or "with respect to the $L^\infty$ norm". So, say: can you find a function $h$ like that such that $|f-g\ast h|_1<\epsilon$ and $|h|_1<C$?
(Note $h$ is not required to be non-negative.)
 A: This is an extended comment. Passing to Fourier transforms,
we have the following problem: $\hat{f}$ and $\hat{g}$ are two functions analytic in some strip $-a<\mathrm{Im}\, z<a$,
and bounded from above (this is the translation of "exponential decrease of $f$ and $g$" on both sides.
Then you want to approximate $\hat{f}/\hat{g}$ on the real line by the boundary values of some bounded function $\hat{h}$ analytic in $0<\mathrm{Im}\, z<\epsilon$ or in $-\epsilon<\mathrm{Im}\, z<0$.
Evidently this is impossible when $\hat{g}$ has a real zero which is not a zero of $\hat{f}$. Condition that $\hat{g}$ has no real zeros does not help, since $\hat{g}$ can tend to zero on the real line faster than $\hat{f}$.
On the other hand, you can not only approximate but find
$h$ such that $f=g*h$, if $\hat{f}/\hat{g}$ is bounded in
some horizontal strip whose boundary contains the real line.
The question is not clearly stated (what does it mean "when I can approximate"?) Are you looking for conditions on both $f$ and $g$ so that $h$ exists? Or conditions on $f$ which imply that for every $g$ in some class $h$ exists? In what class?
