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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.)

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    $\begingroup$ Note here that $|f-g\ast h|_1<\epsilon$ implies $\| \hat{f} -\hat{g}\hat{h} \|_\infty <\eps$. This means that if $\hat{g}$ is zero at a point $x$ where $|\hat{f}(x)|>\epsilon$, this is not possible $\endgroup$
    – Nick S
    Nov 6, 2022 at 0:03
  • $\begingroup$ What if $\widehat{g}(x)$ is non-vanishing on the real line? $\endgroup$
    – H A Helfgott
    Nov 6, 2022 at 0:35
  • $\begingroup$ What do you exactly mean by "When"? You want a condition on $g$ which implies that all $f$ from some class can be approximated? Or a condition on both f and g, which implies that $h$ exists? Or what? $\endgroup$
    – Alexandre Eremenko
    Nov 6, 2022 at 13:14

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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?

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