What I did so far is letting $g(x)=e^{-|x|}$ then by taking the fourier transform of the given equation I get $\widehat{g*f}(x)=\widehat{g}(x)\widehat{f}(x)=\widehat{g}(x)$ which leads to $\widehat{f}=1$
Does this lead to something ?
2026-04-07 16:15:14.1775578514
Does there exist a function $f\in L^1(\mathbb{R})$ such that $\int_{-\infty}^{+\infty}e^{-|y|}f(x-y)dy=e^{-|x|}$ for all $x\in \mathbb{R}$
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