Let $p$ be a smooth (say, $C^\infty$, but this is not crucial) density on the interval $I=[0,1]$ and $g_\sigma$ be the density of $N(0,\sigma^2)$. Define $f=p\ast g_\sigma$. To what extent does the behavior of $f$ on $I$ determine its behavior off of $I$? In other words, is it possible that $f=g$ on $I$ but $f\ne g$ on $I^c$? The idea is to determine the tail behavior of $f$ given only information on $I$.
This question is clearly nonsensical without the convolution structure on $f$. Note also that we do not know $p$, but $f$, on the interval $I$. My intuition is that since $p$ fully determines the behavior of $f$ globally, the "rough" information given by $f$ on $I$ might be enough to tell us about its tails.
(Note: I am also curious what happens if $p$ is not smooth or has weaker regularity in general, but this is not essential at the moment.)
