  We study the approximation properties of pseudo-differential operators with
small time-frequency dispersion, meaning that their spreading functions are
supported in a small neighborhood of the origin. It is commonly assumed that
for such operators $H$, the output $Hf$ can differ only a little from a scalar
multiple of the input $f$. However, we disprove this heuristic statement, hence
revealing some unexpected behaviors of such operators.

Classical Analysis and ODEs