  We point out that by the ``smoothness means fast decay'' principle in Fourier
analysis, it is possible to infer the smoothness (or nonsmoothness) of the
autocorrelation function from a mere glimpse of the initial state.
Specifically, for a generic system with smooth eigenstates, the smoother an
initial state is, the faster its decomposition coefficients with respect to the
eigenstates of the system decay, and in turn the smoother the autocorrelation
function is. The idea is illustrated with three increasingly smooth functions
in an infinite square well. By using the Mellin transform, we also find that
the nonsmoothness or singularity of the initial state affects the short-time
behavior of its autocorrelation function. In particular, a sufficiently
nonsmooth or singular initial state could decay in a nonquadratic power law,
with the exponent continuously tunable. We now understand the periodic cusps of
the autocorrelation function in the quench dynamics of a Bloch state, which was
observed previously [Zhang and Yang, EPL 114, 60001 (2016)].
