  While spike trains are obviously not band-limited, the theory of
super-resolution tells us that perfect recovery of unknown spike locations and
weights from low-pass Fourier transform measurements is possible provided that
the minimum spacing, $\Delta$, between spikes is not too small. Specifically,
for a cutoff frequency of $f_c$, Donoho [2] shows that exact recovery is
possible if $\Delta > 1/f_c$, but does not specify a corresponding recovery
method. On the other hand, Cand\`es and Fernandez-Granda [3] provide a recovery
method based on convex optimization, which provably succeeds as long as $\Delta
> 2/f_c$. In practical applications one often has access to windowed Fourier
transform measurements, i.e., short-time Fourier transform (STFT) measurements,
only. In this paper, we develop a theory of super-resolution from STFT
measurements, and we propose a method that provably succeeds in recovering
spike trains from STFT measurements provided that $\Delta > 1/f_c$.
