  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 measurement cutoff frequency of $f_c$, Donoho [2] showed that exact
recovery is possible if the spikes (on $\mathbb{R}$) lie on a lattice and
$\Delta > 1/f_c$, but does not specify a corresponding recovery method.
Cand$\text{\`e}$s and Fernandez-Granda [3, 4] provide a convex programming
method for the recovery of periodic spike trains (i.e., spike trains on the
torus $\mathbb{T}$), which succeeds provably if $\Delta > 2/f_c$ and $f_c \geq
128$ or if $\Delta > 1.26/f_c$ and $f_c \geq 10^3$, and does not need the
spikes within the fundamental period to lie on a lattice. In this paper, we
develop a theory of super-resolution from short-time Fourier transform (STFT)
measurements. Specifically, we present a recovery method similar in spirit to
the one in [3] for pure Fourier measurements. For a STFT Gaussian window
function of width $\sigma = 1/(4f_c)$ this method succeeds provably if $\Delta
> 1/f_c$, without restrictions on $f_c$. Our theory is based on a
measure-theoretic formulation of the recovery problem, which leads to
considerable generality in the sense of the results being grid-free and
applying to spike trains on both $\mathbb{R}$ and $\mathbb{T}$. The case of
spike trains on $\mathbb{R}$ comes with significant technical challenges. For
recovery of spike trains on $\mathbb{T}$ we prove that the correct solution can
be approximated---in weak-* topology---by solving a sequence of
finite-dimensional convex programming problems.
