  This paper studies sparse spikes deconvolution over the space of measures. We
focus our attention to the recovery properties of the support of the measure,
i.e. the location of the Dirac masses. For non-degenerate sums of Diracs, we
show that, when the signal-to-noise ratio is large enough, total variation
regularization (which is the natural extension of the L1 norm of vectors to the
setting of measures) recovers the exact same number of Diracs. We also show
that both the locations and the heights of these Diracs converge toward those
of the input measure when the noise drops to zero. The exact speed of
convergence is governed by a specific dual certificate, which can be computed
by solving a linear system. We draw connections between the support of the
recovered measure on a continuous domain and on a discretized grid. We show
that when the signal-to-noise level is large enough, the solution of the
discretized problem is supported on pairs of Diracs which are neighbors of the
Diracs of the input measure. This gives a precise description of the
convergence of the solution of the discretized problem toward the solution of
the continuous grid-free problem, as the grid size tends to zero.
