  The stability of spike deconvolution, which aims at recovering point sources
from their convolution with a point spread function (PSF), is known to be
related to the separation between those sources. When the observations are
noisy, it is critical to ensure support stability, where the deconvolution does
not lead to spurious, or oppositely, missing estimates of the point sources. In
this paper, we study the resolution limit of stably recovering the support of
two closely located point sources using the Beurling-LASSO estimator, which is
a convex optimization approach based on total variation regularization. We
establish a sufficient separation criterion between the sources, depending only
on the PSF, above which the Beurling-LASSO estimator is guaranteed to return a
stable estimate of the point sources, with the same number of estimated
elements as that of the ground truth. Our result highlights the impact of PSF
on the resolution limit in the noisy setting, which was not evident in previous
studies of the noiseless setting. Towards the end, we show that the same
resolution limit applies to resolving two close-located sources in conjunction
of other well-separated sources.
