  This article analyzes the recovery performance of two popular finite
dimensional approximations of the sparse spikes deconvolution problem over
Radon measures. We examine in a unified framework both the L1 regularization
(often referred to as Lasso or Basis-Pursuit) and the Continuous Basis-Pursuit
(C-BP) methods. The Lasso is the de-facto standard for the sparse
regularization of inverse problems in imaging. It performs a nearest neighbor
interpolation of the spikes locations on the sampling grid. The C-BP method,
introduced by Ekanadham, Tranchina and Simoncelli, uses a linear interpolation
of the locations to perform a better approximation of the infinite-dimensional
optimization problem, for positive measures. We show that, in the small noise
regime, both methods estimate twice the number of spikes as the number of
original spikes. Indeed, we show that they both detect two neighboring spikes
around the locations of an original spikes. These results for deconvolution
problems are based on an abstract analysis of the so-called extended support of
the solutions of L1-type problems (including as special cases the Lasso and
C-BP for deconvolution), which are of an independent interest. They precisely
characterize the support of the solutions when the noise is small and the
regularization parameter is selected accordingly. We illustrate these findings
to analyze for the first time the support instability of compressed sensing
recovery when the number of measurements is below the critical limit (well
documented in the literature) where the support is provably stable.
