  This paper studies sparse super-resolution in arbitrary dimensions. More
precisely, it develops a theoretical analysis of support recovery for the
so-called BLASSO method, which is an off-the-grid generalisation of l1
regularization (also known as the LASSO). While super-resolution is of
paramount importance in overcoming the limitations of many imaging devices, its
theoretical analysis is still lacking beyond the 1-dimensional (1-D) case. The
reason is that in the 2-dimensional (2-D) case and beyond, the relative
position of the spikes enters the picture, and different geometrical
configurations lead to different stability properties. Our first main
contribution is a connection, in the limit where the spikes cluster at a given
point, between solutions of the dual of the BLASSO problem and Hermite
polynomial interpolation ideals. Polynomial bases for these ideals, introduced
by De Boor, can be computed by Gaussian elimination, and lead to an algorithmic
description of limiting solutions to the dual problem. With this construction
at hand, our second main contribution is a detailed analysis of the support
stability and super-resolution effect in the case of a pair of spikes. This
includes in particular a sharp analysis of how the signal-to-noise ratio should
scale with respect to the separation distance between the spikes. Lastly,
numerical simulations on different classes of kernels show the applicability of
this theory and highlight the richness of super-resolution in 2-D.
