  We study array imaging of a sparse scene of point-like sources or scatterers
in a homogeneous medium. For source imaging the sensors in the array are
receivers that collect measurements of the wave field. For imaging scatterers
the array probes the medium with waves and records the echoes. In either case
the image formation is stated as a sparsity promoting $\ell_1$ optimization
problem, and the goal of the paper is to quantify the resolution. We consider
both narrow-band and broad-band imaging, and a geometric setup with a small
array. We take first the case of the unknowns lying on the imaging grid, and
derive resolution limits that depend on the sparsity of the scene. Then we
consider the general case with the unknowns at arbitrary locations. The
analysis is based on estimates of the cumulative mutual coherence and a related
concept, which we call interaction coefficient. It complements recent results
in compressed sensing by deriving deterministic resolution limits that account
for worse case scenarios in terms of locations of the unknowns in the imaging
region, and also by interpreting the results in some cases where uniqueness of
the solution does not hold. We demonstrate the theoretical predictions with
numerical simulations.
