Superresolution via sparsity constraints

The problem of recovering a measure mu supported on a lattice of span Delta is considered under the condition that measurements are only available concerning the Fourier Transform at frequencies of Omega or less. If Omega is much smaller than the Nyquist frequency pi/Delta and the measurements are noisy, then stable recovery of mu is generally impossible. It is shown here that if, in addition, it is known that mu satisfies certain sparsity constraints, then stable recovery is possible. This finding validates practical efforts in spectroscopy, seismic prospecting, and astronomy to provide superresolution by imposing support limitations in reconstruction.