  We present a pursuit-like algorithm that we call the "superset method" for
recovery of sparse vectors from consecutive Fourier measurements in the
super-resolution regime. The algorithm has a subspace identification step that
hinges on the translation invariance of the Fourier transform, followed by a
removal step to estimate the solution's support. The superset method is always
successful in the noiseless regime (unlike L1-minimization) and generalizes to
higher dimensions (unlike the matrix pencil method). Relative robustness to
noise is demonstrated numerically.
