  We present a fast two-phase algorithm for super-resolution with strong
theoretical guarantees. Given the low-frequency part of the spectrum of a
sequence of impulses, Phase I consists of a greedy algorithm that roughly
estimates the impulse positions. These estimates are then refined by local
optimization in Phase II.
  In contrast to the convex relaxation proposed by Cand\`es et al., our
approach has a low computational complexity but requires the impulses to be
separated by an additional logarithmic factor to succeed. The backbone of our
work is the fundamental work of Slepian et al. involving discrete prolate
spheroidal wave functions and their unique properties.
