  Signal processing techniques have been developed that use different
strategies to bypass the Nyquist sampling theorem in order to recover more
information than a traditional discrete Fourier transform. Here we examine
three such methods: filter diagonalization, compressed sensing, and
super-resolution. We apply them to a broad range of signal forms commonly found
in science and engineering in order to discover when and how each method can be
used most profitably. We find that filter diagonalization provides the best
results for Lorentzian signals, while compressed sensing and super-resolution
perform better for arbitrary signals.
