  Recent results from compressive sampling (CS) have demonstrated that accurate
reconstruction of sparse signals often requires far fewer samples than
suggested by the classical Nyquist--Shannon sampling theorem. Typically, signal
reconstruction errors are measured in the $\ell^2$ norm and the signal is
assumed to be sparse, compressible or having a prior distribution. Our spectrum
estimation by sparse optimization (SpESO) method uses prior information about
isotropic homogeneous turbulent flows with power law energy spectra and applies
the methods of CS to 1-D and 2-D turbulence signals to estimate their energy
spectra with small logarithmic errors. SpESO is distinct from existing energy
spectrum estimation methods which are based on sparse support of the signal in
Fourier space. SpESO approximates energy spectra with an order of magnitude
fewer samples than needed with Shannon sampling. Our results demonstrate that
SpESO performs much better than lumped orthogonal matching pursuit (LOMP), and
as well or better than wavelet-based best M-term or M/2-term methods, even
though these methods require complete sampling of the signal before
compression.
