Markovian high resolution spectral analysis

When short data records are available, spectral analysis is basically an undetermined linear inverse problem. One usually considers the theoretical setting of regularization to solve such ill-posed problems. In this paper, we first show that "nonparametric" and "high resolution" are not incompatible in the field of spectral analysis. To this end, we introduce non quadratic convex penalization functions, like in low level image processing. The spectral amplitudes estimate is then defined as the unique minimizer of a compound convex criterion. An original scheme of regularization to simultaneously retrieve narrow-band and wide-band spectral features is finally proposed.