A unified method for super-resolution recovery and real exponential-sum separation

In this paper, motivated by diffraction of traveling light waves, a simple mathematical model is proposed, both for the multivariate super-resolution problem and the problem of blind-source separation of real-valued exponential sums. This model facilitates the development of a unified theory and a unified solution of both problems in this paper. Our consideration of the super-resolution problem is aimed at applications to fluorescence microscopy and observational astronomy, and the motivation for our consideration of the second problem is the current need of extracting multivariate exponential features in magnetic resonance spectroscopy (MRS) for the neurologist and radiologist as well as for providing a mathematical tool for isotope separation in Nuclear Chemistry. The unified method introduced in this paper can be easily realized by processing only finitely many data, sampled at locations that are not necessarily prescribed in advance, with computational scheme consisting only of matrix vector multiplication, peak finding, and clustering.