Super-resolution by means of Beurling minimal extrapolation

Abstract We investigate the super-resolution capabilities of total variation minimization. Namely, given a finite set Λ ⊆ Z d and spectral data F = μ ˆ | Λ , where μ is an unknown bounded Radon measure on the torus T d , the problem is to find the measures with smallest norm whose Fourier transforms agree with F on Λ. Our main theorem shows that solutions to the problem depend crucially on a set Γ ⊆ Λ , defined in terms of F and Λ. For example, when # Γ = 0 , the solutions are singular measures supported in the zero set of an analytic function, and when # Γ ≥ 2 , the solutions are singular measures supported in the intersection of ( # Γ 2 ) hyperplanes. By theory and example, we show that the case # Γ = 1 is different from other cases, and is deeply connected with the existence of positive solutions. This theorem has implications to the possibility and impossibility of uniquely recovering μ from F on Λ. We illustrate how to apply our theory to both directions, by computing pertinent analytical examples. These examples are of interest in both super-resolution and deterministic compressed sensing. Our concept of an admissibility range fundamentally connects Beurling's theory of minimal extrapolation [7] , [8] with Candes and Fernandez-Granda's work on super-resolution [12] . This connection is exploited to address situations where current algorithms fail to compute a numerical solution to the total variation minimization problem.