2016/01/21 by John J. Benedetto, Benedetto, John J., Weilin Li +1
Engineering · Mathematics · Medicine · #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Medical Imaging Techniques and Applications #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1601.05761
openalex publication_date 2016/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M(\mathbbTd) be the space of complex bounded Radon measures defined on the d-dimensional torus group (ℝ/ℤ)d=\mathbbTd, equipped with the total variation norm ‖⋅‖; and let μ denote the Fourier transform of μ∈ M(\mathbbTd). We address the super-resolution problem: For given spectral (Fourier transform) data defined on a finite set Λ⊂ℤd, determine if there is a unique μ∈ M(\mathbbTd) of minimal norm for which μ equals this data on Λ. Without additional assumptions on μ and Λ, our main theorem shows that the solutions to the super-resolution problem, which we call minimal extrapolations, depend crucially on the set Γ⊂Λ, defined in terms of μ and Λ. For example, when Γ=0, the minimal extrapolations are singular measures supported in the zero set of an analytic function, and when Γ≥ 2, the minimal extrapolations are singular measures supported in the intersection of Γ\choose 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 minimal extrapolations. This theorem has implications to the possibility and impossibility of uniquely recovering μ from 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 with Candes and Fernandez-Granda's work on super-resolution. This connection is exploited to address situations where current algorithms fail to compute a numerical solution to the super-resolution problem.