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Super-resolution by means of Beurling minimal extrapolation. (English) Zbl 1439.43004
Summary: We investigate the super-resolution capabilities of total variation minimization. Namely, given a finite set $$\Lambda\subseteq\mathbb{Z}^d$$ and spectral data $$F=\hat{\mu}|_{\Lambda}$$, where $$\mu$$ is an unknown bounded Radon measure on the torus $$\mathbb{T}^d$$, the problem is to find the measures with smallest norm whose Fourier transforms agree with $$F$$ on $$\Lambda$$. Our main theorem shows that solutions to the problem depend crucially on a set $$\Gamma\subseteq\Lambda$$, defined in terms of $$F$$ and $$\Lambda$$. For example, when $$\#\Gamma=0$$, the solutions are singular measures supported in the zero set of an analytic function, and when $$\#\Gamma\geq 2$$, the solutions are singular measures supported in the intersection of $$\binom{\#\Gamma}{2}$$ hyperplanes. By theory and example, we show that the case $$\#\Gamma=1$$ 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 $$\mu$$ from $$F$$ on $$\Lambda$$. 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 [A. Beurling, The collected works of Arne Beurling. Volume 1: Complex analysis. Volume 2: Harmonic analysis. Ed. by Lennart Carleson, Paul Malliavin, John Neuberger, John Wermer. Boston etc.: Birkhäuser Verlag (1989; Zbl 0732.01042)] with Candès and Fernandez-Granda’s work on super-resolution [E. J. Candès and C. Fernandez-Granda, J. Fourier Anal. Appl. 19, No. 6, 1229–1254 (2013; Zbl 1312.94015)]. This connection is exploited to address situations where current algorithms fail to compute a numerical solution to the total variation minimization problem.

##### MSC:
 43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups 46E27 Spaces of measures 46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics 94A12 Signal theory (characterization, reconstruction, filtering, etc.)
PDCO
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