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# Super-resolution by means of Beurling minimal extrapolation

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### Abstract

Let $$M(\mathbb{T}^d)$$ be the space of complex bounded Radon measures defined on the $$d$$-dimensional torus group $$(\mathbb{R}/\mathbb{Z})^d=\mathbb{T}^d$$, equipped with the total variation norm $$\|\cdot\|$$; and let $$\hat\mu$$ denote the Fourier transform of $$\mu\in M(\mathbb{T}^d)$$. We address the super-resolution problem: For given spectral (Fourier transform) data defined on a finite set $$\Lambda\subset\mathbb{Z}^d$$, determine if there is a unique $$\mu\in M(\mathbb{T}^d)$$ for which $$\hat\mu$$ equals this data on $$\Lambda$$. Without additional assumptions on $$\mu$$ and $$\Lambda$$, our main theorem shows that the solutions to the super-resolution problem, which we call minimal extrapolations, depend crucially on the set $$\Gamma\subset\Lambda$$, defined in terms of $$\mu$$ and $$\Lambda$$. For example, when $$\Gamma=0$$, the minimal extrapolations are singular measures supported in the zero set of an analytic function, and when $$\Gamma\geq 2$$, the minimal extrapolations are singular measures supported in the intersection of $$\Gamma\choose 2$$ hyperplanes. This theorem has implications to the possibility and impossibility of uniquely recovering $$\mu$$ from $$\Lambda$$. 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. 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 minimal extrapolations. By applying our theorem and examples, we study whether the minimal extrapolations of $$\mu$$ and $$T\mu$$ are related, for different types of linear operators $$T\colon M(\mathbb{T}^d)\to M(\mathbb{T}^d)$$. Additionally, our concept of an admissibility range fundamentally connects...

### Author and article information

###### Journal
2016-01-21
2016-03-07
###### Article
1601.05761