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Preprint

2016-01-21

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...

http://arxiv.org/licenses/nonexclusive-distrib/1.0/