Sparse non-negative super-resolution -- simplified and stabilised

The convolution of a discrete measure, x=sumi=1kaiδti, with a local window function, φ(s-t), is a common model for a measurement device whose resolution is substantially lower than that of the objects being observed. Super-resolution concerns localising the point sources ai,tii=1k with an accuracy beyond the essential support of φ(s-t), typically from m samples y(sj)=sumi=1k aiφ(sj-ti)+ηj, where ηj indicates an inexactness in the sample value. We consider the setting of x being non-negative and seek to characterise all non-negative measures approximately consistent with the samples. We first show that x is the unique non-negative measure consistent with the samples provided the samples are exact, i.e. ηj=0, m≥ 2k+1 samples are available, and φ(s-t) generates a Chebyshev system. This is independent of how close the sample locations are and em does not rely on any regulariser beyond non-negativity; as such, it extends and clarifies the work by Schiebinger et al. and De Castro et al., who achieve the same results but require a total variation regulariser, which we show is unnecessary. Moreover, we characterise non-negative solutions hatx consistent with the samples within the bound sumj=1mηj2≤ δ2. Any such non-negative measure is within mathcal O(δ1/7) of the discrete measure x generating the samples in the generalised Wasserstein distance, converging to one another as δ approaches zero. We also show how to make these general results, for windows that form a Chebyshev system, precise for the case of φ(s-t) being a Gaussian window. The main innovation of these results is that non-negativity alone is sufficient to localise point sources beyond the essential sensor resolution.