Super-resolution of positive near-colliding point sources

Abstract

In this paper, we analyze the capacity of super-resolution of one-dimensional positive sources. In particular, we consider the same setting as in [arXiv:1904.09186v2 [math.NA]] and generalize the results there to the case of super-resolving positive sources. To be more specific, we consider resolving $d$ positive point sources with $p⩽d$ nodes closely spaced and forming a cluster, while the rest of the nodes are well separated. Similarly to [arXiv:1904.09186v2 [math.NA]], our results show that when the noise level $\hat{I}\mu \lesssim {\mathrm{SRF}}^{-2p+1}$, where $\mathrm{SRF}=(\hat{I}\text{c◯}\hat{I}”{)}^{-1}$ with $\hat{I}\text{c◯}$ being the cutoff frequency and $\hat{I}”$ the minimal separation between the nodes, the minimax error rate for reconstructing the cluster nodes is of order $\frac{1}{\hat{I}}\text{c◯}{\mathrm{SRF}}^{2p-2}\hat{I}\mu$, while for recovering the corresponding amplitudes $\left\{a_j\right\}$ the rate is of order ${\mathrm{SRF}}^{2p-1}\hat{I}\mu$. For the non-cluster nodes, the corresponding minimax rates for the recovery of nodes and amplitudes are of order $\frac{\hat{I}}{\mu }\hat{I}\text{c◯}$ and $\hat{I}\mu$, respectively. Our numerical experiments show that the Matrix Pencil method achieves the above optimal bounds when resolving the positive sources.