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 SRF^-2 p+1, where SRF=( )^-1 with being the cutoff frequency and the minimal separation between the nodes, the minimax error rate for reconstructing the cluster nodes is of order 1 SRF^2 p-2 , while for recovering the corresponding amplitudes _j\ the rate is of order SRF^2 p-1 . For the non-cluster nodes, the corresponding minimax rates for the recovery of nodes and amplitudes are of order and , respectively. Our numerical experiments show that the Matrix Pencil method achieves the above optimal bounds when resolving the positive sources.

Tasks

Reproductions