  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 $p \leqslant d$ 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
$\epsilon \lesssim \mathrm{SRF}^{-2 p+1}$, where $\mathrm{SRF}=(\Omega
\Delta)^{-1}$ with $\Omega$ being the cutoff frequency and $\Delta$ the minimal
separation between the nodes, the minimax error rate for reconstructing the
cluster nodes is of order $\frac{1}{\Omega} \mathrm{SRF}^{2 p-2} \epsilon$,
while for recovering the corresponding amplitudes $\left\{a_j\right\}$ the rate
is of order $\mathrm{SRF}^{2 p-1} \epsilon$. For the non-cluster nodes, the
corresponding minimax rates for the recovery of nodes and amplitudes are of
order $\frac{\epsilon}{\Omega}$ and $\epsilon$, respectively. Our numerical
experiments show that the Matrix Pencil method achieves the above optimal
bounds when resolving the positive sources.
