Accuracy of spike-train Fourier reconstruction for colliding nodes

We consider a signal reconstruction problem for signals F of the form F(x) = Σ
^d
_j=1
a
_j
δ(x-x
_j
) from their Fourier transform F(F)(s) = ∫
^∞
_-∞
F(x)e
^-isx
dx. We assume F(F)(s) to be known for each s ε [-N,N] with an absolute error not exceeding ε > 0. We give an absolute lower bound (which is valid with any reconstruction method) for the “worst case” reconstruction error of F from F(F) for situations where the x
_j
nodes are known to form an I elements cluster contained in an interval of length h <;<; 1. Using “decimation” algorithm of [6], [7] we provide an upper bound for the reconstruction error, essentially of the same form as the lower one. Roughly, our main result states that for h of order 1/N ϵ
^1/2l-1
the worst case reconstruction error of the cluster nodes is of the same order 1/N ϵ
^1/2l-1
, and hence the inside configuration of the cluster nodes (in the worst case scenario) cannot be reconstructed at all. On the other hand, decimation algorithm reconstructs F with the accuracy of order 1/N ϵ
^1/2l
.