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 .

Andrey Akinshin

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