We consider a signal reconstruction problem for signals F of the form F(x) = Sigma(d)(j=1) a(j)delta (x - x(j)) from their Fourier transform F(F)(s) = integral(infinity)(-infinity) F(x)e(-isx) dx. We assume F(F)(s) to be known for each s is an element of [-N, N] with an absolute error not exceeding epsilon > 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 1 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 epsilon(1/2l - 1) the worst case reconstruction error of the cluster nodes is of the same order 1/N epsilon(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 epsilon(1/2l) .


Akinshin, Andrey;  Batenkov, Dmitry;  Yomdin, Yosef

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