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In this paper we establish accuracy bounds of Prony's method (PM) forrecovery of sparse measures from incomplete and noisy frequency measurements,or the so-called problem of super-resolution, when the minimal separationbetween the points in the support of the measure may be much smaller than theRayleigh limit. In particular, we show that PM is optimal with respect to thepreviously established min-max bound for the problem, in the setting when themeasurement bandwidth is constant, with the minimal separation going to zero.Our main technical contribution is an accurate analysis of the inter-relationsbetween the different errors in each step of PM, resulting in previouslyunnoticed cancellations. We also prove that PM is numerically stable infinite-precision arithmetic. We believe our analysis will pave the way toproviding accurate analysis of known algorithms for the super-resolutionproblem in full generality.

On the accuracy of Prony's method for recovery of exponential sums with closely spaced exponents