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

翻译：本文建立了普罗尼方法（PM）从不完整且含噪的频率测量中恢复稀疏测度的精度界，即当支撑点间最小间隔可能远小于瑞利极限时所谓的超分辨问题。特别地，我们证明在测量带宽恒定且最小间隔趋近于零的条件下，PM方法对于该问题先前建立的极小极大误差界具有最优性。本文的主要技术贡献在于精确分析了PM方法各步骤中不同误差之间的相互关联，揭示了先前未被注意到的误差抵消现象。我们还证明了PM在有限精度算术中具有数值稳定性。我们相信该分析将为全面建立超分辨问题已知算法的精确分析奠定基础。