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.

中文翻译：

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论Prony方法恢复具有紧密间隔指数的指数和的准确性


在本文中，我们建立了 Prony 方法 (PM) 的精度范围，用于从不完整和噪声频率测量中恢复稀疏测量，或所谓的超分辨率问题，当测量支持中的点之间的最小间隔可能是远小于瑞利极限。特别是，我们表明，在测量带宽恒定且最小间隔为零的情况下，PM 相对于之前为该问题建立的最小-最大界限而言是最佳的。我们的主要技术贡献是准确分析 PM 每个步骤中不同错误之间的相互关系，从而导致以前未被注意到的取消。我们还证明 PM 在有限精度算术中是数值稳定的。我们相信我们的分析将为全面通用地对超分辨率问题的已知算法进行准确分析铺平道路。