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：在本文中,我们确定了普罗尼方法(PM)的准确性界限,以便从不完全和噪音的频率测量中恢复稀少的测量,或者所谓的超分辨率问题,而支持该测量的点之间的最小分离可能小于雷利限值。特别是,我们表明,在测量带宽不变、最小分离为零的环境下,普罗尼方法(PM)相对于先前为问题设定的最小最大约束值而言是最佳的。我们的主要技术贡献是对PM每一步中不同错误之间的相互关系的准确分析,从而导致先前未引起注意的取消。我们还证明,PM在限定精度的算术中,PM在数字上是稳定的。我们相信,我们的分析将为全面全面分析超分辨率问题的已知算法铺平铺平了道路。