本文提出一种快速稳定的非均匀离散傅里叶变换算法，利用低秩近似将非均匀变换矩阵分解为对角缩放的傅里叶矩阵之和，实现O(N log N log(1/ε)/log log(1/ε))次运算，适用于任意精度。
摘要：
通过将非均匀离散傅里叶变换（NUDFT）视为均匀离散傅里叶变换的扰动版本，本文提出了一种快速、稳定且简单的计算NUDFT的算法，其运算复杂度为O(N log N log(1/ε)/log log(1/ε))，其中N是变换大小，0<ε<1是工作精度。关键观察是，非均匀变换矩阵与DFT矩阵逐元素相除通常可以被低秩矩阵很好地近似，从而将非均匀变换矩阵表示为对角缩放的DFT矩阵之和。该算法易于实现，自动适应任何工作精度，并与最先进的算法竞争。在完全均匀情况下，该算法本质上就是FFT。此外，本文还描述了逆非均匀变换和二维非均匀变换的准最优算法。
英文摘要：
By viewing the nonuniform discrete Fourier transform (NUDFT) as a perturbed version of a uniform discrete Fourier transform, we propose a fast, stable, and simple algorithm for computing the NUDFT that costs $\mathcal{O}(N\log N\log(1/ε)/\log\!\log(1/ε))$ operations based on the fast Fourier transform, where $N$ is the size of the transform and $0<ε<1$ is a working precision. Our key observation is that a NUDFT and DFT matrix divided entry-by-entry is often well-approximated by a low rank matrix, allowing us to express a NUDFT matrix as a sum of diagonally-scaled DFT matrices. Our algorithm is simple to implement, automatically adapts to any working precision, and is competitive with state-of-the-art algorithms. In the fully uniform case, our algorithm is essentially the FFT. We also describe quasi-optimal algorithms for the inverse NUDFT and two-dimensional NUDFTs.
A Superfast Direct Solver for 2D Type-II Inverse Nonuniform Discrete Fourier Transform Based on Hierarchically Semiseparable Matrices
School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）；Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University（复旦大学当代应用数学上海市重点实验室）
Nonuniform Iterative Phasing Framework and Sampling Requirements for 3D Dynamical Inversion from Coherent Surface Scattering Imaging
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Advanced Photon Source, Argonne National Laboratory（阿贡国家实验室先进光子源）；University of California, Berkeley（加州大学伯克利分校）
Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT
National University of Singapore（新加坡国立大学）；New Jersey Institute of Technology（新泽理工学院）；Université de Lorraine（洛林大学）；Inria Nancy Grand-Est（法国国家信息与自动化研究院南斯特大东部中心）；University Wien（维也纳大学）
