The FFT algorithm that implements the discrete Fourier transform is considered one of the top ten algorithms of the $20$th century. Its main strengths are the low computational cost of $\mathcal{O}(n \log n$) and its stability. It is one of the most commonly used algorithms to analyze signals with a dense frequency representation. In recent years there has been an increasing interest in sparse signal representations and a need for algorithms that exploit such structure. We propose a new technique that combines the properties of the discrete Fourier transform with the sparsity of the signal. This is achieved by integrating ideas of Prony's method into Fourier's method. The resulting technique has the same frequency resolution as the original FFT algorithm but uses fewer samples and can achieve a lower computational cost. Moreover, the proposed algorithm is well suited for a parallel implementation.
Sparse phase retrieval of one-dimensional signals by Prony's method
Institut für Mathematik und Wissenschaftliches Rechnen Karl-Franzens-Universität Graz（格拉茨大学数学与科学计算研究所）；Institut für Numerische und Angewandte Mathematik Georg-August-Universität Göttingen（哥廷根大学数值与应用数学研究所）
Accelerating Phase Field Simulations Through a Hybrid Adaptive Fourier Neural Operator with U-Net Backbone
Sandia National Laboratories（桑迪亚国家实验室）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University of California, Berkeley（加州大学伯克利分校）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
Contour-integral and Fourier transform based multivariable quantum eigenvalue transformation for commuting matrices
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）
Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation
Aston University（阿斯顿大学）；B. Verkin Institute for Low Temperature Physics and Engineering（B.维尔金低温物理与工程研究所）；V. N. Karazin Kharkiv National University（V.N.卡拉津哈尔科夫国立大学）
