标题： 广义傅里叶级数：一种针对非周期函数的N log2(N)扩展，可消除吉布斯振荡 显示英文标题

标题： Generalized Fourier Series: An N log2(N) extension for aperiodic functions that eliminates Gibbs oscillations

摘要： 本文介绍了广义傅里叶级数（GFS），这是一种新颖的谱方法，将经典的傅里叶级数扩展到非周期函数。 GFS通过将函数分解为周期性和非周期性分量，解决了非周期情况下出现的吉布斯现象和收敛性差等关键问题。 周期部分使用标准傅里叶模态表示，并通过快速傅里叶变换（FFT）高效计算。 非周期部分采用自适应、低秩的正弦函数，具有非谐波模式，动态调整以捕捉域边界处的不连续性和导数跳跃。 与传统傅里叶扩展方法不同， GFS在不需要计算域扩展的情况下实现高精度，提供了非周期函数的紧凑且高效的表示。 自适应低秩方法在确保精度的同时最小化计算开销，通常涉及非周期部分的额外复杂数值模式。 此外，GFS表现出高分辨率能力，在周期域中的自由度与FFT相当，并保持N log2(N)的计算复杂度。 通过数值实验验证了GFS的有效性，展示了其在非周期域中准确逼近函数及其导数的能力。 凭借其稳健的框架和最小的计算成本，GFS在数值偏微分方程、信号处理、机器学习和计算物理等应用领域具有显著潜力，为高精度函数逼近提供了一个稳健且高效的工具。 显示英文摘要

摘要： This article introduces the Generalized Fourier Series (GFS), a novel spectral method that extends the clas- sical Fourier series to non-periodic functions. GFS addresses key challenges such as the Gibbs phenomenon and poor convergence in non-periodic settings by decomposing functions into periodic and aperiodic com- ponents. The periodic part is represented using standard Fourier modes and efficiently computed via the Fast Fourier Transform (FFT). The aperiodic component employs adaptive, low-rank sinusoidal functions with non-harmonic modes, dynamically tuned to capture discontinuities and derivative jumps across domain boundaries. Unlike conventional Fourier extension methods, GFS achieves high accuracy without requiring compu- tational domain extensions, offering a compact and efficient representation of non-periodic functions. The adaptive low-rank approach ensures accuracy while minimizing computational overhead, typically involving additional complex modes for the aperiodic part. Furthermore, GFS demonstrates a high-resolution power, with degrees of freedom comparable to FFT in periodic domains, and maintains N log2(N) computational complexity. The effectiveness of GFS is validated through numerical experiments, showcasing its ability to approximate functions and their derivatives in non-periodic domains accurately. With its robust framework and minimal computational cost, GFS holds significant potential for advancing applications in numerical PDEs, signal processing, machine learning, and computational physics by providing a robust and efficient tool for high-accuracy function approximations.