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.

翻译：本文介绍了广义傅里叶级数（GFS），这是一种新颖的谱方法，将经典傅里叶级数推广至非周期函数。GFS 通过将函数分解为周期分量与非周期分量，解决了非周期情形下的关键挑战，如吉布斯现象和收敛性差的问题。周期部分采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。非周期分量则采用自适应的低秩正弦函数，其非谐波模态经过动态调谐，以捕捉定义域边界处的间断点和导数跳跃。与传统的傅里叶延拓方法不同，GFS 无需扩展计算域即可实现高精度，为表示非周期函数提供了一种紧凑而高效的方案。这种自适应低秩方法在保证精度的同时，最小化了计算开销，通常仅需为周期部分引入额外的复模态。此外，GFS 展现出高分辨率能力，其自由度在周期域中与 FFT 相当，并保持 N log₂(N) 的计算复杂度。数值实验验证了 GFS 的有效性，展示了其在非周期域中精确逼近函数及其导数的能力。凭借其鲁棒的框架和极小的计算成本，GFS 为高精度函数逼近提供了一个强大而高效的工具，在数值偏微分方程、信号处理、机器学习以及计算物理等领域具有重要的应用潜力。