Generalized Fourier Series: An Nlog2(N) extension for aperiodic functions that eliminates Gibbs oscillations

This article introduces the Generalized Fourier Series (GFS), a novel spectral method that extends the classical 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 components. The periodic part is represented using standard Fourier modes and efficiently computed via the Fast Fourier Transform (FFT). The aperiodic component employs a low-rank adaptive set of sinusoidal functions with non-harmonic complex modes, dynamically tuned to capture discontinuities and derivative jumps across domain boundaries while enabling both oscillatory and exponential (growth/decay) behavior. Unlike conventional Fourier extension methods, GFS achieves high accuracy without requiring computational 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 only a small number of 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 log 2 ( 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.