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

Narsimha Reddy Rapaka and Mohamed Kamel Riahi

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 249, issue C, 208-228

Abstract: 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.

Keywords: Fourier spectral method; Non-periodic domain; Gibbs oscillations; Non-harmonic modes; Complex Fourier modes; Fast solvers (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:249:y:2026:i:c:p:208-228

DOI: 10.1016/j.matcom.2026.05.013

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