Gibbs phenomenon

The persistent overshoot and oscillation near a jump discontinuity in truncated Fourier series. Increasing the number of terms narrows the oscillatory region but does not eliminate the limiting overshoot.

Fourier series: A representation of a periodic function as a sum of sine and cosine terms. The phenomenon arises when a Fourier series is cut off at a finite number of terms.
Henry Wilbraham: A British mathematician who described the Fourier-series overshoot later associated with Gibbs. Wilbraham published an early account of the phenomenon decades before Gibbs analyzed it.
Jump discontinuity: A discontinuity where the one-sided limits of a function exist but differ. This is the kind of singularity that triggers the characteristic Fourier overshoot.
Fejér summation: A method that averages successive Fourier partial sums to improve convergence. Averaging suppresses the persistent overshoot that appears in ordinary partial sums.
Spectral method: A numerical method that approximates solutions using global basis functions such as Fourier modes. Discontinuities in computed solutions can trigger Gibbs oscillations and reduce accuracy.
Fourier partial sum: The finite sum of the first terms of a Fourier series. Its behavior near a jump produces the characteristic overshoot and oscillation.
Josiah Willard Gibbs: An American mathematical physicist known for foundational work in thermodynamics and mathematical physics. Gibbs's 1899 analysis brought the overshoot to wider mathematical attention.
Periodic function: A function whose values repeat after a fixed nonzero interval. Fourier series represent periodic extensions, so endpoint jumps can create the same effect.
Lanczos sigma factor: A spectral filtering factor that damps high-frequency Fourier coefficients. It reduces truncation ripples by tapering modes instead of abruptly cutting them off.
Fourier transform: An integral transform that expresses a signal in terms of its frequencies. Finite-bandwidth Fourier reconstruction exhibits ringing analogous to the series phenomenon.

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