Partial Fourier series overshoot at discontinuities — the ~9% that never vanishes
5
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Max Overshoot
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% of Discontinuity
~8.9%
Gibbs Limit
The Gibbs phenomenon (Wilbraham 1848, rediscovered Gibbs 1899): partial Fourier sums near a jump discontinuity overshoot by ~9% of the jump height, regardless of how many terms are added.
The overshoot converges to exactly (Si(π)/π − ½)·2 ≈ 8.9% of the jump. More terms push the spike closer to the discontinuity but cannot eliminate it.
This matters in signal processing (ringing artifacts), image compression (JPEG ringing), and numerical PDE solvers.