Fourier Series & Gibbs Phenomenon

Gibbs Phenomenon (1899): When approximating a function with a jump discontinuity by its Fourier series, the partial sum overshoots the jump by ~8.9% (exactly π⁻¹∫₀^π sinc(t)dt − ½ ≈ 0.0895) regardless of how many terms are added. This is NOT a numerical error — it is an intrinsic mathematical property. Adding more terms makes the overshoot narrower but not smaller. The anomaly was first noticed by Wilbraham (1848) and later independently by Gibbs. It has implications for signal processing: Gibbs ringing affects JPEG compression and MRI reconstruction. The Lanczos σ-factor (sinc windowing) can suppress it.