Partial Fourier sums approximating discontinuous functions. Near discontinuities, the overshoot converges to ≈8.9% of the jump — the Gibbs phenomenon — regardless of how many terms are added.
N = 1 | overshoot: 0.00%
Square wave: f(x) = (4/π)Σ sin((2k-1)x)/(2k-1). Sawtooth: f(x) = (2/π)Σ (-1)^{k+1} sin(kx)/k. The Gibbs phenomenon: as N→∞, the overshoot does not shrink but converges to (2/π)∫₀^π sinc(t)dt - 1 ≈ 8.9%. Cesàro summation (Fejér kernel) removes the Gibbs phenomenon by averaging partial sums.