7.3 The Gibbs phenomenon

Near a jump, the partial sums converge pointwise but not uniformly. For the square wave, \(\operatorname {sign}x\) on \((-\pi , \pi )\), with series \(\frac 4\pi \sum _{k\ \text {odd}}\frac {\sin kx}k\), the partial sums overshoot to about \(1.179\) just to the right of \(0\), however many terms are taken (Figure 10). The peak of \(S_N\) is near \(x = \pi /N\), and as \(N \to \infty \) its height tends to \[ \frac 2\pi \int _0^\pi \frac {\sin s}s\,ds = \frac 2\pi \operatorname {Si}(\pi ) = 1.17898 , \] an overshoot of \(8.95\%\) of the jump of \(2\). Wilbraham found this in 1848; Gibbs described it again in 1899. The overshoot moves towards the jump and gets narrower, so it does not prevent mean-square convergence, but it matters wherever discontinuous signals are truncated in frequency, as in filtering and spectral numerical methods. Averaging the partial sums (Fejér’s sums) removes it.

1
x1NNpk123456789sk.at17==rr=9tin352iag,5l6 sp|,ulub9mc| ssk∕bilee1nodtf tathe3 s:quare wave
k — Figure 10: Left: partial sums \(S_5\) and \(S_{25}\) of the Fourier series of the square wave \(\operatorname {sign}x\) (dashed). Near the jump each overshoots to about \(1.18\); more terms make the overshoot narrower, not lower. Right: the sine coefficients of a string plucked at one third of its length, relative to the first. They decay like \(1/k^2\), and every third harmonic is missing, because the pluck is at a node of those modes.

Example 7.3.1. A string of length \(1\) plucked at \(x = p\) to height \(h\) has initial shape \(hx/p\) on \([0, p]\) and \(h(1 - x)/(1 - p)\) on \([p, 1]\), and sine coefficients \[ b_k = \frac {2h\sin k\pi p}{k^2\pi ^2p(1 - p)} . \] They decay like \(1/k^2\), since the shape is continuous with a corner. The harmonic \(k\) is absent whenever \(kp\) is an integer: plucking at the middle silences all the even harmonics, and plucking at one third every third one (Figure 10). Guitarists choose where to pluck to control the tone.

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