Fourier Series Convergence at Discontinuity, Gibbs Phenomenon

Cuthbert Nyack

The square wave has discontinuities at -T/4 and +T/4. At either discontinuity, the Fourier Series converges to the mid point of the "jump". About either side of the jump the series oscillate. The height of the peaks of the oscillation decreases away from the jump, but the height of peak1, peak2 etc remain the same as the number of terms summed increases. The effect is referred to as the Gibbs phenomenon and is illustrated by the applet below.

The applet below illustrates the Gibbs phenomenon.
Fn = 1 shows the square wave reconstructed to Ns1, Ns2 terms and the corresponding errors.
Fn = 2 shows what happens if the red waveform is expanded horizontally by 3 (Ns1/Ns2 = 3). Near the edge the red sits on the green waveform.
Fn = 3,4; 5,6; 7,8; 9,10; 11,12; 13,14; and 15,16; shows cases with different Ns1/Ns2.

Fn = 17 shows a square wave with height of the discontinuity being set by 'a' and 'b'.
Fn = 18, 19 shows a case with the discontinuity = 0.5
Fn = 20, 21 shows a case with the discontinuity = 0.01
Fn = 22, 23 shows a case with the discontinuity = 0.05

Fn = 24 to 34 shows cases with a discontinuity in the first derivative, in these cases the Series does eventually converge the the correct limit.

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