The final subject we shall consider
is the convergence of Fourier series. I shall show two
examples, closely linked, but with radically different
behaviour.
for ; for . |
for ; for . |
Note that is the derivative of .
It is not very hard to find the relevant Fourier series,
Let us compare the partial sums, where we let the sum in the Fourier series run from to instead of . We note a marked difference between the two cases. The convergence of the Fourier series of is uneventful, and after a few steps it is hard to see a difference between the partial sums, as well as between the partial sums and . For , the square wave, we see a surprising result: Even though the approximation gets better and better in the (flat) middle, there is a finite (and constant!) overshoot near the jump. The area of this overshoot becomes smaller and smaller as we increase . This is called the Gibbs phenomenon (after its discoverer). It can be shown that for any function with a discontinuity such an effect is present, and that the size of the overshoot only depends on the size of the discontinuity! A final, slightly more interesting version of this picture, is shown in Fig. 4.6 .