Note: For a given number, N, the partial sum S_N of the Fourier series for the top function provides a "better" approximation to the function than the F.S. for the bottom one. This illustrates the negative effect of "singularities", in this case discontinuities, on the convergence of Fourier series. The partial sums S_N of a Fourier series are finite linear combinations of continuous functions and therefore continuous functions themselves. In the bottom case, these continuous functions are trying to approximate a discontinuous function. The effect is clearly seen in the bottom plots. Near the points of discontinuity, the partial sums exhibit oscillations of higher amplitude. These are examples of the so-called "Gibbs phenomenon", often referred to as "ringing". This "ringing" phenomenon is well known in signal and image processing, an example of which is shown on the next page.
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Experiment No. 1: Truncation of high frequencies
Top left: n = 160. Top right: n = 200. Bottom left: n = 220. Bottom right: n = 240.
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Here is an example of "ringing" exhibited in two-dimensional Fourier series-type approximations to image functions. In the above figures, as n is increased, we are removing more and more "high frequency" terms from the Fourier series (corresponding to partial sums S_N of lower N value). You can see the effects of these truncations on the image near edges, in particular, the various masts on the boat. After all, edges represent discontinuities of greyscale values in an image.