XKCD and Fourier

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] If you’ve done Fourier series approximations of functions, you had three reactions to today’s XKCD comic: First, of course, was a smile of recognition. Then came a “wait a minute” as you realized that the Gibbs phenomenon doesn’t get smaller as you add more terms to the Fourier expansion; it just concentrates itself at the corners: That ringing never goes away. Here, for example, is the plot for ∑k=1,3,5,…994kπsin⁡(kπx) Finally, you realized that thinking like this ruins the joke.