XKCD and Fourier

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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

Finally, you realized that thinking like this ruins the joke.