A good mathematical explanation of Gibbs phenomenon

I was explaining to someone how Fourier series work in context of constructing signals that are not everywhere differentiable, e.g. square waves, sawtooth waves, etc. When I mentioned the Gibbs phenomenon however, I realized that I never really learned of why it happens. In fact, as the story goes, not everyone even realized that it's an actual mathematical property of infinite series of periodic signals and not a computational fluke, and it turns out that most proofs are fairly laborious and elaborate.

After reading several of them, I started realizing why such phenomenon might occur, but I have a background in real and complex analysis, topology and so on. The question is can I fully explain and rigorously prove Gibbs phenomenon mathematically to someone with only the basic undergraduate calculus courses in their arsenal (or any other general prerequisites for an undergraduate signal processing course)? If so, then how?

• IMHO, the Wikipedia article on the Gibbs phenomenon is actually quite well written. Is that what you are looking for or do you need something else? en.wikipedia.org/wiki/Gibbs_phenomenon – Hilmar Jan 16 '12 at 0:06
• I've always found the phenomenon fascinating. One of the more surprising details with respect to a Fourier series truncated to a finite length is that as you increase the number of terms in the sum, the Gibbs oscillations get compressed in time, but the magnitude of overshoot is constant. A long time ago, I was given a great explanation of why in an undergraduate course, but I don't think I wrote it down. – Jason R Jan 16 '12 at 13:56

You can always say that sin and cos has curved shape, and you need infinite amount of frequencies to form a sharp edge from a many curved shapes.