Question on fourier series convergence

[member 428835]

hey pf!

if we have a piecewise-smooth function $f(x)$ and we create a Fourier series $f_n(x)$ for it, will our Fourier series always have the 9% overshoot (gibbs phenomenon), and thus $\lim_{n \rightarrow \infty} f_n(x) \neq f(x)$?

thanks!

 

[jbunniii]

First, not every Fourier series exhibits the overshoot/Gibbs phenomenon. If the function is smooth (a special case of piecewise smooth), then the convergence of the Fourier series is uniform, and there is no Gibbs effect.

The Gibbs effect occurs at jump discontinuities. There are many examples of functions with jump discontinuities where the Fourier series converges pointwise at all points of continuity. For example, a "square wave" exhibits the Gibbs effect but the series converges pointwise everywhere. At the points of continuity, it converges to the original function, and at the points of discontinuity, it converges to the midpoint between the upper and lower values of the "square wave." Any partial sum will show the Gibbs effect, but as you sum more and more terms, the main part of the "ripple" is limited to smaller and smaller neighborhoods around the discontinuities. In the limit, it goes away entirely.