ca.analysis and odes - What does Gibbs phenomenon shows the nature of Fourier Series

Hi there,

I think there is a bit more to answering your question than considering just the strict $L^2$ convergence however. The Gibbs phenomenon is important when considering the pointwise convergence of the partial sums of the fourier series. When $f^{prime}$ is continuous on a compact interval, you will get pointwise convergence of the partial sums $S_N$ so $S_N(x) to f(x)$ as $N to infty$. (You only get uniform convergence if in addition the function $f$ is compatible with the boundary conditions for your expansion but this is not your question anyway).

Now what happens when $f$ is discontinuous? It turns out that $S_N(x) to frac{1}{2}[f(x_-) + f(x_+)]$, the average of the left and right limits of the function. However, this is really only true for $N to infty$. Otherwise for any finite $N$ there is a small width of order $1/N$ around your discontinuous point $x$ where your partial sums are uniformly bounded away from either value $f(x_+)$, $f(x_-)$ by some fixed percentage (I recall $9$% of the jump size or something but don't quote me on that). Check out the photos at:http://en.wikipedia.org/wiki/Gibbs_phenomenon
That little wiggle of the wave where the jump of $f$ occurs stays uniformly bounded away from the value of the function but the width of this region ($1/N$) goes to zero as $N to infty$ so that technically you still get the full pointwise convergence.