What does Gibbs phenomenon shows the nature of Fourier Series

As the title shows,we know that there is some points the series not approaching to the function.

Now,take the convergence theorem into consideration.As there is some the first break-points，the series is still convergent.And,the Gibbs phenomenon always takes place on the first break-points.

Why does Gibbs phenomenon take place?What does it show the nature of Fourier Series?

• I'm having trouble understanding the 3rd sentence. Could you explain further what you mean? – Yemon Choi Mar 7 '10 at 19:15
• @Yemon Choi: 我也没看明白。 – Sunni Mar 8 '10 at 1:27
• this means the Gibbs phenomenon always takes place on the first break-points. – DarkLight May 19 '10 at 8:34
• Do you mean discontinuities? – bright-star Mar 28 '14 at 2:04

A Fourier series truncated to order $n$ is the best approximation to the given function in the $L^2$ sense using trigonometric polynomials of order $n$. As such, small rapid deviations don't matter much. Since there is a limit to how big the derivatives of a trigonometric polynomial of fixed order can be (without the coefficients being big), in order to fit such a polynomial to a discontinuity it pays to overshoot a bit on each side of the discontinuity in order to “gather speed” so you can get from one value to the other fast. When I say it “pays”, i mean to say that you what you lose by not approximating the function too well at the overshoot, you more than gain back by doing the jump faster.
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.