Gibbs Phenomenon

Solution 1:

Chapter 3 of the book The Gibbs phenomenon in Fourier analysis, splines, and wavelet approximations by Abdul J. Jerri seems to cover the case of a general orthogonal expansion along eigenfunctions of a Sturm-Liouville problem.

Solution 2:

The periodic jump function defined by $j(t):=(\pi -t)/2$ on $]0,2\pi[$ has $\sum_{k=1}^\infty \sin(kt)/k$ as its Fourier series. We now compute the value of the partial sum $s_N$ at the point $t_N:=\pi/N$: $$s_N(t_N)=\sum_{k=1}^N{\sin(k\pi/N)\over k}= \sum_{k=1}^N{\pi\over N}{\sin(k\pi/N)\over k\pi/N} \doteq \int_0^\pi {\rm sinc}(t)dt\doteq 1.852$$ (the last sum can be interpreted as a Riemann sum for the sinc-integral). On the other hand $\lim_{t\to 0+} j(t)=\pi/2\doteq 1.571$ so that we have a guaranteed overshoot of about 17%.

The choice of the point $t_N$ is nearly "optimal", but I won't go into this here.

Solution 3:

Here is my attempted proof, sorry I don't know how to convert Word into Latex