Gibbs phenomenon of Fourier series

Let \(f(x)\) be an appropriate one-periodic function of \(x\in\mathbb{R}\). Its fourier series is given by \[ f(x) \sim \sum_{n=-\infty}^\infty C_ne^{2\pi i nx}, \quad C_n := \int_0^1 e^{-2\pi i nx} f(x) dx. \] The \(N\)-th partial sum of the series is denoted by \[ S_N(x) := \sum_{n=-N}^N C_ne^{2\pi i nx}. \] The well-known facts on the behavior of \(S_N\) as (\(N\rightarrow\infty\)) are the following:

There exists a continuous one-periodic function \(f(x)\) such that \(S_N(0)\rightarrow\infty\) (\(N\rightarrow\infty\)). This example shows that the continuity does not guarantee even the pointwise convergence of Fourier series.
However, if \(f(x)\) is Hörder continuous, that is, there exists an exponent \(\alpha\in(0,1]\) such that \(\lvert{f(x)-f(y)}\rvert\leqq\lvert{x-y}\rvert^\alpha\) for any \(x,y\in\mathbb{R}\), then \(S_N(x)\) converges to \(f(x)\) uniformly in \(x\in\mathbb[0,1]\): \[ \max_{x\in[0,1]} \lvert{S_N(x)-f(x)}\rvert \rightarrow 0 \quad (N\rightarrow\infty). \] This results show that if \(f(x)\) is a little bit smoother than continuous functions, then its Fourier series converges to \(f(x)\) uniformly. In particular,
Suppose that \(f(x)\) is piecewise \(C^1\) one-periodic funtion. The discontinuous points of \(f(x)\) in \([0,1)\) is denoted by \[0\leqq p_1 < \dotsb < p_m <1.\] Set \[ p_0:=p_m-1, \quad p_{m+1}:=p_1+1, \quad \delta_0 :=\frac{\min\{p_1-p_0,\dotsc,p_m-p_{m-1}\}}{2}. \] Then for any \(\delta\in(0,\delta_0]\) \[ \max_{x \in I(\delta)} \lvert{S_N(x)-f(x)}\rvert \rightarrow 0 \quad (N\rightarrow\infty), \quad I(\delta) := [0,1] \setminus \left( \bigcup_{k=0}^{m+1} (p_k-\delta,p_k+\delta) \right), \] \[ S_N(p_k) \rightarrow \frac{f(p_k-0)+f(p_k+0)}{2} \quad (N\rightarrow\infty), \quad k=1,\dotsc,m. \] Moreover \(S_N(x)\) vibrates violently near \(x=p_k\). This is said to be Gibbs phenomenon.

A MATLAB code gibbs.m. creates the animation of the Gibbs phenomenon of the Fourier series of one-periodic piecewise \(C^1\) functions given by \[ f(x) := \begin{cases} 1 &\ (n \leqq x \leqq n+1/2), \\ 0 &\ (n+1/2 < x < n+1), \end{cases} \quad g(x) := x-n \quad (n \leqq x < n+1), \quad n\in\mathbb{Z}. \] Their Fourier series are given by \[ f(x) \sim \sum_{l=0}^\infty \frac{2\sin\bigl(2\pi(2l+1)x\bigr)}{\pi(2l+1)}, \quad g(x) \sim 1 - \sum_{n=1}^\infty \frac{\sin(2\pi nx)}{\pi n}. \]