## 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 $f(x) \sim \sum_{l=0}^\infty \frac{2\sin\bigl(2\pi(2l+1)x\bigr)}{\pi(2l+1)}, \quad g(x) \sim 1 - \sum_{n=1}^\infty \frac{\sin(2\pi nx)}{\pi n}.$