Gibbs phenomenon of Fourier series
Let \(f(x)\) be an appropriate oneperiodic 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 wellknown facts on the behavior of \(S_N\) as (\(N\rightarrow\infty\))
are the following:

There exists a continuous oneperiodic 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
\(\lvert{f(x)f(y)}\rvert\leqq\lvert{xy}\rvert^\alpha\)
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 \(C^1\) oneperiodic funtion.
The discontinuous points of \(f(x)\) in \([0,1)\) is denoted by
\[0\leqq p_1 < \dotsb < p_m <1.\]
Set
\[
p_0:=p_m1,
\quad
p_{m+1}:=p_1+1,
\quad
\delta_0
:=\frac{\min\{p_1p_0,\dotsc,p_mp_{m1}\}}{2}.
\]
Then for any \(\delta\in(0,\delta_0]\)
\[
\max_{x \in I(\delta)}
\lvert{S_N(x)f(x)}\rvert
\rightarrow
0
\quad
(N\rightarrow\infty),
\quad
I(\delta)
:=
[0,1]
\setminus
\left(
\bigcup_{k=0}^{m+1}
(p_k\delta,p_k+\delta)
\right),
\]
\[
S_N(p_k)
\rightarrow
\frac{f(p_k0)+f(p_k+0)}{2}
\quad
(N\rightarrow\infty),
\quad
k=1,\dotsc,m.
\]
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 oneperiodic piecewise \(C^1\) functions given by
\[
f(x)
:=
\begin{cases}
1
&\ (n \leqq x \leqq n+1/2),
\\
0
&\ (n+1/2 < x < n+1),
\end{cases}
\quad
g(x)
:=
xn
\quad (n \leqq x < n+1),
\quad
n\in\mathbb{Z}.
\]
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}.
\]