Theorem: If a periodic function of period \( 2\ell \) is square-integrable on any finite interval, then the Fourier series converges to the function at almost every point:
\[
f(x) \,\sim \, \frac{a_0}{2} + \sum_{k\ge 0} \left[ a_k \cos \frac{k \pi x}{\ell} + b_k \sin \frac{k \pi x}{\ell} \right] ,
\]
where its coefficients are determined via Euler--Fourier formulas:
\[
\begin{split} a_k &= \frac{1}{\ell} \, \int_{-\ell}^{\ell} f(x)\,\cos \frac{k \pi x}{\ell} \, {\text d} x , \quad k=0,1,2,\ldots ,
\\
b_k &= \frac{1}{\ell} \, \int_{-\ell}^{\ell} f(x)\,\sin \frac{k \pi x}{\ell} \, {\text d} x , \quad k=1,2,\ldots .
\end{split} \qquad\qquad ■
\]
Some example:
\[
\sum_{n\ge 1} \frac{(-1)^{n+1}}{2n-1} \, \cos (2n-1) x =
\begin{cases}
\frac{\pi}{4} , & \ \mbox{for } |x|< \frac{\pi}{2} , \\ -\frac{\pi}{4} , & \ \mbox{for } |x|> \frac{\pi}{2} ,
\end{cases} \qquad \mbox{on interval } \ -\pi < x < \pi . \]
\[
\sum_{n\ge 1} \frac{(-1)^{n+1}}{2n-1} \, \sin (2n-1) x = \frac{x}{2} , \qquad \mbox{on interval } \ -\pi < x < \pi . \]
\[
e^x = \frac{2\,\sinh \pi}{\pi} \left[ \frac{1}{2} + \sum_{n\ge 0} \frac{(-1)^n}{1+n^2} \left( \cos nx -n\,\sin nx \right) \right] , \qquad \mbox{on interval } \ |x|< \pi . \]
\[
\sin x = \frac{2}{\pi} - \frac{4}{\pi} \,\sum_{n\ge 1} \frac{1}{4n^2 -1} \, \cos 2nx , \qquad \mbox{on interval } \ 0\le x < \pi . \]
\[
\sum_{n\ge 1} \frac{1}{2n-1} \, \cos (2n-1) x = \frac{1}{2}\,\ln \cot \frac{x}{2}
\qquad \mbox{on interval } \ 0 < x < \pi . \]
\[
\sum_{n\ge 1} \frac{1}{2n-1} \, \sin (2n-1) x = \frac{\pi}{4} \times \begin{cases}
1 , & \ \mbox{for } 0< x , \\ - 1 , & \ \mbox{for } x<0 , \end{cases} \qquad \mbox{on interval } \ -\pi < x < \pi . \]
\[
\frac{4}{\pi}\,\sum_{n\ge 1} \frac{1}{2n-1} \, \sin \frac{(2n-1)\pi x}{\ell} = 2 \left[ H\left( \frac{x}{\ell} \right) - H\left( \frac{x}{\ell} -1 \right) \right] -1 .
\]
\[
\sum_{n\ge 1} \frac{1}{n} \, \cos n x = -\frac{1}{2} \, \ln \left[ 2 \left( 1 - \cos x \right) \right] , \qquad \mbox{on interval } \ 0 < x < 2\pi ; \]
\[
\sum_{n\ge 1} \frac{1}{n} \, \sin n x = \begin{cases}
\frac{\pi -x}{2} , & \ \mbox{for } 0< x , \\ - \frac{\pi +x}{2} , & \ \mbox{for } x<0 , \end{cases} \qquad \mbox{on interval } \ -\pi < x < \pi . \]
\[
\sum_{n\ge 1} \frac{1}{n} \, \sin \frac{n\pi x}{\ell} = \frac{\ell-x}{2\ell}
.
\]
\[
\begin{split}
\sum_{n\ge 1} \frac{(-1)^{n+1}}{n} \, \cos n x &= \ln \left( 2 \, \cos \frac{x}{2} \right) , \qquad \mbox{on interval } \ |x| < \pi ; \\ \sum_{n\ge 1} \frac{(-1)^{n+1}}{n} \, \sin n x &=\frac{x}{2} , \qquad \mbox{on interval } \ |x| < \pi .
\end{split} \]
\[
\begin{split}
\frac{\pi^2}{12} + \sum_{n\ge 1} \frac{(-1)^{n+1}}{n^2} \, \cos n x &= x^2 , \qquad \mbox{on interval } \ |x| < \pi ; \\ \sum_{n\ge 1} \frac{(-1)^{n+1}}{n^2} \, \sin n x &=?? , \qquad \mbox{on interval } \ |x| < \pi . \end{split} \]
\[
\begin{split}
\frac{\pi^2}{16} + \sum_{n\ge 1} \frac{1}{n^2} \, \cos n x &= \frac{x^2}{4} - \frac{\pi x}{4} , \qquad \mbox{on interval } \ 0< x < < 2\pi ; \\ \sum_{n\ge 1} \frac{1}{n^2} \, \sin n x &=?? , \qquad \mbox{on interval } \ |x| < \pi .
\end{split} \]
\[
\begin{split}
\sum_{n\ge 1} \frac{1}{(2n-1)^2} \, \cos (2n-1) x &= \frac{1}{2} - \frac{\pi |x|}{4} , \qquad \mbox{on interval } \ |x|< < \pi ; \\ \sum_{n\ge 1} \frac{1}{(2n-1)^2} \, \sin (2n-1) x &=?? , \qquad \mbox{on interval } \ |x| < \pi .
\end{split} \]
\[
\begin{split}
\sum_{n\ge 1} \frac{1}{n^3} \, \cos n x &= ?? , \qquad \mbox{on interval } \ 0< x < < 2\pi ; \\ \sum_{n\ge 1} \frac{1}{n^3} \, \sin n x &=\frac{\pi^2 x}{6} - \frac{\pi x^2}{4} + \frac{x^3}{12} , \qquad \mbox{on interval } \ 0 < x < 2\pi
. \end{split} \]
Cesaro Approximation
Gibbs Phenomenon
Even and Odd Functions
Chebyshev expantion
Legendre expansion
Bessel--Fourier Series
Hermite Expansion
Laguerre Expansion
Motivated examples