(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (194)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

common Fourier series

(Example)

This entry gives some examples of commonly encountered periodic functions and their Fourier series, with graphs to show the speed of convergence.

We take as the model interval, with the th Fourier coefficient of a function defined as

$\displaystyle \widehat{f}(n) = \int_0^1 f(t) e^{-2\pi i n t} dt\„ \quad n \in \mathbb{Z} .$

The parameters of the functions in the examples have been chosen to attempt to minimize the complexity of $\widehat{f}(0)$ . But the Fourier coefficients for the most common variations of the functions given below are easily derived by taking the appropriate linear transformations on the coefficients given.

We do not dwell on the convergence of the Fourier series for each function, although we note that by a theorem of Dirichlet, the Fourier series for each function (each of bounded variation) converges uniformly on any compact interval where the function is continuous.

Square wave

$\displaystyle f(t) = \begin{cases} -\frac{1}{2}\„ & t < \frac{1}{2} \ +\frac{1}{2}\„ & t > \frac{1}{2} . \end{cases}$

$\begin{align*} \widehat{f}(n) &= \begin{cases} 0\„ & n = 0\ \dfrac{1 - e^{i\p... ...\sum_{n \in \mathbb{N}\text{ odd }} \frac{1}{n} \sin 2\pi n t . \end{align*}$

**Figure:** Square wave function
$\includegraphics{graphs-1.eps}$

Sawtooth wave

$\displaystyle f(t) = t - \frac{1}{2} .$

$\begin{align*} \widehat{f}(n) &= \begin{cases} 0\„ & n=0 \ -\dfrac{1}{2\pi in... ...{1}{\pi} \sum_{n \in \mathbb{N}} \frac{1}{n} \sin 2\pi n t . \end{align*}$

**Figure:** Sawtooth wave function
$\includegraphics{graphs-2.eps}$

Triangular wave

$\displaystyle f(t) = \begin{cases} t - \frac{1}{4} \„ & t < \frac{1}{2} \ -t + \frac{3}{4} \„ & t > \frac{1}{2} . \end{cases}$

$\begin{align*} \widehat{f}(n) &= \begin{cases} 0\„ & n=0 \ - \dfrac{1-e^{i\pi... ...sum_{n \in \mathbb{N}\text{ odd}} \frac{1}{n^2} \cos 2\pi n t . \end{align*}$

**Figure:** Triangular wave function
$\includegraphics{graphs-3.eps}$

Anyone with an account can edit this entry. Please help improve it!

"common Fourier series" is owned by stevecheng.

(view preamble)