common Fourier series

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

We take $[0,1]$ as the model interval, with the $n$ th Fourier coefficient of a function $f$ defined as

$\widehat{f}(n)=\int_{0}^{1}f(t)\,e^{-2\pi int}\,dt\,,\quad n\in\mathbb{Z}\,.$

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 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.

1 Square wave

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

{makeimage}

	$\displaystyle\widehat{f}(n)$	$\displaystyle=\begin{cases}0\,,&n=0\\ \dfrac{1-e^{i\pi n}}{-2\pi in}\,,&n\neq 0\end{cases}$
		$\displaystyle=\begin{cases}0\,,&\text{$n$ is even}\\ -\dfrac{1}{\pi in}\,,&\text{$n$ is odd.}\end{cases}$
	$\displaystyle f(x)$	$\displaystyle\sim-\frac{2}{\pi}\,\sum_{n\in\mathbb{N}\text{ odd }}\,\frac{1}{n% }\sin 2\pi nt\,.$

2 Sawtooth wave

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

{makeimage}

	$\displaystyle\widehat{f}(n)$	$\displaystyle=\begin{cases}0\,,&n=0\\ -\dfrac{1}{2\pi in}\,,&n\neq 0\,.\end{cases}$
	$\displaystyle f(t)$	$\displaystyle\sim-\frac{1}{\pi}\,\sum_{n\in\mathbb{N}}\,\frac{1}{n}\sin 2\pi nt\,.$

3 Triangular wave

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

{makeimage}

	$\displaystyle\widehat{f}(n)$	$\displaystyle=\begin{cases}0\,,&n=0\\ -\dfrac{1-e^{i\pi n}}{2\pi^{2}n^{2}}\,,&n\neq 0\,.\end{cases}$
		$\displaystyle=\begin{cases}0\,,&\text{$n$ is even}\\ -\dfrac{1}{\pi^{2}n^{2}}\,,&\text{$n$ is odd}\,.\end{cases}$
	$\displaystyle f(t)$	$\displaystyle\sim-\frac{2}{\pi^{2}}\,\sum_{n\in\mathbb{N}\text{ odd}}\,\frac{1% }{n^{2}}\cos 2\pi nt\,.$

Title	common Fourier series
Canonical name	CommonFourierSeries
Date of creation	2013-03-22 15:41:51
Last modified on	2013-03-22 15:41:51
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	8
Author	stevecheng (10074)
Entry type	Example
Classification	msc 42A16
Related topic	SawBladeFunction
Related topic	MinimalityPropertyOfFourierCoefficients