# common Fourier series

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.

## 1 Square wave

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

## 2 Sawtooth wave

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

## 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}
 $\displaystyle\widehat{f}(n)$ $\displaystyle=\begin{cases}0\,,&n=0\\ -\dfrac{1-e^{i\pi n}}{2\pi^{2}n^{2}}\,,&n\neq 0\,.\end{cases}$ $\displaystyle=\begin{cases}0\,,&\text{n is even}\\ -\dfrac{1}{\pi^{2}n^{2}}\,,&\text{n is odd}\,.\end{cases}$ $\displaystyle f(t)$ $\displaystyle\sim-\frac{2}{\pi^{2}}\,\sum_{n\in\mathbb{N}\text{ odd}}\,\frac{1% }{n^{2}}\cos 2\pi nt\,.$ Figure 3: Triangular wave function
Title common Fourier series CommonFourierSeries 2013-03-22 15:41:51 2013-03-22 15:41:51 stevecheng (10074) stevecheng (10074) 8 stevecheng (10074) Example msc 42A16 SawBladeFunction MinimalityPropertyOfFourierCoefficients