common Fourier series

This entry gives some examples of commonly encountered periodic functionsand 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

$$\widehat{f}(n)={\int }_0^{1}f(t)e^{-2\pi int}𝑑t,n\in \mathbb{Z}.$$

The parameters of the functions in the exampleshave been chosen to attempt to minimizethe complexity of $\widehat{f}(0)$. But the Fourier coefficients forthe most common of the functions given below are easily derivedby 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

$\widehat{f}(n)$

$f(x)$$\sim -{\displaystyle \frac{2}{\pi }}{\displaystyle \sum _{n\in \mathbb{N}\text{ odd }}}{\displaystyle \frac{1}{n}}\sin 2\pi nt.$

Figure 1: Square wave function

2 Sawtooth wave

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

$\widehat{f}(n)$

$f(t)$$\sim -{\displaystyle \frac{1}{\pi }}{\displaystyle \sum _{n\in \mathbb{N}}}{\displaystyle \frac{1}{n}}\sin 2\pi nt.$

Figure 2: Sawtooth wave function

3 Triangular wave

$\widehat{f}(n)$

$f(t)$$\sim -{\displaystyle \frac{2}{{\pi }^2}}{\displaystyle \sum _{n\in \mathbb{N}\text{ odd}}}{\displaystyle \frac{1}{n^2}}\cos 2\pi nt.$

Figure 3: Triangular wave function