common Fourier series

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

We take [0,1] as the model interval, with the nth Fourier coefficient of a functionMathworldPlanetmath f defined as


The parameters of the functions in the examples have been chosen to attempt to minimize the complexity of 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 variationMathworldPlanetmath) converges uniformly on any compact interval where the function is continuousMathworldPlanetmath.

1 Square wave

f^(n) ={0,n=01-eiπn-2πin,n0
={0,n is even-1πin,n is odd.
f(x) -2πn odd 1nsin2πnt.
Figure 1: Square wave function

2 Sawtooth wave

f^(n) ={0,n=0-12πin,n0.
f(t) -1πn1nsin2πnt.
Figure 2: Sawtooth wave function

3 Triangular wave

f^(n) ={0,n=0-1-eiπn2π2n2,n0.
={0,n is even-1π2n2,n is odd.
f(t) -2π2n odd1n2cos2πnt.
Figure 3: Triangular wave function
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