# Fourier series of a periodic function with infinite number of extreme points

My formula booklet in signals analysis states that a condition for the Fourier series of a periodic function $x(t)$ to exist is that one period of $x(t)$ contains a finite number of maxima and minima.

Is this true, and does it mean that I can't find the Fourier series of a square wave? (Because I just did.)

• The series will converge to the right value at all continuous points. At discontinuous ones, it converges to the Cauchy principal value (average of the right- and left- sided limits). You can check that your Fourier series gives exactly this type of behaviour. – FusRoDah Oct 8 '17 at 19:29

## 1 Answer

1. The ("formal") Fourier series $\sum_k\ c_k\ e^{ikt}$ of a given $2\pi$-periodic function $t\mapsto x(t)$ "exists" as soon as $x(\cdot)$ is integrable over an interval of length $2\pi$, so that the coefficients $c_k$ can be computed.

2. The difficult question is: For which $t$ this series converges to the given function value $x(t)\$? The series may be useful for the study of $x(\cdot)$ even if for some $t$ it does not converge to the desired value (as, e.g., in the case of a square wave).

3. A sufficient condition for $\sum_k\ c_k\ e^{ikt}=x(t)$ for all $t$ is the following: The given function should be continuous and of bounded variation on ${\mathbb R}/(2\pi)$. (A square wave is not continuous.) Such a function may very well (but usuallly doesn't) have infinitely many proper local maxima and minima, let alone be constant on subintervals. Consider, e.g., the following function, extended periodically to all of ${\mathbb R}$:

$$x(t)\ :=\ t^2\ \cos{1\over t}\quad\bigl(0<|t|\leq\pi\bigr), \qquad x(0):=0\ .$$