Example 2

As a second example we consider the step function defined as :

$$f(x)=\begin{cases} -1 & -1\le x<0 \\ +1 & 0\le x<1 \end{cases}$$

This function has a discontinuity at \( x = 0 \)!

We again use SymPy to analytically calculate its Fourier series.

hat = sp.Piecewise((-1, sp.Interval(-1,0).contains(x)), 
                   (+1, sp.Interval(0,1).contains(x)))
fourier = sp.fourier_series(hat, (x, -1, 1))
approx = []
norder = 10

for order in range(norder):
    approx.append(fourier.truncate(n=order))
    display(approx[-1])

$$ 0 $$ $$ \frac{4 \sin{\left(\pi x \right)}}{\pi} $$ $$ \frac{4 \sin{\left(\pi x \right)}}{\pi} + \frac{4 \sin{\left(3 \pi x \right)}}{3 \pi} $$ $$ \frac{4 \sin{\left(\pi x \right)}}{\pi} + \frac{4 \sin{\left(3 \pi x \right)}}{3 \pi} + \frac{4 \sin{\left(5 \pi x \right)}}{5 \pi} $$ $$ \frac{4 \sin{\left(\pi x \right)}}{\pi} + \frac{4 \sin{\left(3 \pi x \right)}}{3 \pi} + \frac{4 \sin{\left(5 \pi x \right)}}{5 \pi} + \frac{4 \sin{\left(7 \pi x \right)}}{7 \pi} $$ $$ \frac{4 \sin{\left(\pi x \right)}}{\pi} + \frac{4 \sin{\left(3 \pi x \right)}}{3 \pi} + \frac{4 \sin{\left(5 \pi x \right)}}{5 \pi} + \frac{4 \sin{\left(7 \pi x \right)}}{7 \pi} + \frac{4 \sin{\left(9 \pi x \right)}}{9 \pi} $$ $$ \frac{4 \sin{\left(\pi x \right)}}{\pi} + \frac{4 \sin{\left(3 \pi x \right)}}{3 \pi} + \frac{4 \sin{\left(5 \pi x \right)}}{5 \pi} + \frac{4 \sin{\left(7 \pi x \right)}}{7 \pi} + \frac{4 \sin{\left(9 \pi x \right)}}{9 \pi} + \frac{4 \sin{\left(11 \pi x \right)}}{11 \pi} $$ $$ \frac{4 \sin{\left(\pi x \right)}}{\pi} + \frac{4 \sin{\left(3 \pi x \right)}}{3 \pi} + \frac{4 \sin{\left(5 \pi x \right)}}{5 \pi} + \frac{4 \sin{\left(7 \pi x \right)}}{7 \pi} + \frac{4 \sin{\left(9 \pi x \right)}}{9 \pi} + \frac{4 \sin{\left(11 \pi x \right)}}{11 \pi} + \frac{4 \sin{\left(13 \pi x \right)}}{13 \pi} $$ $$ \frac{4 \sin{\left(\pi x \right)}}{\pi} + \frac{4 \sin{\left(3 \pi x \right)}}{3 \pi} + \frac{4 \sin{\left(5 \pi x \right)}}{5 \pi} + \frac{4 \sin{\left(7 \pi x \right)}}{7 \pi} + \frac{4 \sin{\left(9 \pi x \right)}}{9 \pi} + \frac{4 \sin{\left(11 \pi x \right)}}{11 \pi} + \frac{4 \sin{\left(13 \pi x \right)}}{13 \pi} + \frac{4 \sin{\left(15 \pi x \right)}}{15 \pi} $$ $$ \frac{4 \sin{\left(\pi x \right)}}{\pi} + \frac{4 \sin{\left(3 \pi x \right)}}{3 \pi} + \frac{4 \sin{\left(5 \pi x \right)}}{5 \pi} + \frac{4 \sin{\left(7 \pi x \right)}}{7 \pi} + \frac{4 \sin{\left(9 \pi x \right)}}{9 \pi} + \frac{4 \sin{\left(11 \pi x \right)}}{11 \pi} + \frac{4 \sin{\left(13 \pi x \right)}}{13 \pi} + \frac{4 \sin{\left(15 \pi x \right)}}{15 \pi} + \frac{4 \sin{\left(17 \pi x \right)}}{17 \pi} $$

s = sp.plot(hat, (x, -1, 1), show=True, legend=False)
s[0].line_color = "red"

for series in approx:
    fs = sp.plot(series, (x, -1, 1),show = False, legend = False)
    fs[0].line_color = "black"
    s.append(fs[0])

s.show()

In these two numerical examples we have seen that both continous as well as discontinous functions can be approximated by Fourier series (trigonometric polynomials). In general the convergence of the Fourier series to a given function is assured if the following conditions named after Lejeune Dirichlet are fulfilled:

Dirichlet's conditions:

1. \( f \) must be absolutely integrable over a period.
2. \( f \) must be of bounded variation in any given bounded interval.
3. \( f \) must have a finite number of discontinuities in any given bounded interval, and the discontinuities cannot be infinite.