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Chapter 22
Visualizing Fourier series for periodic functions.
Truncated Fourier series approximations.
Let $\omega = 2\pi/L$. For an arbitrary $L$-periodic $f(x)$, we define the numbers $a_k$ and $b_k$ by
$$a_0 = \frac{1}{L} \int_0^{L} f(x)\, dx, $$
$$a_k = \frac{1}{L/2} \int_0^{L} f(x) \cos k\omega x\, dx,$$
$$b_k = \frac{1}{L/2} \int_0^{L} f(x) \sin k \omega x\, dx.$$
We then use these to define the Fourier series associated to $f(x)$ to be
$$f(x) = a_0 + \sum_{k=1}^\infty (a_k \cos k \omega x + b_k \sin k \omega x).$$
Let $f(x)$ be an $L$-periodic function whose discontinuities are only jump discontinuities
(i..e., limits exist both from the left and from the right at each discontinuity). Assume $f(x)$
is differentiable away from those points and away from perhaps finitely many other points, with $f'$
continuous elsewhere, and assume $f'$ has at most jump discontinuities too.
Then the associated Fourier series as defined above converges everywhere, and it coincides with $f(x)$ except possibly at the jump discontinuities for $f$, where its value is
the average of the limits from the left and right.
To give you some feeling about this result, below we give two examples.
In each example, you can see a $2\pi$-periodic function $f(x)$ with jump continuities,
and you can see how the partial sums of the Fourier series for $f(x)$ approximates $f(x)$.
As you increase the truncation parameter(s), observe how the Gibbs phenomenon appears!
Example 1
Function
Let $F(x)$ be the $2\pi$-periodic step function that alternates between the values 1 and 0 over
successive intervals of length $\pi$. The graph of $F(x)$ is shown in blue.
Example 2
Function
Let $F(x)$ be the $2\pi$-periodic “sawtooth” function as below. The graph of $F(x)$ is shown in blue.
Partial Sums
$F_{M,N}(x): = \frac{\pi}{4} + \sum_{m=0}^M \frac{2}{(2m+1)^2 \, \pi}\, \cos \left(
(2m+1)\, x \right) \> -\>
\sum_{n=1}^N \frac{1}{n }\, \sin\left(
n\, x \right)$