Fourier Series/Absolute Value Function over Symmetric Range

Theorem
Let $\lambda \in \R_{>0}$ be a strictly positive real number.

Let $\map f x: \openint {-\lambda} \lambda \to \R$ be the absolute value function on the open real interval $\openint {-\lambda} \lambda$:
 * $\forall x \in \openint {-\lambda} \lambda: \map f x = \size x$

The Fourier series of $f$ over $\openint {-\lambda} \lambda$ can be given as:

Proof
From Absolute Value Function is Even Function, $\map f x$ is an even function.

By Fourier Series for Even Function over Symmetric Range, we have:


 * $\displaystyle \map f x \sim \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} \lambda$

where:

Hence from Half-Range Fourier Cosine Series for Identity Function:


 * $\map f x \sim \dfrac \lambda 2 - \displaystyle \frac {4 \lambda} {\pi^2} \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^2} \cos \dfrac {\paren {2 n + 1} \pi x} \lambda$

Hence the result.