Fourier Series/Triangle Wave

Theorem
Let $\map T x$ be the triangle wave defined on the real numbers $\R$ as:


 * $\forall x \in \R: \map T x = \begin {cases}

\size x & : x \in \closedint {-l} l \\ \map T {x + 2 l} & : x < -l \\ \map T {x - 2 l} & : x > +l \end {cases}$ where:
 * $l$ is a given real constant
 * $\size x$ denotes the absolute value of $x$.

Then its Fourier series can be expressed as:

Proof
Let $\map f x: \openint {-l} l \to \R$ denote the absolute value function on the open interval $\openint {-l} l$:
 * $\map f x = \size x = \begin{cases}

x & : x > 0 \\ 0 & : x = 0 \\ -x & : x < 0 \end{cases}$

From Fourier Series for Absolute Value Function over Symmetric Range, $\map f x$ can immediately be expressed as:


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