Half-Range Fourier Cosine Series over Negative Range

Theorem
Let $f \left({x}\right)$ be a real function defined on the interval $\left[{0 \,.\,.\, l}\right]$.

Let $f$ be expressed using the half-range Fourier cosine series over $\left[{0 \,.\,.\, l}\right]$:


 * $\displaystyle f \left({x}\right) \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} l$

where:
 * $a_n = \displaystyle \frac 2 l \int_0^l f \left({x}\right) \cos \frac {n \pi x} l \, \mathrm d x$

for all $n \in \Z_{\ge 0}$.

Then over the interval $\left[{-l \,.\,.\, 0}\right]$, $C \left({x}\right)$ takes the values:
 * $S \left({x}\right) = f \left({-x}\right)$

That is, the real function expressed by the half-range Fourier cosine series over $\left[{0 \,.\,.\, l}\right]$ is an even function over $\left[{-l \,.\,.\, l}\right]$.

Proof
From Fourier Series for Even Function over Symmetric Range, $C \left({x}\right)$ is the Fourier series of an even real function over the interval $\left[{0 \,.\,.\, l}\right]$.

We have that $C \left({x}\right) \sim f \left({x}\right)$ over $\left[{0 \,.\,.\, l}\right]$.

Thus over $\left[{-l \,.\,.\, 0}\right]$ it follows that $S \left({x}\right) = f \left({-x}\right)$.