Fourier Series over General Range from Specific

Theorem
Let $a, b \in \R$ be real numbers.

Let $f: \R \to \R$ be a function such that $\displaystyle \int_a^b \map f x \rd x$ converges absolutely.

Then $f$ can be expressed by a Fourier series of the form:


 * $\displaystyle \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty \paren {A_m \cos \frac {2 m \pi \paren {x - a} } {b - a} + B_m \sin \frac {2 m \pi \paren {x - a} } {b - a} }$

where:

Proof
Consider the Fourier series:


 * $(1): \quad \displaystyle \map S x = \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty \paren {A_m \cos \frac {2 m \pi \paren {x - a} } {b - a} + B_m \sin \frac {2 m \pi \paren {x - a} } {b - a} }$

Let $\xi = \dfrac {2 \pi \paren {x - a} } {b - a}$.

Then:
 * $\dfrac {\d \xi} {\d x} = \dfrac {2 \pi} {b - a}$

Then:

and:

Substituting $\xi$ for $x$ in $(1)$:


 * $(2): \quad \map S \xi = \dfrac {A_0} 2 + \displaystyle \sum_{m \mathop = 1}^\infty \paren {A_m \cos m \xi + B_m \sin m \xi}$

which is a Fourier series for a real function $\map g \xi$ such that $\displaystyle \int_0^{2 \pi} \map g \xi \rd x$ converges absolutely.

The Fourier coefficients of $\map g \xi$ are given by:

Substituting $x$ for $\xi$:

and:

Thus while $A_n$ and $B_n$ are exactly the Fourier coefficients of $\map g \xi$, they are also exactly the Fourier coefficients of $\map S x$.

Thus $\map S x$ is the Fourier series for $\map f x$.