How do you find Fourier Coefficients with non-pi periodic functions? So during lecture, we learnt how to find the Fourier Series for piecewise functions that have a period of $\space 2 \pi \space$.
Now I'm looking through the exercises and the lecturer has thrown a function with period $\space 4 \space$.
How can I find the coefficients?
 A: If $\ell > 0$ is a real number, the functions
$$
1,\quad
c_{n}(x) = \cos \frac{2\pi nx}{\ell},\quad
s_{n}(x) = \sin \frac{2\pi nx}{\ell},\qquad n > 0,
$$
are an orthonormal basis for the space of square-integrable functions on $[0, \ell]$ (in the sense that the set of finite linear combinations of these functions is dense). Loosely, these functions are "the Fourier basis for $\ell$-periodic functions".
It's a sneaky but important detail that the "fundamental period", i.e., the period of $c_{1}$ and $s_{1}$, has to "match" $\ell$, the length of the interval, a.k.a., the period of the function being expanded.

At risk of answering a question you didn't ask, at some stage you're likely to be given a function, say $f(x) = x$, on $[0, \pi]$ and asked to expand $f$ as:


*

*A sine series,

*A cosine series.
If you've encountered the fact that "functions have unique Fourier series", the question will cause puzzlement. The resolution is, $f$ is only specified on an interval of length $\pi$, while the "standard" Fourier basis has fundamental period $2\pi$. That means you can't expand $f$ at all until you've extended $f$ to $[-\pi, \pi]$ (or some interval of length $2\pi$).
Expanding $f$ as a cosine series implicitly extends $f$ to be even: $f(x) = |x|$ on $[-\pi, \pi]$. Expanding $f$ as a sine series implicitly extends $f$ to be odd: $f(x) = x$ on $[-\pi, \pi]$. The Fourier series are different because they're expansions of different functions.
