We will first start by reviewing Lecture 19 and our introduction of the Fourier convergence theorem. This involves Theorem 12.5 in the notes. In particular, the theorem states that if $f$ is a $2\pi$-periodic function with $f$ and $f'$ continuous on the interval $(-\pi, \pi)$, then the Fourier series of $f$ at $x$ converges to the average of the left- and right-limits. Thus $$ \frac{1}{2}[f(x_-) + f(x_+)] = \frac{a_0}{2} + \sum_{1}^\infty [a_n \cos(nx) + b_n \sin(nx)] $$

We'll draw some pictures of how to visualise the theorem. We will also discuss again this notion of pointwise convergence vs. uniform convergence and the notion of the Gibbs' Phenomenon.

Next, we want to simply note that the Fourier series you've derived for $2\pi$-periodic functions on $[-\pi, \pi]$ can be easily extended to functions defined on $[-L, L]$. The truth is that we should really just have done the derivation like this from the get-go! This leads to:

Theorem 12.7: (Fourier coefficients for $2L$-periodic function) Let $f$ be a periodic function with period $2L$. Then $$ f(x) \sim \frac{a_0}{2} + \sum_{n=1}^\infty \left[ a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right)\right] $$