Parallel between Fourier series and orthogonal projections

My professor made an analogy between Fourier series and orthogonal projections, and I was hoping someone could explain that somewhat more. Basically, as I understand it:

$$a_n = \frac{1}{L} \int_L^L f(x) \cos\left(\frac{ n\pi x}{L}\right) \ dx \longleftrightarrow c_1 = \frac{v \cdot b_i}{b_i \cdot b_i},$$

where $$\dfrac{1}{L}$$ can be thought of as normalizing the projection, and the integrand is the inner product (equivalent to the dot product on the right side).

Am I understanding this correctly? And can someone clarify this for me?

• "Paralell..." Not a pun?
– Pedro
May 1, 2014 at 23:35

He is referring to the inner product induced by the space of square-integrable functions $L^2 [-L, L]$.
$$a_n = \frac{ \left <f(x), \cos \left ( \frac{n \pi x}{L} \right )\right >}{ \left < \cos \left ( \frac{n \pi x}{L} \right ),\cos \left ( \frac{m \pi x}{L} \right )\right >} = \frac{ \int_{-L}^{L} f(x) \cos \left ( \frac{n \pi x}{L} \right ) dx }{ \int_{-L}^{L} \cos \left ( \frac{n \pi x}{L} \right ) \cos \left ( \frac{m \pi x}{L} \right ) dx }.$$
where $\left < \cos \left ( \frac{n \pi x}{L} \right ),\cos \left ( \frac{m \pi x}{L} \right )\right > = \left\{\begin{matrix} 1& \text{if} \;n = m \neq 0\\ 0& \text{if} \;n \neq m \end{matrix}\right.$