Understanding fourier coefficients

Question

In my text book it says that if $c_n$ are fourier coefficients of $f$ and $f_n$ is an orthonormal set then

$\bigg( \sum_{n=1}^N c_n f_n, f - \sum_{n=1}^N c_n f_n\bigg) = 0$

This is not immediately obvious to me and I am wondering if someone has a proof of this?

Answer 1

We have:

$$\left( \sum_{n=1}^N c_n f_n, f - \sum_{n=1}^N c_n f_n\right) = \left(\sum_{n=1}^N c_nf_n, f\right) - \left(\sum_{n=1}^N c_n f_n , \sum_{m=1}^N c_m f_m\right) \\ = \sum_{n=1}^N c_n (f_n,f) - \sum_{n=1}^N\sum_{m=1}^Nc_n c_m \underbrace{(f_n,f_m)}_{= \delta_{nm}} \\ = \sum_{n=1}^N c_n^2 - \sum_{n=1}^Nc_n^2 = 0$$

Answer 2

This is a slightly stronger result from which the other follows. It says that the error in approximating $f$ by $\sum_n c_n f_n$ is perpendicular to the linear subspace spanned by the $f_n$.

If $c_n = \langle f, f_n \rangle$, and $k \le n$, then $\langle f-\sum_n c_n f_n , f_k \rangle = \langle f, f_k \rangle - \langle \sum_n c_n f_n , f_k \rangle = c_k - c_k = 0$