# Fourier series & orthogonality

1. Feb 5, 2012

### e.arroway

I'm studying the Fourier series and trying to get an intuitive sense for orthogonality. With the $\hat{x}$ and $\hat{y}$ basis functions in 2-D Euclidean space, this makes a lot of sense: no matter how you change your x-coordinate, you have no effect on your y-coordinate.

With the complex exponentials as basis functions, orthogonality is less intuitive. I can see that the product of two different basis functions integrates to zero over a single period:
$\int_0^L e^{2\pi iax/L}\cdot e^{-2\pi ibx/L}\,dx = 0 \text{ if } a \neq b$
but that doesn't give me a good analogy to my simple statement about Euclidean space above.

What if we consider an incomplete set of basis functions where the "1" and "-1" basis functions $e^{\pm2\pi ix/L}$ are missing. Then if we try to approximate $f(x) = e^{2\pi ix/L}$ using the remaining basis functions, how close can we get?

I tried defining "close" in the "least squares" sense, but I wasn't able to make much progress answering this question on my own. It's not a homework question - just a thought experiment, so I was hoping someone might be able to point me in the right direction.