I'm studying the Fourier series and trying to get an intuitive sense for orthogonality. With the [itex]\hat{x}[/itex] and [itex]\hat{y}[/itex] 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.(adsbygoogle = window.adsbygoogle || []).push({});

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:

[itex]\int_0^L e^{2\pi iax/L}\cdot e^{-2\pi ibx/L}\,dx = 0 \text{ if } a \neq b[/itex]

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 [itex]e^{\pm2\pi ix/L}[/itex] are missing. Then if we try to approximate [itex]f(x) = e^{2\pi ix/L}[/itex] 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Fourier series & orthogonality

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**