Exploring the Orthogonality of Sine and Cosine Functions in Fourier Series

[Wicketer]

Could someone kindly explain whether the 90 degree phase difference between sine & cosine functions contribute to the fact that they are orthogonal? I just studied Fourier series and treating sines and cosines as vectors is fine for my brain to handle, but I can't tell whether the phase difference of 90 degrees is a coincidence relative to their orthogonality in an infinite dimension space. Just trying to understand what I learn.

Cheers.

 

[LCKurtz]

I would answer that with a no, but I'm open to a contrary view. Phase shifting doesn't explain the orthogonality of {sin(nx)} on (0,pi), for example. To really get a grip on it I think you need to look at Sturm-Liouville theory about eigenvalue problems and orthogonality of eigenfunctions. A good place to look is

http://en.wikipedia.org/wiki/Sturm–Liouville_theory

The sequence {sin(nx)} are eigenfunctions of the S-L problem

y'' + λy = 0
y(0)=0, y(pi)=0