Playing some more with orthogonal functions to try and get a better handle of why a Fourier transform works. I decided I would try some simple functions I can work with easily to test orthogonality. The definition for as orthogonal function is one such that:
So I decided to use two very simple functions over the interval b = -a.
Remember that ab even power b is the same for a and -a. Thus:
So my two functions are orthogonal. Now for something crazy.
I want to extract the coefficient for b. So I can to a projection for x2.
Notice how c does not show up in this result.
The same for x3.
Here, b does not show up in the result.
I verified this using a discrete check in a speadsheet. While I expected it would work, watching it actually do so was interesting.