I'm wanting to research into how functions can be broken down into coefficients of simple basis functions. In one dimension I know this can be done with the Fourier Transform and an extension into 3d can be used with Spherical Harmonics. However I'm wanting to start right at the beginning.

I think the simplest method is just integrate the product of your basis functions and your target functions to produce the coefficients.

$\displaystyle c[n] = \int f()basis[n]()$

$\displaystyle f(x) = \sum c[n]basis[n]()$

I've tried this with many different basis functions, such as :

- sin waves, where each basis iteration is a higher frequency
- sin waves of the same frequency but shifted
- triangle waves
- step functions

But to be honest, I'm doing it blindly, and don't know which functions are better and why.

- Should I use the whole domain of sin?

- Should the basis function have negative values in its range?

- Should the basis functions just be shifted versions of a single type?

- Should the basis functions just be increasing frequencies of a single type?

All I know is that each basis function needs to be normalized, ie the integral over its domain needs to be 1.0, where the function is squared. I've also read that the basis functions should be orthogonal, however I'm not sure why, as non orthogonal basis functions seem to work ok.

Does anyone have any suggestions for suitable material I can read? preferably a book?