It's a bit weirder than this suggests. You are right as long as you have a complete basis you can transform code your function by a weighted sum of bases vecto

It's a bit weirder than this suggests. You are right as long as you have a complete basis you can transform code your function by a weighted sum of bases vectors, where the weights are inner products of the function against he dual vectors of that basis, and the basis vector form a complete orthonormal basis (we can relax that last bit). So that's great, but what do the basis vectors look like? You need a family of functions so that scalings and translations of them have inner product 0 against each other. So this is easy if the shared support is 0, or if the +ve and -ve parts cancel out. It doesn't take too much playing about to see how this works with Haar basis, or sin/cos (i.e. FFT), so it looks pretty intuitive. It also seem kind of intuitive that these might be the only way to do this... but that's wrong. It turns out you can construct other families that work, unlike the nicely behaved sin/cos, or step functions, they are not smooth, not symmetric (at least for orthonormal basis) - quite odd. They don't have closed forms, so if you want to see what one looks like you'll have to realized it as a fixed point or by some other approximation method: e.g. https://en.wikipedia.org/wiki/Daubechies_wavelet#/media/File:Daubechies4-functions.svg https://en.wikipedia.org/wiki/Daubechies_wavelet#/media/File...