> you can replace the 'N' sin/cos functions used in Fourier analysis with any set of 'N' orthogonal functions But why? It rarely makes sense to use such arbit

> you can replace the 'N' sin/cos functions used in Fourier analysis with any set of 'N' orthogonal functions But why? It rarely makes sense to use such arbitrary functions that don't satisfy further properties! Typically you'd also want your functions to be the eigenfunctions of a symmetric linear operator which is naturally associated to your problem. Most often, the Laplacian (second derivative operator) of your domain. Thus the natural linear PDE on your space (diffusion, potential, wave equations and their variations) can be readily solved in the Fourier domain. The sequence of eigenvalues provides a lot of geometrical information about the shape of the domain.