trying to formulate this question more exactly, but, is there a convenient generic method for reconstructing a function from its discrete fourier transform using a different basis (perhaps a completely unreasonable one like a sufficiently short sample of the sound of a metal pipe falling)
the way that i can think of is that you simply go through the result, extract the phase and amplitude of each frequency component, and then do a summation of those. but this seems awfully heavy? is there a way to do this efficiently