TITLE: How many linear combinations of harmonics or normal modes can describe the same periodic function as a Fourier series?
QUESTION [0 upvotes]: Please note that I am not asking how many terms in a linear combination can describe a specific periodic function but if given that there exist a set or linear combination of normal modes that describe a periodic function one can find another diferent set of normal modes that as could also describe the same periodic function.
I would like to add a physics example to make my question more specific, given a set of electrons in vacuum oscillating like armonic oscillators with different frequencies, the resulting electromagnetic wave will be the Fourier sum of the EM waves produced by each oscilating electron, if that last sentence if not wrong, then I could find another set of electrons with different frequencies that produce the same electromagnetic wave?

REPLY [1 votes]: For a fixed  orthonormal basis there is only one way to express the function. This is what it means for a set of functions or vectors to be linearly independent.
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