# How many linear combinations of harmonics or normal modes can describe the same periodic function as a Fourier series?

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?

• What does "normal mode" mean in your question? Jul 10, 2019 at 17:04
This is like asking in how many ways can a given vector be constructed from components. The answer is equal to the number of sets of basis vectors you are allowed to choose from. That is infinity if you allow any set of basis vectors, and just $$1$$ (i.e. a unique solution) if you have just one set of basis vectors (and we are assuming the basis vectors are themselves a linearly independent set, which is how one would usually choose them).