Given the product of two functions defined explicitly through their Fourier coefficients (of unknown undeveloped form):
\sum_k{c_k e^{i k t}} \cdot \sum_k{c'_k e^{i k t}}
Is there any way to express it as a Fourier series? (Edit: approximated using a finite number of terms of the original)
That is: \sum_k{c''_k e^{i k t}} where each c''_k could be explicitly defined from a finite sum of $c$ and c'.
I feel the convolution theorem should be of some help here, but I can't see how for the life of me...
(probably not relevant, but my goal is to use this product's equality with a third Fourier series and use coefficient identity in order to extract a set of optimisation constraints based on the terms of all three original series)
Edit: since I am trying to identify coefficients, what I'm really hoping for is an approximated expression of the product, based on a limited number of terms... In the absence of any particular properties of $c$ and c' that would simplify the convolution, is there any way to achieve this?
(thanks a lot to people who already answered and made me realise the issue with my original formulation)