I am trying to express the Fourier coefficients of $f(x+y)$ in terms of the Fourier coefficients of $f(x)$. I have
$$\int_{-\pi}^{\pi} f(x+y) e^{-inx} dx = \int_{-\pi+y}^{\pi+y} f(u) e^{-in(u-y)} du = e^{iny}\int_{-\pi +y}^{\pi+y}f(u)e^{-inu}du $$ $=2 \pi e^{iny}\hat{f}(n)$.
Is this correct? I was under the impression that the Fourier coefficients were supposed to be translation-invariant, which is the main reason for why I am doubting this result.
First of all, the $2\pi$ is wrong.
Second, if you have a function that is an outcome of interference, then the same function but translated will have the same spectrum (meaning the absolute value of the Fourier coefficients will be the same) but you need to add to each wave a phase element for making the interference happen later or earlier.