I'd like to know whether Fourier coefficients (and series) of $f(x):\mathbb{R} \to \mathbb{C}$ and $f(x+c)$ for $c \in \mathbb{R}$ are the same and if so, why?
I have this question where $f$ is a $ 2 \pi$ periodic and Continuously differential which it's Fourier coefficient is $3^{-n^2}$, I need to compute $g$'s Fourier coefficient, where g(x)=\pi f'(x+2011). eventually I got that $\hat{g(n)}=\pi i n \frac{1}{2 \pi}\int_{0}^{2\pi}e^{-inx}f(x+2011 )$ and I would like to say that $\hat{g(n)}=\pi i n\hat{f(n)}=\pi i n3^{-n^2}$, I'm just not sure if it is correct. Thanks a lot.