I have a problem with the following:
If $f \in L^2 (T)$ and $\displaystyle f(x)= \sum_{n=-\infty}^{\infty} a_n e^{2 \pi i nx } $ in $L^2(T)$ the Fourier expansion, then why the Fourier expand of $ f \circ R_\alpha $ is given by:
$$ f \circ R_\alpha (x) = \sum_{n =-\infty}^{\infty} e^{2 \pi i nx} e^{2 \pi i n \alpha} $$
where $ \displaystyle R_\alpha(x) = x+ \alpha (mod 1) = \{x + \alpha \}$ ?
$T$ is the $1-$dimensional torus.
I think I am missing something obvious, but I can't understand what.
Thank you!