Some Inequality from Fourier Series

Question

Let $f \in L^2(\mathbb{T})$.

Let $t \in \mathbb{R}$. Show that $$\|f-f_t\|_2 ^2 =\sum_{n \in \mathbb{Z}}|c_n(f)(1-e^{-int})|^2.$$ Here $$c_n(f)= \frac{1}{2\pi} \int_{[-\pi,\pi]}f(x)e^{-inx}dx.$$

Answer 1

Try Parseval's theorem:

$$\|f-f_t\|_2^2= \sum_{n=-\infty}^\infty |c_n(f)-c_n(f_t)|^2.$$

Now use the fact that $c_n(f_t)=e^{-int}c_n(f)$, by substitution in the integral and periodicity of $f$. Can you finish from here?