How to use Parseval' identity( Plancherel)?

Question

(May be this is very basic question for MO)

Let $f\in L^{2} (\mathbb R)$ with $\lim_{t\to \pm \infty} f(t)=0.$ Put $$F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt \ (n=1,2,...)$$ Fix $\alpha \in (0, \infty)$ and we define $H_{n}(x)$ as follows: $$\frac{1}{2\pi}\int_{-n}^{n} e^{itx} (f(t+\alpha) -f(t-\alpha))dt = (e^{-i\alpha x}- e^{i\alpha x})F_{n}(x) + H_{n}(x)$$

My Question: Can we expect to prove: $H_{n}(x) \to 0$ as $n\to \infty$ in $L^{2}(\mathbb R)$ ?

I guess some where we need to use Parseval' identity( Plancherel); but I am bit confused, how to use it.

Thanks

Answer 1

As proposed in (How to use Parseval' s( Plancherel' s) identity?), You can use that

$$\int_{-n}^ne^{itx} f(t+α)\,dt=\int_{-n+α}^{n+α}e^{i(t-α)x}f(t)\,dt =e^{-iαx}\int_{-n+α}^{n+α}e^{itx}f(t)\,dt$$

etc. to reduce the expression for $H_n$ to integrals over the segments $[\pm n-α,\pm n+α]$. Then apply the asymptotic behavior of $f$ for large arguments.