There is a certain function $S_n(x)=\frac{\sin x \sin nx}{x^2}$ for $x\neq 0$, and $S_n(0)=n$. There is another function $T_n$, such that $\widehat{T_n}=4S_n$, where $\widehat{\cdot}$ is the Fourier transform operator. We can assume that $S_n\in L_1$. Everything is defined on $\mathbb{R}$. The question asks to find the Fourier transform $\widehat{S}_n(\eta)$. The answer is apparently $\widehat{S}_n=\frac{\pi}{2} T_n$. This is how I proceeded:
$$\int_\mathbb{R}e^{-inx}T_n(x)dx=\widehat{T}_n=4S_n=\frac{1}{2\pi}\int_\mathbb{R}e^{in\xi}\widehat{S}_n(\xi)d\xi.$$
The problem is that on the right-hand side we have $e^{in\xi}$ and on the left-hand side we have $e^{-inx}$. If the signs were the same we could conclude immediately. Can I get some help on how to proceed? Is there some property of Fourier transforms that I am not aware of?
Update: This is the solution we have in the book, but I don't understand where the first equality comes from.
