Could anyone show me how to prove the following results about Fourier Transform, please? It is stated in my book without proof. Thank you.
Let $\mathcal F$ denote the Fourier linear operator and $f$ be a $\mathcal L^1(\mathbb R)$ function. Then $$\mathcal F^2 (f) = f (-x).$$ That is, if we apply the Fourier transform twice, we get a spatially reversed version of the function.
If you use the following (unitary) definition of the Fourier transform
$$\mathcal{F}\{f(x)\}=F(\xi)=\int_{-\infty}^{\infty}f(x)e^{-2\pi ix\xi}dx\\ f(x)=\int_{-\infty}^{\infty}F(\xi)e^{2\pi ix\xi}d\xi$$
you have
$$\mathcal{F}^2\{f(x)\}=\mathcal{F}\{F(\xi)\}=\int_{-\infty}^{\infty}F(\xi)e^{-2\pi i x\xi}d\xi=f(-x)$$