I need some help... I am asked to prove the following property of the Fourier transform, when $F[f(x)]=\widetilde{f}(x)$, where $F[f(x)]$ is the Fourier transform of $f(x)$:
$$F[ \widetilde{f}(x) ]= \frac{f(-k)}{2 \pi}$$
We know that: $F[ \widetilde{f}(x) ]=\int_{- \infty}^{+ \infty}{ \widetilde{f}(x) e^{-i k x}}dx$.
But how can I prove this? I got stuck.. :/ Could you give me a hint?
Then $\int_{-\infty}^\infty \bar {f}(x)e^{-ikx}dx=\int_{-\infty}^{\infty} \bar {f}(x)e^{i(-k)x}dx=f(-k)$
I realise this is out by a $2\pi$ but you get the idea. (Your definition of a fourier transform needs a $2\pi $ somewhere).