I saw the proof of this identity in a question about Fourier transforms :
$F(f(−t))w=F(f(t))(−w)$
Can someone give the intuition behind it ?
What I understand of Fourier transform of a function at a point $w$ is that : it gives the amplitude of the wave of frequency $w$ in the fourier series of the function $f(t)$. When we flip the function why should there be a change of sign in the frequency ?
A Fourier transform gives complex-valued frequencies. These are $\mathrm{e}^{\mathrm{i}\omega \theta}$ for various values of $\omega$. If $\omega$ is positive, these circulate anti-clockwise in the complex plane. If $\omega$ is negative, these circulate clockwise in the complex plane.
What the identity you call out is saying is that under time reversal the two types of circulation are swapped.