Translation property of the Fourier transform ($\widehat{f(\cdot -a)}(k)=e^{-ika}\hat{f}(k)$)

For an integrable function and the angular-frequency convention,so a translation multiplies each frequency component by a factor of modulus one. The amplitude of every component stays the same, while its wave phase changes by $-ka$. For example, delaying $\cos (2\pi \nu t)$ by $a$ produces $\cos (2\pi \nu t-2\pi \nu a)$: faster oscillations accumulate a larger wave phase shift for the same delay. Substituting $y=x-a$ in the defining Fourier transform is the entire proof. Conversely, this multiplier in the Fourier transform produces a translation after Fourier inversion.