= Translation property of the Fourier transform
{c}
{title2=$\widehat{f(\cdot-a)}(k)=e^{-ika}\widehat f(k)$}
For an <integrable function> and the angular-frequency convention,
$$
\mathcal F[f(x-a)](k)=e^{-ika}\widehat f(k),
$$
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>.
Back to article page
