Properties of Fourier Transform/Translation

Theorem
Let $\map f x$ be a Lebesgue integrable function.

Let $x_0$ be a real number.

Let $\map h x$ be a Lebesgue integrable function such that:


 * $\map h x = \map f {x - x_0}$

Then:
 * $\map {\hat h} \zeta = e^{-2 \pi i x_0 \zeta} \map {\hat f} \zeta$

where $\map {\hat h} \zeta$ and $\map {\hat f} \zeta$ are the Fourier transforms of $\map h x$ and $\map f x$ respectively.