Properties of Fourier Transform/Modulation

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

Let $\zeta_0$ be a real number.

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


 * $\map h x = e^{2 \pi i x \zeta_0} \map f x$

Then:


 * $\map {\hat h} \zeta = \map {\hat f} {\zeta - \zeta_0}$

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