Properties of Fourier Transform/Scaling

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

Let $a$ be a non-zero real number.

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


 * $\map h x = \map f {a x}$

Then:


 * $\map {\hat h} \zeta = \dfrac 1 {\size a} \map {\hat f} {\dfrac \zeta a}$

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