Properties of Fourier Transform/Linearity

Theorem
Let $\map f x$ and $\map g x$ be Lebesgue integrable functions.

Let $a$ and $b$ be complex numbers.

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


 * $\map h x = a \map f x + b \map g x$

Then:
 * $\map {\hat h} \zeta = a \map {\hat f} \zeta + b \map {\hat g} \zeta$

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