Properties of Fourier Transform

Theorem

In the following:

$\map f x$ and $\map g x$ denote Lebesgue integrable functions

for a given $\map f x$, $\map {\hat f} \zeta$ denotes its Fourier transform.

Linearity

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:

$\map {\hat h} s = a \map {\hat f} s + b \map {\hat g} s$

Translation

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:

$\map {\hat h} s = e^{-2 \pi i x_0 s} \map {\hat f} s$

Modulation

Let $s_0$ be a real number.

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

$\map h x = e^{2 \pi i x s_0} \map f x$

Then:

$\map {\hat h} s = \map {\hat f} {s - s_0}$

Time Scaling

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:

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