Table of Fourier Transform Properties

Fourier Transform Properties

Property (contributor)	Expanation
Convolution (Ben Henry)	If $h(x)=\left(f*g\right)(x)$, becomes   ${\hat {h}}(\xi )={\hat {f}}(\xi )\cdot {\hat {g}}(\xi ).$
Scaling (Chris Lau)	Given a, which is non-zero and real, and $\ h(x)=f(ax)$, then ${\hat {h}}(\xi )={\frac {1}{|a|}}{\hat {f}}\left({\frac {\xi }{a}}\right)$. If a=−1, then the time-reversal property states: if $\ h(x)=f(-x)$, then ${\hat {h}}(\xi )={\hat {f}}(-\xi )$.
Linearity (Victor Shepherd)	${\mathcal {F}}\{ax(t)+by(t)\}=a{F}\{x(t)\}+b{F}\{y(t)\}$