Table of Fourier Transform Properties

Scaling For a non-zero real number a, if h(x) = ƒ(ax), then  . The case a = −1 leads to the time-reversal property, which states: if h(x) = ƒ(−x), then 

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)	For a non-zero real number a, if h(x) = ƒ(ax), then  ${\hat {h}}(\xi )={\frac {1}{|a|}}{\hat {f}}\left({\frac {\xi }{a}}\right)$.     The case a = −1 leads to the time-reversal property, which states: if h(x) = ƒ(−x), then  ${\hat {h}}(\xi )={\hat {f}}(-\xi )$.