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 $h(x)=(f*g)(x)$, 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  $\hat{h}(\xi )=\hat{f}(-\xi )$.