Table of Fourier Transform Properties

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)	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)	${ℱ}\{ax(t)+by(t)\}=aF\{x(t)\}+bF\{y(t)\}$