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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle / h(x)=ƒ(ax), then&thinsp; <math>\hat{h}(\xi)=\frac{1}{|a|}\hat{f}\left(\frac{\xi}{a}\right)} .     If a = −1, then the time-reversal property states: if Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle / h(x)=ƒ(−x), then&thinsp; <math>\hat{h}(\xi)=\hat{f}(-\xi)} .