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 ${\hat {h}}(\xi )={\frac {1}{|a|}}{\hat {f}}\left({\frac {\xi }{a}}\right)$. If a=−1, then the time-reversal property states: if Failed to parse (syntax error): {\displaystyle \ h(x)=ƒ(−x), then&thinsp; <math>\hat{h}(\xi)=\hat{f}(-\xi)} .