The Fourier Transforms

The Fourier transform was named after Joseph Fourier, a French mathematician. A Fourier Transform takes a function to its frequency components.

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Properties of a Fourier Transform:

Convolution

$f(t)*g(t)\,$	${\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\sqrt {2\pi }}\cdot F(\omega )\cdot G(\omega )\,$	(unitary convention)
	${\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad F(\omega )\cdot G(\omega )\,$	(non-unitary convention)
	${\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad F(f)\cdot G(f)\,$	(ordinary frequency)