The Fourier Transform

The Fourier transform of a function is defined as

and can be shown to exist for any function for which

The Fourier transform is invertible, i.e. given , can be obtained using the inverse Fourier transform, viz.

Some important properties of the Fourier transform are listed below (where by convention capitalized functions refer to the Fourier transform)

- Linearity

where are arbitrary complex constants. - Similarity

where is an arbitrary real constant. - Shift

where is an arbitrary real constant. - Parseval's Theorem

- Convolution Theorem

- Autocorrelation Theorem