This is a list of linear transformations of functions related to Fourier analysis.
(These transforms are generally designed to be invertible.)
In the case of the Fourier transform, each basis function corresponds to a single frequency component.
Applied to functions of continuous arguments, Fourier-related transforms include: For usage on computers, number theory and algebra, discrete arguments (e.g. functions of a series of discrete samples) are often more appropriate, and are handled by the transforms (analogous to the continuous cases above): The use of all of these transforms is greatly facilitated by the existence of efficient algorithms based on a fast Fourier transform (FFT).
The Nyquist–Shannon sampling theorem is critical for understanding the output of such discrete transforms.