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tabulated transforms of specific functions

The following table recaps the four basic forms discussed above, highlighting the duality of the properties of discreteness and periodicity.  I.e., if the signal representation in one domain has either (or both) of those properties, then its transform representation to the other domain has the other property (or both).

Fourier transforms on arbitrary locally compact abelian topological groups

The Fourier variants can also be generalized to Fourier transforms on arbitrary locally compact abelian topological groups, which are studied in harmonic analysis; there, the Fourier transform takes functions on a group to functions on the dual group. This tréatment also allows a general formulation of the convolution theorem, which relates Fourier transforms and convolutions.  See also the Pontryagin duality for the generalized underpinnings of the Fourier transform.