Editing Harmonic analysis (section)

== Fourier analysis ==

{{main|Fourier analysis}}
The classical [[Fourier transform]] on '''[[Real number|R]]'''<sup>''n''</sup> is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as [[distribution (mathematics)#Tempered distributions and Fourier transform|tempered distributions]]. For instance, if we impose some requirements on a distribution ''f'', we can attempt to translate these requirements into the Fourier transform of ''f''. The [[Paley–Wiener theorem]] is an example. The Paley–Wiener theorem immediately implies that if ''f'' is a nonzero [[Distribution (mathematics)|distribution]] of [[compact support]] (these include functions of compact support), then its Fourier transform is never compactly supported (i.e., if a signal is limited in one domain, it is unlimited in the other). This is an elementary form of an [[uncertainty principle]] in a harmonic-analysis setting.

Fourier series can be conveniently studied in the context of [[Hilbert space]]s, which provides a connection between harmonic analysis and [[functional analysis]]. There are four versions of the Fourier transform, dependent on the spaces that are mapped by the transformation:
* Discrete/periodic–discrete/periodic: [[Discrete Fourier transform]]
* Continuous/periodic–discrete/aperiodic: [[Fourier series]]
* Discrete/aperiodic–continuous/periodic: [[Discrete-time Fourier transform]]
* Continuous/aperiodic–continuous/aperiodic: [[Fourier transform]]
As the spaces mapped by the Fourier transform are, in particular, subspaces of the space of tempered distributions it can be shown that the four versions of the Fourier transform are particular cases of the Fourier transform on tempered distributions.