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The classical Fourier transform on R < sup > n </ sup > is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered distributions.

For instance, if we impose some requirements on a distribution f, we can attempt to translate these requirements in terms of the Fourier transform of f. The Paley – Wiener theorem is an example of this.

The Paley – Wiener theorem immediately implies that if f is a nonzero distribution of compact support ( these include functions of compact support ), then its Fourier transform is never compactly supported.

This is a very elementary form of an uncertainty principle in a harmonic analysis setting.

See also Convergence of Fourier series.