Paley-Wiener theorem

A function vanishes almost everywhere outside an interval if and only if its Fourier transform

satisfies

and is the restriction to the real line of a certain entire analytic function of a complex variable satisfying for all (see [1]). A description of the image of a certain space of functions or generalized functions on a locally compact group under the Fourier transform or under some other injective integral transform is called an analogue of the Paley–Wiener theorem; the most frequently encountered analogues of the Paley–Wiener theorem are a description of the image of the space of infinitely-differentiable functions of compact support and a description of the image of the space of rapidly-decreasing infinitely-differentiable functions on a locally compact group under the Fourier transform on . Such analogues are known, in particular, for Abelian locally compact groups, for certain connected Lie groups, for certain subalgebras of the algebra on real semi-simple Lie groups, and also for certain other integral transforms.

References

[1]	N. Wiener, R.E.A.C. Paley, "Fourier transforms in the complex domain" , Amer. Math. Soc. (1934)
[2]	V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1977) (Translated from Russian)
[3]	I.M. Gel'fand, M.I. Graev, N.Ya. Vilenkin, "Generalized functions" , 5. Integral geometry and representation theory , Acad. Press (1966) (Translated from Russian)
[4]	D.P. Zhelobenko, "Harmonic analysis of functions on semi-simple complex Lie groups" , Moscow (1974) (In Russian)
[5]	W. Rudin, "Functional analysis" , McGraw-Hill (1973)

Comments

Let with . Then the Fourier transform of can be extended to an entire analytic function on satisfying: for any integer there is a constant such that for all ,

(*)

Conversely, let be an entire function which satisfies (*) (replacing with ), for some . Then there exists a with and .

References

[a1]	F. Trèves, "Topological vectorspaces, distributions and kernels" , Acad. Press (1967)
[a2]	G. Warner, "Harmonic analysis on semi-simple Lie groups" , II , Springer (1972)
[a3]	S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4
[a4]	Y. Katznelson, "An introduction to harmonic analysis" , Dover, reprint (1976)