Paley-Wiener theorem

A function $ f \in L _ {2} ( - \infty , + \infty ) $ vanishes almost everywhere outside an interval $ [ - A , A ] $ if and only if its Fourier transform

$$ F ( y) = \ \int\limits _ {- \infty } ^ { {+ } \infty } f ( x) e ^ {ixy} d x ,\ y \in \mathbf R , $$

satisfies

$$ \int\limits _ {- \infty } ^ { {+ } \infty } | F ( y) | ^ {2} d y < \infty $$

and is the restriction to the real line of a certain entire analytic function $ F ( z) $ of a complex variable $ z $ satisfying $ | F ( z) | \leq e ^ {A | z | } $ for all $ z \in \mathbf C $( 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 $ C _ {0} ^ \infty ( G) $ of infinitely-differentiable functions of compact support and a description of the image of the space $ S ( G) $ of rapidly-decreasing infinitely-differentiable functions on a locally compact group $ G $ under the Fourier transform on $ G $. Such analogues are known, in particular, for Abelian locally compact groups, for certain connected Lie groups, for certain subalgebras of the algebra $ C _ {0} ^ \infty ( G) $ 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 $ \phi \in C _ {0} ^ \infty ( \mathbf R ) $ with $ \supp \phi \subset [- A, A] $. Then the Fourier transform $ \widehat \phi $ of $ \phi $ can be extended to an entire analytic function on $ \mathbf C $ satisfying: for any integer $ m \geq 0 $ there is a constant $ c _ {m} > 0 $ such that for all $ w \in \mathbf C $,

$$ \tag{* } | \widehat \phi ( w) | \leq c _ {m} ( 1+ | w | ) ^ {-} m e ^ {2 \pi A | \mathop{\rm Im} w | } . $$

Conversely, let $ F: \mathbf C \rightarrow \mathbf C $ be an entire function which satisfies (*) (replacing $ \widehat \phi $ with $ F $), for some $ A > 0 $. Then there exists a $ \phi \in C _ {0} ^ \infty ( \mathbf R ) $ with $ \supp \phi \subset [- A, A] $ and $ \widehat \phi = F $.

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)