View source for Paley-Wiener theorem

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p0711201.png" /> vanishes almost everywhere outside an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p0711202.png" /> if and only if its [[Fourier transform|Fourier transform]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p0711203.png" /></td> </tr></table>

satisfies

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p0711204.png" /></td> </tr></table>

and is the restriction to the real line of a certain entire analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p0711205.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p0711206.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p0711207.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p0711208.png" /> (see [[#References|[1]]]). A description of the image of a certain space of functions or generalized functions on a locally compact group under the [[Fourier transform|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p0711209.png" /> of infinitely-differentiable functions of compact support and a description of the image of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112011.png" /> of rapidly-decreasing infinitely-differentiable functions on a locally compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112012.png" /> under the Fourier transform on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112013.png" />. Such analogues are known, in particular, for Abelian locally compact groups, for certain connected Lie groups, for certain subalgebras of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112014.png" /> on real semi-simple Lie groups, and also for certain other integral transforms.

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



====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112015.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112016.png" />. Then the Fourier transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112018.png" /> can be extended to an entire analytic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112019.png" /> satisfying: for any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112020.png" /> there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112021.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112022.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

Conversely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112024.png" /> be an entire function which satisfies (*) (replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112025.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112026.png" />), for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112027.png" />. Then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112028.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071120/p07112030.png" />.

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