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| In [[mathematics]], a '''Paley–Wiener theorem''' is any theorem that relates decay properties of a function or [[distribution (mathematics)|distribution]] at infinity with [[analytic function|analyticity]] of its [[Fourier transform]]. The theorem is named for [[Raymond Paley]] (1907–1933) and [[Norbert Wiener]] (1894–1964). The original theorems did not use the language of [[generalized function|distributions]], and instead applied to [[Lp space|square-integrable functions]]. The first such theorem using distributions was due to [[Laurent Schwartz]].
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| ==Holomorphic Fourier transforms==
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| The classical Paley–Wiener theorems make use of the holomorphic Fourier transform on classes of [[square-integrable function]]s supported on the real line. Formally, the idea is to take the integral defining the (inverse) Fourier transform
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| :<math>f(\zeta) = \int_{-\infty}^\infty F(x)e^{i x \zeta}\,dx</math>
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| and allow ''ζ'' to be a [[complex number]] in the [[upper half-plane]]. One may then expect to differentiate under the integral in order to verify that the [[Cauchy–Riemann equations]] hold, and thus that ''f'' defines an analytic function. Of course, this integral may not be well-defined, even for ''F'' in ''L''<sup>2</sup>('''R'''), and so differentiation under the sign of the integral is out of the question. One must impose further restrictions on ''F'' in order that this integral be well-defined.
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| The first such restriction is that ''F'' be supported on '''R'''<sub>+</sub>: that is, ''F'' ∈ ''L''<sup>2</sup>('''R'''<sub>+</sub>). The Paley–Wiener theorem now asserts the following:<ref>{{harvnb|Rudin|1973|loc=Theorem 19.2}}; {{harvnb|Strichartz|1994|loc=Theorem 7.2.4}}; {{harvnb|Yosida|1968|loc=§VI.4}}</ref> The holomorphic Fourier transform of ''F'', defined by
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| :<math>f(\zeta) = \int_0^\infty F(x) e^{i x\zeta}\, dx</math>
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| for ζ in the [[upper half-plane]] is a holomorphic function. Moreover, by [[Plancherel's theorem]], one has
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| :<math>\int_{-\infty}^\infty \left |f(\xi+i\eta) \right|^2\, d\xi \le \int_0^\infty |F(x)|^2\, dx</math>
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| and by [[dominated convergence]], | |
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| :<math>\lim_{\eta\to 0^+}\int_{-\infty}^\infty \left|f(\xi+i\eta)-f(\xi) \right|^2\,d\xi = 0.</math>
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| Conversely, if ''f'' is a holomorphic function in the upper half-plane satisfying
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| :<math>\sup_{\eta>0} \int_{-\infty}^\infty \left |f(\xi+i\eta) \right|^2\,d\xi = C < \infty</math>
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| then there exists ''F'' in ''L''<sup>2</sup>('''R'''<sub>+</sub>) such that ''f'' is the holomorphic Fourier transform of ''F''.
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| In abstract terms, this version of the theorem explicitly describes the [[Hardy space]] [[H square|''H''<sup>2</sup>('''R''')]]. The theorem states that
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| :<math> \mathcal{F}H^2(\mathbf{R})=L^2(\mathbf{R_+}).</math>
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| This is a very useful result as it enables one pass to the Fourier transform of a function in the Hardy space and perform calculations in the easily understood space ''L''<sup>2</sup>('''R'''<sub>+</sub>) of square-integrable functions supported on the positive axis.
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| By imposing the alternative restriction that ''F'' be [[compact support|compactly supported]], one obtains another Paley–Wiener theorem.<ref>{{harvnb|Rudin|1973|loc=Theorem 19.3}}; {{harvnb|Strichartz|1994|loc=Theorem 7.2.1}}</ref> Suppose that ''F'' is supported in [−''A'', ''A''], so that ''F'' ∈ ''L''<sup>2</sup>(−''A'',''A''). Then the holomorphic Fourier transform
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| :<math>f(\zeta) = \int_{-A}^A F(x)e^{i x\zeta}\,dx</math>
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| is an [[entire function]] of [[exponential type]] ''A'', meaning that there is a constant ''C'' such that
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| :<math>|f(\zeta)|\le Ce^{A|\zeta|},</math> | |
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| and moreover, ''f'' is square-integrable over horizontal lines:
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| :<math>\int_{-\infty}^{\infty} |f(\xi+i\eta)|^2\,d\xi < \infty.</math>
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| Conversely, any entire function of exponential type ''A'' which is square-integrable over horizontal lines is the holomorphic Fourier transform of an ''L''<sup>2</sup> function supported in [−''A'', ''A''].
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| ==Schwartz's Paley–Wiener theorem==
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| Schwartz's Paley–Wiener theorem asserts that the Fourier transform of a [[distribution (mathematics)|distribution]] of [[compact support]] on '''R'''<sup>''n''</sup> is an [[entire function]] on '''C'''<sup>''n''</sup> and gives estimates on its growth at infinity. It was proven by [[Laurent Schwartz]] ([[#CITEREFSchwartz1952|1952]]). The formulation presented here is from {{harvtxt|Hörmander|1976}}.
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| Generally, the Fourier transform can be defined for any [[tempered distribution]]; moreover, any distribution of compact support ''v'' is a tempered distribution. If ''v'' is a distribution of compact support and ''f'' is an infinitely differentiable function, the expression
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| :<math> v(f) = v_x(f(x)) </math> | |
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| is well defined. In the above expression the variable ''x'' in ''v<sub>x</sub>'' is a dummy variable and indicates that the distribution is to be applied with the argument function considered as a function of ''x''.
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| It can be shown that the Fourier transform of ''v'' is a function (as opposed to a general tempered distribution) given at the value ''s'' by
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| :<math> \hat{v}(s) = (2 \pi)^{-\frac{n}{2}} v_x\left(e^{-i \langle x, s\rangle}\right)</math>
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| and that this function can be extended to values of ''s'' in the complex space '''C'''<sup>''n''</sup>. This extension of the Fourier transform to the complex domain is called the [[Fourier–Laplace transform]]. | |
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| <blockquote>'''Schwartz's Theorem.''' An entire function ''F'' on '''C'''<sup>''n''</sup> is the Fourier–Laplace transform of distribution ''v'' of compact support if and only if for all ''z'' ∈ '''C'''<sup>''n''</sup>,
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| :<math> |F(z)| \leq C (1 + |z|)^N e^{B|\text{Im}(z)|} </math>
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| for some constants ''C'', ''N'', ''B''. The distribution ''v'' in fact will be supported in the closed ball of center 0 and radius ''B''.</blockquote> | |
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| Additional growth conditions on the entire function ''F'' impose regularity properties on the distribution ''v''. For instance:<ref>{{harvnb|Strichartz|1994|loc=Theorem 7.2.2}}; {{harvnb|Hörmander|1976|loc=Theorem 7.3.1}}</ref>
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| <blockquote>'''Theorem.''' If for every positive ''N'' there is a constant ''C<sub>N</sub>'' such that for all ''z'' ∈ '''C'''<sup>''n''</sup>,
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| :<math> |F(z)| \leq C_N (1 + |z|)^{-N} e^{B|\text{Im}(z)|} </math>
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| then ''v'' is infinitely differentiable, and conversely.</blockquote>
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| Sharper results giving good control over the [[singular support]] of ''v'' have been formulated by {{harvtxt|Hörmander|1976}}. In particular,<ref>{{harvnb|Hörmander|1976|loc=Theorem 7.3.8}}</ref> let ''K'' be a convex compact set in '''R'''<sup>''n''</sup> with supporting function ''H'', defined by
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| :<math>H(x) = \sup_{y\in K} \langle x,y\rangle.</math>
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| Then the singular support of ''v'' is contained in ''K'' if and only if there is a constant ''N'' and sequence of constants ''C<sub>m</sub>'' such that
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| :<math>|\hat{v}(\zeta)| \le C_m(1+|\zeta|)^Ne^{H(\text{Im}(\zeta))}</math>
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| for |Im(ζ)| ≤ ''m''log(|ζ|+1).
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{citation|first=L.|last=Hörmander|authorlink=Lars Hörmander|title=Linear Partial Differential Operators|publisher=Springer Verlag|year=1976}}.
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| *{{Citation | last1=Rudin | first1=Walter | author1-link=Walter Rudin | title=Real and complex analysis | publisher=[[McGraw-Hill]] | location=New York | edition=3rd | isbn=978-0-07-054234-1 | id={{MathSciNet | id = 924157}} | year=1987}}.
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| * {{Citation | last1=Schwartz | first1=Laurent | authorlink=Laurent Schwartz | title=Transformation de Laplace des distributions | id={{MathSciNet | id = 0052555}} | year=1952 | journal=Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] | volume=1952 | pages=196–206}}
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| * {{citation|first=R.|last=Strichartz|year=1994|title=A Guide to Distribution Theory and Fourier Transforms|publisher=CRC Press|isbn=0-8493-8273-4}}.
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| * {{citation|first=K.|last=Yosida|authorlink=Kōsaku Yosida|title=Functional Analysis|publisher=Academic Press|year=1968}}.
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| {{DEFAULTSORT:Paley-Wiener theorem}}
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| [[Category:Theorems in Fourier analysis]]
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| [[Category:Generalized functions]]
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| [[Category:Theorems in complex analysis]]
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| [[Category:Hardy spaces]]
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