Paley–Wiener theorem

2025-08-05T17:12:12Z

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{{Short description|Mathematical theorem}}
In [[mathematics]], a '''Paley–Wiener theorem''' is a theorem that relates decay properties of a function or [[distribution (mathematics)|distribution]] at infinity with [[analytic function|analyticity]] of its [[Fourier transform]]. It is named after [[Raymond Paley]] (1907–1933) and [[Norbert Wiener]] (1894–1964) who, in 1934, introduced various versions of the theorem.{{sfn | Paley | Wiener | 1934}} 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]]. These theorems heavily rely on the [[triangle inequality]] (to interchange the absolute value and integration).

The original work by Paley and Wiener is also used as a namesake in the fields of [[control theory]] and [[harmonic analysis]]; introducing the '''[[Jensen%27s_formula#Applications|Paley–Wiener condition]]''' for [[polynomial_matrix_spectral_factorization|spectral factorization]] and the '''[[Riesz_sequence#Paley-Wiener_criterion|Paley–Wiener criterion]]''' for [[Frame_(linear_algebra)#Non-harmonic_Fourier_series|non-harmonic Fourier series]] respectively.{{sfn | Paley | Wiener | 1934|pp=14-20,100}} These are related mathematical concepts that place the decay properties of a function in context of [[stability theory|stability problems]].

==Holomorphic Fourier transforms==
The classical Paley–Wiener theorems make use of the [[Holomorphic function|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

:<math>f(\zeta) = \int_{-\infty}^\infty F(x)e^{i x \zeta}\,dx</math>

and allow <math>\zeta</math> 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 <math>f</math> defines an analytic function. However, this integral may not be well-defined, even for <math>F</math> in <math>L^2(\mathbb{R})</math>; indeed, since <math>\zeta</math> is in the upper half plane, the modulus of <math>e^{ix\zeta}</math> grows exponentially as <math>x \to -\infty</math>; so [[Leibniz integral rule|differentiation under the integral sign]] is out of the question. One must impose further restrictions on <math>F</math> in order to ensure that this integral is well-defined.

The first such restriction is that <math>F</math> be supported on <math>\mathbb{R}_+</math>: that is, <math>F\in L^2(\mathbb{R}_+)</math>. The Paley–Wiener theorem now asserts the following:<ref>{{harvnb|Rudin|1987|loc=Theorem 19.2}}; {{harvnb|Strichartz|1994|loc=Theorem 7.2.4}}; {{harvnb|Yosida|1968|loc=§VI.4}}</ref> The holomorphic Fourier transform of <math>F</math>, defined by

:<math>f(\zeta) = \int_0^\infty F(x) e^{i x\zeta}\, dx</math>

for <math>\zeta</math> in the [[upper half-plane]] is a holomorphic function. Moreover, by [[Plancherel's theorem]], one has

:<math>\int_{-\infty}^\infty \left |f(\xi+i\eta) \right|^2\, d\xi \le \int_0^\infty |F(x)|^2\, dx</math>

and by [[dominated convergence]],

:<math>\lim_{\eta\to 0^+}\int_{-\infty}^\infty \left|f(\xi+i\eta)-f(\xi) \right|^2\,d\xi = 0.</math>

Conversely, if <math>f</math> is a holomorphic function in the upper half-plane satisfying

:<math>\sup_{\eta>0} \int_{-\infty}^\infty \left |f(\xi+i\eta) \right|^2\,d\xi = C < \infty</math>

then there exists <math>F\in L^2(\mathbb{R}_+)</math> such that <math>f</math> is the holomorphic Fourier transform of <math>F</math>.

In abstract terms, this version of the theorem explicitly describes the [[Hardy space]] [[H square|<math>H^2(\mathbb{R})</math>]]. The theorem states that

:<math> \mathcal{F}H^2(\mathbb{R})=L^2(\mathbb{R_+}).</math>

This is a very useful result as it enables one to pass to the Fourier transform of a function in the Hardy space and perform calculations in the easily understood space
<math>L^2(\mathbb{R}_+)</math> of square-integrable functions supported on the positive axis.

By imposing the alternative restriction that <math>F</math> be [[compact support|compactly supported]], one obtains another Paley–Wiener theorem.<ref>{{harvnb|Rudin|1987|loc=Theorem 19.3}}; {{harvnb|Strichartz|1994|loc=Theorem 7.2.1}}</ref> Suppose that <math>F</math> is supported in <math>[-A,A]</math>, so that <math>F\in L^2(-A,A)</math>. Then the holomorphic Fourier transform

:<math>f(\zeta) = \int_{-A}^A F(x)e^{i x\zeta}\,dx</math>

is an [[entire function]] of [[exponential type]] <math>A</math>, meaning that there is a constant <math>C</math> such that

:<math>|f(\zeta)|\le Ce^{A|\zeta|},</math>

and moreover, <math>f</math> is square-integrable over horizontal lines:

:<math>\int_{-\infty}^{\infty} |f(\xi+i\eta)|^2\,d\xi < \infty.</math>

Conversely, any entire function of exponential type <math>A</math> which is square-integrable over horizontal lines is the holomorphic Fourier transform of an
<math>L^2</math> function supported in <math>[-A,A]</math>.

==Schwartz's Paley–Wiener theorem==
Schwartz's Paley–Wiener theorem asserts that the Fourier transform of a [[distribution (mathematics)|distribution]] of [[compact support]] on <math>\mathbb{R}^n</math> is an [[entire function]] on <math>\mathbb{C}^n</math> 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}}.

Generally, the Fourier transform can be defined for any [[distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]]; moreover, any distribution of compact support <math>v</math> is a tempered distribution. If <math>v</math> is a distribution of compact support and <math>f</math> is an infinitely differentiable function, the expression

:<math> v(f) = v(x\mapsto f(x)) </math>

is well defined.

It can be shown that the Fourier transform of <math>v</math> is a function (as opposed to a general tempered distribution) given at the value <math>s</math> by

:<math> \hat{v}(s) = (2 \pi)^{-\frac{n}{2}} v\left(x\mapsto e^{-i \langle x, s\rangle}\right)</math>

and that this function can be extended to values of <math>s</math> in the complex space <math>\mathbb{C}^n</math>. This extension of the Fourier transform to the complex domain is called the [[Fourier–Laplace transform]].

{{math theorem|name=Schwartz's theorem| math_statement=An entire function <math>F</math> on <math>\mathbb{C}^n</math> is the Fourier–Laplace transform of a distribution <math>v</math> of compact support if and only if for all <math>z\in\mathbb{C}^n</math>,

:<math> |F(z)| \leq C (1 + |z|)^{-N} e^{B|\text{Im}(z)|} </math>

for some constants <math>C</math>, <math>N</math>, <math>B</math>. The distribution <math>v</math> in fact will be supported in the closed ball of center <math>0</math>
and radius <math>B</math>.}}

Additional growth conditions on the entire function <math>F</math> impose regularity properties on the distribution <math>v</math>.
For instance:<ref>{{harvnb|Strichartz|1994|loc=Theorem 7.2.2}}; {{harvnb|Hörmander|1990|loc=Theorem 7.3.1}}</ref>

{{math theorem|math_statement= If for every positive <math>N</math> there is a constant <math>C_N</math> such that for all <math>z\in\mathbb{C}^n</math>,

:<math> |F(z)| \leq C_N (1 + |z|)^{-N} e^{B|\mathrm{Im}(z)|} </math>

then <math>v</math> is an infinitely differentiable function, and vice versa.}}

Sharper results giving good control over the [[singular support]] of <math>v</math> have been formulated by {{harvtxt|Hörmander|1990}}. In particular,<ref>{{harvnb|Hörmander|1990|loc=Theorem 7.3.8}}</ref> let <math>K</math> be a convex [[Compact space|compact set]] in <math>\mathbb{R}^n</math> with supporting function <math>H</math>, defined by

:<math>H(x) = \sup_{y\in K} \langle x,y\rangle.</math>

Then the singular support of <math>v</math> is contained in <math>K</math> [[if and only if]] there is a constant <math>N</math> and sequence of constants <math>C_m</math> such that

:<math>|\hat{v}(\zeta)| \le C_m(1+|\zeta|)^Ne^{H(\mathrm{Im}(\zeta))}</math>

for <math>|\mathrm{Im}(\zeta)| \le m \log(| \zeta |+1).</math>

==Notes==
{{Reflist}}

==References==
*{{citation
| first=L.
| last=Hörmander
| author-link=Lars Hörmander
| title=Linear Partial Differential Operators, Volume 1
| publisher=Springer
| year=1976
| isbn=978-3-540-00662-6
}}
* {{citation|first=L.|last=Hörmander|authorlink=Lars Hörmander|title=The Analysis of Linear Partial Differential Operators I|publisher=Springer Verlag|year=1990}}.
* {{cite book |authorlink1=Raymond Paley | authorlink2=Norbert Wiener| last=Paley | first=Raymond E. A. C. | last2=Wiener | first2=Norbert | title=Fourier Transforms in the Complex Domain | publisher=American Mathematical Soc. | publication-place=Providence, RI | date=1934 | isbn=978-0-8218-1019-4}}
*{{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 |mr=924157 | year=1987}}.
* {{Citation | last1=Schwartz | first1=Laurent | authorlink=Laurent Schwartz | title=Transformation de Laplace des distributions |mr=0052555 | year=1952 | journal=Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] | volume=1952 | pages=196–206}}
* {{citation|first=R.|last=Strichartz|year=1994|title=A Guide to Distribution Theory and Fourier Transforms|publisher=CRC Press|isbn=0-8493-8273-4}}.
* {{citation|first=K.|last=Yosida|authorlink=Kōsaku Yosida|title=Functional Analysis|publisher=Academic Press|year=1968}}.

{{DEFAULTSORT:Paley-Wiener theorem}}
[[Category:Theorems in Fourier analysis]]
[[Category:Generalized functions]]
[[Category:Theorems in complex analysis]]
[[Category:Hardy spaces]]

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Paley–Wiener theorem