Paley–Wiener theorem
Encyclopedia
In mathematics
, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution
at infinity with 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 distributions
, and instead applied to square-integrable functions
. The first such theorem using distributions was due to Laurent Schwartz
.
s supported on the real line. Formally, the idea is to take the integral defining the (inverse) Fourier transform
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 ƒ defines an analytic function. Of course, this integral may not be well-defined, even for F ∈ L2(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.
The first such restriction is that F be supported on (0,∞): that is, F ∈ L2(0,∞). The Paley–Wiener theorem now asserts the following: The holomorphic Fourier transform of F, defined by
for ζ in the upper half-plane is a holomorphic function. Moreover, by Plancherel's theorem, one has
and by dominated convergence,
Conversely, if ƒ is a holomorphic function in the upper half-plane satisfying
then there exists F ∈ L2(0,∞) such that ƒ is the holomorphic Fourier transform of F.
In abstract terms, this version of the theorem explicitly describes the Hardy space
H2(R)
. The theorem states that
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
of square-integrable functions supported on the positive axis.
By imposing the alternative restriction that F be compactly supported, one obtains another Paley–Wiener theorem. Suppose that F is supported in [−A,A], so that F ∈ L2(-A,A). Then the holomorphic Fourier transform
is an entire function
of exponential type A, meaning that there is a constant C such that
and moreover, ƒ is square-integrable over horizontal lines:
Conversely, any entire function of exponential type A which is square-integrable over horizontal lines is the holomorphic Fourier transform of an L2 function supported in [−A,A].
of compact support on Rn is an entire function
on Cn and gives estimates on its growth at infinity. It was proven by Laurent Schwartz
(1952). The formulation presented here is from .
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
is well defined. In the above expression the variable x in vx is a dummy variable and indicates that the distribution is to be applied with the argument function considered as a function of x.
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
and that this function can be extended to values of s in the complex space Cn. This extension of the Fourier transform to the complex domain is called the Fourier–Laplace transform.
Schwartz's theorem is the following statement:
Additional growth conditions on the entire function F impose regularity properties on the distribution v. For instance:
Sharper results giving good control over the singular support of v have been formulated by . In particular, let K be a convex compact set in Rn with supporting function H (defined by
). Then the singular support of v is contained in K if and only if there is a constant N and sequence of constants Cm such that
for
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
at infinity with analyticity
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
of its Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
. The theorem is named for Raymond Paley
Raymond Paley
Raymond Edward Alan Christopher Paley was an English mathematician. Paley was born in Bournemouth, England. He was educated at Eton. From there he entered Trinity College, Cambridge where he showed himself the most brilliant student among a remarkable collection of fellow undergraduates...
(1907–1933) and Norbert Wiener
Norbert Wiener
Norbert Wiener was an American mathematician.A famous child prodigy, Wiener later became an early researcher in stochastic and noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems.Wiener is regarded as the originator of cybernetics, a...
(1894–1964). The original theorems did not use the language of distributions
Generalized function
In mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognized theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing physical phenomena such as point charges...
, and instead applied to square-integrable functions
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
. The first such theorem using distributions was due to Laurent Schwartz
Laurent Schwartz
Laurent-Moïse Schwartz was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields medal in 1950 for his work...
.
Holomorphic Fourier transforms
The classical Paley–Wiener theorems make use of the holomorphic Fourier transform on classes of square-integrable functionSquare-integrable function
In mathematics, a quadratically integrable function, also called a square-integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite...
s supported on the real line. Formally, the idea is to take the integral defining the (inverse) Fourier transform
and allow ζ to be a complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
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 ƒ defines an analytic function. Of course, this integral may not be well-defined, even for F ∈ L2(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.
The first such restriction is that F be supported on (0,∞): that is, F ∈ L2(0,∞). The Paley–Wiener theorem now asserts the following: The holomorphic Fourier transform of F, defined by
for ζ in the upper half-plane is a holomorphic function. Moreover, by Plancherel's theorem, one has
and by dominated convergence,
Conversely, if ƒ is a holomorphic function in the upper half-plane satisfying
then there exists F ∈ L2(0,∞) such that ƒ is the holomorphic Fourier transform of F.
In abstract terms, this version of the theorem explicitly describes the Hardy space
Hardy space
In complex analysis, the Hardy spaces Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper...
H2(R)
H square
In mathematics and control theory, H2, or H-square is a Hardy space with square norm. It is a subspace of L2 space, and is thus a Hilbert space...
. The theorem states that
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
of square-integrable functions supported on the positive axis.By imposing the alternative restriction that F be compactly supported, one obtains another Paley–Wiener theorem. Suppose that F is supported in [−A,A], so that F ∈ L2(-A,A). Then the holomorphic Fourier transform
is an entire function
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
of exponential type A, meaning that there is a constant C such that
and moreover, ƒ is square-integrable over horizontal lines:
Conversely, any entire function of exponential type A which is square-integrable over horizontal lines is the holomorphic Fourier transform of an L2 function supported in [−A,A].
Schwartz's Paley–Wiener theorem
Schwartz's Paley–Wiener theorem asserts that the Fourier transform of a distributionDistribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
of compact support on Rn is an entire function
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
on Cn and gives estimates on its growth at infinity. It was proven by Laurent Schwartz
Laurent Schwartz
Laurent-Moïse Schwartz was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields medal in 1950 for his work...
(1952). The formulation presented here is from .
Generally, the Fourier transform can be defined for any tempered distribution
Tempered distribution
*Distribution *Tempered representation...
; 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
is well defined. In the above expression the variable x in vx is a dummy variable and indicates that the distribution is to be applied with the argument function considered as a function of x.
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
and that this function can be extended to values of s in the complex space Cn. This extension of the Fourier transform to the complex domain is called the Fourier–Laplace transform.
Schwartz's theorem is the following statement:
- An entire function F on Cn is the Fourier–Laplace transform of distribution v of compact support if and only if for all z ∈ Cn,
- for some constants C, N, B. The distribution v in fact will be supported in the closed ball of center 0 and radius B.
Additional growth conditions on the entire function F impose regularity properties on the distribution v. For instance:
- If for every positive N there is a constant CN such that for all z ∈ Cn,
- then v is infinitely differentiable, and conversely.
Sharper results giving good control over the singular support of v have been formulated by . In particular, let K be a convex compact set in Rn with supporting function H (defined by
). Then the singular support of v is contained in K if and only if there is a constant N and sequence of constants Cm such thatfor
.