Paley–Wiener theorem

The classical Paley–Wiener theorems make use of the holomorphic Fourier transform on classes of square-integrable functions supported on the real line. Formally, the idea is to take the integral defining the (inverse) Fourier transform

${\displaystyle f(\zeta )=\int _{-\infty }^{\infty }F(x)e^{ix\zeta }\,dx}$

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

The first such restriction is that ${\displaystyle F}$ be supported on ${\displaystyle \mathbb {R} _{+}}$: that is, ${\displaystyle F\in L^{2}(\mathbb {R} _{+})}$. The Paley–Wiener theorem now asserts the following:[1] The holomorphic Fourier transform of ${\displaystyle F}$, defined by

${\displaystyle f(\zeta )=\int _{0}^{\infty }F(x)e^{ix\zeta }\,dx}$

for ${\displaystyle \zeta }$ in the upper half-plane is a holomorphic function. Moreover, by Plancherel's theorem, one has

${\displaystyle \int _{-\infty }^{\infty }\left|f(\xi +i\eta )\right|^{2}\,d\xi \leq \int _{0}^{\infty }|F(x)|^{2}\,dx}$

and by dominated convergence,

${\displaystyle \lim _{\eta \to 0^{+}}\int _{-\infty }^{\infty }\left|f(\xi +i\eta )-f(\xi )\right|^{2}\,d\xi =0.}$

Conversely, if ${\displaystyle f}$ is a holomorphic function in the upper half-plane satisfying

${\displaystyle \sup _{\eta >0}\int _{-\infty }^{\infty }\left|f(\xi +i\eta )\right|^{2}\,d\xi =C<\infty }$

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

In abstract terms, this version of the theorem explicitly describes the Hardy space ${\displaystyle H^{2}(\mathbb {R} )}$. The theorem states that

${\displaystyle {\mathcal {F}}H^{2}(\mathbb {R} )=L^{2}(\mathbb {R_{+}} ).}$

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 ${\displaystyle L^{2}(\mathbb {R} _{+})}$ of square-integrable functions supported on the positive axis.

By imposing the alternative restriction that ${\displaystyle F}$ be compactly supported, one obtains another Paley–Wiener theorem.[2] Suppose that ${\displaystyle F}$ is supported in ${\displaystyle [-A,A]}$, so that ${\displaystyle F\in L^{2}(-A,A)}$. Then the holomorphic Fourier transform

${\displaystyle f(\zeta )=\int _{-A}^{A}F(x)e^{ix\zeta }\,dx}$

is an entire function of exponential type ${\displaystyle A}$, meaning that there is a constant ${\displaystyle C}$ such that

${\displaystyle |f(\zeta )|\leq Ce^{A|\zeta |},}$

and moreover, ${\displaystyle f}$ is square-integrable over horizontal lines:

${\displaystyle \int _{-\infty }^{\infty }|f(\xi +i\eta )|^{2}\,d\xi <\infty .}$

Conversely, any entire function of exponential type ${\displaystyle A}$ which is square-integrable over horizontal lines is the holomorphic Fourier transform of an ${\displaystyle L^{2}}$ function supported in ${\displaystyle [-A,A]}$.

Schwartz's Paley–Wiener theorem asserts that the Fourier transform of a distribution of compact support on ${\displaystyle \mathbb {R} ^{n}}$ is an entire function on ${\displaystyle \mathbb {C} ^{n}}$ and gives estimates on its growth at infinity. It was proven by Laurent Schwartz (1952). The formulation presented here is from Hörmander (1976) harvtxt error: no target: CITEREFHörmander1976 (help).

Generally, the Fourier transform can be defined for any tempered distribution; moreover, any distribution of compact support ${\displaystyle v}$ is a tempered distribution. If ${\displaystyle v}$ is a distribution of compact support and ${\displaystyle f}$ is an infinitely differentiable function, the expression

${\displaystyle v(f)=v(x\mapsto f(x))}$

is well defined.

It can be shown that the Fourier transform of ${\displaystyle v}$ is a function (as opposed to a general tempered distribution) given at the value ${\displaystyle s}$ by

${\displaystyle {\hat {v}}(s)=(2\pi )^{-{\frac {n}{2}}}v\left(x\mapsto e^{-i\langle x,s\rangle }\right)}$

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

Additional growth conditions on the entire function ${\displaystyle F}$ impose regularity properties on the distribution ${\displaystyle v}$. For instance:[3]

Sharper results giving good control over the singular support of ${\displaystyle v}$ have been formulated by Hörmander (1990). In particular,[4] let ${\displaystyle K}$ be a convex compact set in ${\displaystyle \mathbb {R} ^{n}}$ with supporting function ${\displaystyle H}$, defined by

${\displaystyle H(x)=\sup _{y\in K}\langle x,y\rangle .}$

Then the singular support of ${\displaystyle v}$ is contained in ${\displaystyle K}$ if and only if there is a constant ${\displaystyle N}$ and sequence of constants ${\displaystyle C_{m}}$ such that

${\displaystyle |{\hat {v}}(\zeta )|\leq C_{m}(1+|\zeta |)^{N}e^{H(\mathrm {Im} (\zeta ))}}$

for ${\displaystyle |\mathrm {Im} (\zeta )|\leq m\log(|\zeta |+1).}$

• Hörmander, L. (1990), The Analysis of Linear Partial Differential Operators I, Springer Verlag.
• Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 0924157.
• Schwartz, Laurent (1952), "Transformation de Laplace des distributions", Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.], 1952: 196–206, MR 0052555
• Strichartz, R. (1994), A Guide to Distribution Theory and Fourier Transforms, CRC Press, ISBN 0-8493-8273-4.
• Yosida, K. (1968), Functional Analysis, Academic Press.