Paley–Wiener theorem

Holomorphic Fourier transforms

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

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

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

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

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

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

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

and by dominated convergence,

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

Conversely, if f is a holomorphic function in the upper half-plane satisfying

:\sup_{\eta0} \int_{-\infty}^\infty \left |f(\xi+i\eta) \right|^2,d\xi = C

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

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

: \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 L^2(\mathbb{R}_+) 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\in L^2(-A,A). Then the holomorphic Fourier transform

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

is an entire function of exponential type A, meaning that there is a constant C such that

:|f(\zeta)|\le Ce^{A|\zeta|},

and moreover, f is square-integrable over horizontal lines:

:\int_{-\infty}^{\infty} |f(\xi+i\eta)|^2,d\xi

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

Schwartz's Paley–Wiener theorem

Schwartz's Paley–Wiener theorem asserts that the Fourier transform of a distribution of compact support on \mathbb{R}^n is an entire function on \mathbb{C}^n 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

: v(f) = v(x\mapsto f(x))

is well defined.

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

: \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 s in the complex space \mathbb{C}^n. This extension of the Fourier transform to the complex domain is called the Fourier–Laplace transform.

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

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:

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

then v is an infinitely differentiable function, and vice versa.}}

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 \mathbb{R}^n with supporting function H, defined by

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

Then the singular support of v is contained in K if and only if there is a constant N and sequence of constants C_m such that

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

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

Notes

References

• {{citation | author-link=Lars Hörmander
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References

1. {{harvnb. Rudin. 1987. Strichartz. 1994. Yosida. 1968
2. {{harvnb. Rudin. 1987. Strichartz. 1994
3. {{harvnb. Strichartz. 1994. Hörmander. 1990
4. {{harvnb. Hörmander. 1990