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 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 f defines an analytic function. However, this integral may not be well-defined, even for F in L2 — indeed, since ζ is in the upper half plane, the modulus of eixζgrows exponentially as — 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 R+: that is, F ∈ L2. 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 f is a holomorphic function in the upper half-plane satisfying then there existsF in L2 such that f is the holomorphic Fourier transform of F. In abstract terms, this version of the theorem explicitly describes the Hardy spaceH2. 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 L2 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 , so that F ∈ L2. Then the holomorphic Fourier transform is an entire function of exponential typeA, meaning that there is a constant C such that and moreover, f is square-integrable over horizontal lines: Conversely, any entire function of exponential typeA which is square-integrable over horizontal lines is the holomorphic Fourier transform of an L2 function supported in .
Schwartz's Paley–Wiener theorem
Schwartz's Paley–Wiener theorem asserts that the Fourier transform of a distribution 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. 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. It can be shown that the Fourier transform of v is a function given at the value s by and that this function can be extended to values of s in the complex spaceCn. This extension of the Fourier transform to the complex domain is called the Fourier–Laplace transform.
Schwartz's theorem. An entire function F on Cn is the Fourier–Laplace transform of a 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:
Theorem. If for every positive N there is a constant CN such that for all z ∈ Cn, 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 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 |Im| ≤ mlog.
