Paley–Wiener theorem

Paley–Wiener theorem 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. These theorems heavily rely on the triangle inequality (to interchange the absolute value and integration).

Contents 1 Holomorphic Fourier transforms 2 Schwartz's Paley–Wiener theorem 3 Notes 4 References 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 {displaystyle f(zeta )=int _{-infty }^{infty }F(x)e^{ixzeta },dx} 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(R) — indeed, since ζ is in the upper half plane, the modulus of eixζ grows exponentially as {displaystyle xrightarrow -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 R+: that is, F ∈ L2(R+). The Paley–Wiener theorem now asserts the following:[1] The holomorphic Fourier transform of F, defined by {displaystyle f(zeta )=int _{0}^{infty }F(x)e^{ixzeta },dx} for ζ in the upper half-plane is a holomorphic function. Moreover, by Plancherel's theorem, one has {displaystyle int _{-infty }^{infty }left|f(xi +ieta )right|^{2},dxi leq int _{0}^{infty }|F(x)|^{2},dx} and by dominated convergence, {displaystyle lim _{eta to 0^{+}}int _{-infty }^{infty }left|f(xi +ieta )-f(xi )right|^{2},dxi =0.} Conversely, if f is a holomorphic function in the upper half-plane satisfying {displaystyle sup _{eta >0}int _{-infty }^{infty }left|f(xi +ieta )right|^{2},dxi =C

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