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Factoring Fourier transforms with zeros in a strip


Author: D. G. Dickson
Journal: Proc. Amer. Math. Soc. 106 (1989), 407-413
MSC: Primary 30D15; Secondary 42A85, 46F12
MathSciNet review: 962242
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Abstract: $ f$ is the Fourier transform of an infinitely differentiable function of compact support on $ {\mathbf{R}}$ if, and only if, $ f$ is entire and of exponential type with $ \left\vert {f\left( x \right)} \right\vert = O\left( {\vert x{\vert^{ - N}}} \right)$ for each $ N > 0$ as $ \vert x\vert \to \infty $ for real $ x$. In some sense, such an $ f$ has its zeros close to the real axis and has positive density of zeros $ F$ with $ n\left( r \right) = Dr + o\left( r \right)$. It is shown here that if the zeros of $ f$ are in a strip parallel to the real axis and if $ n\left( r \right) = Dr + O\left( 1 \right)$, then $ f$ is the product of two such transforms with zero densities $ D/2$ and indicators one-half of the indicator of $ f$. There is a factorable $ f$ in $ \widehat{\mathcal{D}}\left( {\mathbf{R}} \right)$ with zeros on a line and not satisfying the stricter density condition. Analogous results hold for transforms of distributions of compact support on $ {\mathbf{R}}$. The study was motivated by the outstanding problem of Ehrenpreis that asks if $ \mathcal{D}\left( {\mathbf{R}} \right) * \mathcal{D}\left( {\mathbf{R}} \right) = \mathcal{D}\left( {\mathbf{R}} \right)$.


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DOI: https://doi.org/10.1090/S0002-9939-1989-0962242-1
Article copyright: © Copyright 1989 American Mathematical Society