De Bruijn’s question on the zeros of Fourier transforms

Abstract

Letf(z) be a real entire function of genus 1^*, δ≥0, and suppose that for each ε>0, all but a finite number of the zeros off(z) lie in the strip |Imz| ≤δ+ε. Let λ be a positive constant such that$\lim \sup _{r \to \infty } \log M\left( {r;f} \right)/r^2< 1/\left( {4\lambda } \right)$. It is shown that for each ε>0, all but a finite number of the zeros of the entire function$e^{ - \lambda D^2 } f(z) : = \sum {_{m = 1}^\infty ( - \lambda )^m f^{2m} (z)/m!}$ lie in the strip$\left| {\operatorname{Im} z} \right| \leqslant \sqrt {\max \left\{ {\Delta ^2 - 2\lambda ,0} \right\}} + \varepsilon$ and if Δ² < 2λ, then all but a finite number of the zeros of e^−λD2 f(z) are real and simple. As a consequence, de Bruijn's question whether the functions e^γ t ²,λ>0, are strong universal factors is answered affirmatively.