De Bruijn's question on the zeros of Fourier transforms (Q1880938)

The authors solve a fifty-year-old open problem. They provide an affirmative answer to the following question raised by \textit{N. G. de Bruijn} [Duke Math. J. 17, 197--226 (1950; Zbl 0038.23302)]. Let \(F:\mathbb R\to \mathbb C\) be an integrable function. Suppose that for some constant \(b>2\), \(F(t)=O(\exp(-| t| ^b))\) as \(| t| \to\infty\) and that \(F(-t)=\overline{F(t)}\) for \(t\in \mathbb R\). If for each \(\varepsilon >0\), all but a finite number of the zeros of the Fourier transform of \(F(t)\) lie in the strip \(| \text{Im}\, z| \leq \varepsilon \), does it follow that for each \(\lambda > 0\), the Fourier transform of \(e^{\lambda t^2}F(t)\) has only a finite number of nonreal zeros? The main theorem (Theorem 2.2) is deduced from several interesting new results involving the properties of the operators \(e^{-\lambda D^2}\), where \(D=d/dx\) and \(\lambda >0\).