De Bruijn's question on the zeros of Fourier transforms
type: Item
instance of: scholarly article
title: De Bruijn's question on the zeros of Fourier transforms
zbMATH Open document ID: 1072.30019
DOI: 10.1007/BF02788795
author: Haseo Ki
author: Young-One Kim
published in: Journal d'Analyse Mathématique
publication date: 2004-09-27T00:00:00Z
review text: 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 \(\lambda >0\).
Mathematics Subject Classification ID: 30D10
Mathematics Subject Classification ID: 30D15
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 2103502
zbMATH Keywords: zeros of Fourier transforms
zbMATH Keywords: universal factors
zbMATH Keywords: differential operators
zbMATH Keywords: Laguerre-Pólya class
reviewed by: George Csordas
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02788795
OpenAlex ID: W1982481511
description: scientific article; zbMATH DE number 2103502