Power series, uncertainty inequalities and real zeros of Fourier transforms
type: Item
instance of: scholarly article
title: Power series, uncertainty inequalities and real zeros of Fourier transforms
zbMATH Open document ID: 0858.42003
author: Enrico Lang
published in: Expositiones Mathematicae
publication date: 1997-03-19T00:00:00Z
review text: Let \(f(x)\) be a summable function and \(\widehat f(u)\) its Fourier transform. In 1940, A. Beurling claimed that \(f\) and \(\widehat f\) cannot both decay at infinity ``so quickly'' that the following double integral \[ \int^\infty_{-\infty} \int^\infty_{-\infty} |f(x)\widehat f(u)|e^{|xu|} dx du \] converges. A statement of this kind is an instance of an uncertainty inequality. In the paper, the author explores another kind of uncertainty inequality which is related, but not equivalent, to Beurling's result. Let us consider the even function \(f(x)\) that decays rapidly on the real line together with \(\widehat f(u)\), \[ |f(x)|\leq e^{-\alpha|x|^p};\;p>1,\;\alpha>0,\quad |\widehat f(u)|\leq e^{-\beta|u|^q};\;q> 1,\;\beta>0.\tag{1} \] Theorem. If a function \(f\) satisfies (1) with \(1/p+1/q<1\) then it is identically zero. Conjecture. Let \(f(z)\) be an entire function, strongly critical with \(p>2\), monotone decreasing on the positive real axis, even and positive on the whole real axis. Then \(\widehat f\) has infinitely many real zeros.
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 42A20
Mathematics Subject Classification ID: 30D10
zbMATH DE Number: 897949
zbMATH Keywords: Fourier transform
zbMATH Keywords: entire function
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 897949