Differential operators and entire functions with simple real zeros
type: Item
instance of: scholarly article
title: Differential operators and entire functions with simple real zeros
zbMATH Open document ID: 1066.30031
author: S. A. de Gaston
author: David A. Cardon
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-01-26T00:00:00Z
review text: A real entire function \(\varphi(x)\) is said to be in the \textit{Laguerre-Pólya class} if \(\varphi(x)\) can be expressed in the form \[ \varphi(x)=cx^ne^{-\alpha x^2+\beta x} \prod_{k=1}^\infty\left(1+\frac x{x_k}\right)e^{-\frac x{x_k}}, \] where \(c,\beta, x_k \in \mathbb R\), \(\alpha \geq 0\), \(n\) is a nonnegative integer and \(\sum_{k=1}^\infty 1/x_k^2 < \infty\). Let \(D:=\dfrac d{dx}\). If \(\varphi(x)=\sum_{k=0}^\infty \alpha_kx^k\) and \(f(x)\) are functions in the Laguerre-Pólya class, then under suitable hypotheses (Lemma 2) the function \(\varphi(D)f(x)=\sum_{k=0}^\infty \alpha_kf^{(k)}(x)\) also belongs to the Laguerre-Pólya class. In this well-written paper, the authors solve an open problem concerning the simplicity of the zeros of \(\varphi(D)f(x)\). In particular, they prove the following theorem. Let \(\varphi\) and \(f\) be functions in the Laguerre-Pólya class. Set \(\varphi(x)=e^{-\alpha x^2}\varphi_1(x)\) and \(f(x)= e^{-\beta x^2}f_1(x)\), where \(\varphi_1\) and \(f_1\) have genus \(0\) or \(1\), and \(\alpha, \beta\geq 0\). If \(\alpha \beta < 1/4\) and if \(\varphi\) has infinitely many zeros, then \(\varphi(D)f(x)\) has only simple real zeros.
reviewed by: George Csordas
Mathematics Subject Classification ID: 30D15
Mathematics Subject Classification ID: 30C10
zbMATH DE Number: 2131144
zbMATH Keywords: differential operators
zbMATH Keywords: Laguerre-Pólya class
zbMATH Keywords: simplicity of zeros
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.07.032
OpenAlex ID: W2097887908
cites work: Differential operators of infinite order and the distribution of zeros of entire functions
cites work: On inequalities of the Turán type
DOI: 10.1016/J.JMAA.2004.07.032
description: scientific article; zbMATH DE number 2131144